GL arXiv:2105.12621v1 [math.AG] 26 May 2021

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THE GEOMETRY OF POLYNOMIAL REPRESENTATIONS

ARTHUR BIK, JAN DRAISMA, ROB H. EGGERMONT, AND ANDREW SNOWDEN

Abstract. We define a GL-variety to be a (typically infinite dimensional) algebraic varietyequipped with an action of the infinite general linear group under which the coordinate ringforms a polynomial representation. Such varieties have been used to study asymptoticproperties of invariants like strength and tensor rank, and played a key role in two recentproofs of Stillman’s conjecture. We initiate a systematic study of GL-varieties, and establisha number of foundational results about them. For example, we prove a version of Chevalley’stheorem on constructible sets in this setting.

Contents

1. Introduction 12. GL-varieties 73. Generalized orbits 114. The embedding theorem 155. The shift theorem and consequences 186. Some properties of affine spaces 217. The decomposition theorem 248. Theory of types 309. Examples and applications 38References 43

1. Introduction

Infinite dimensional varieties admitting symmetry by the infinite general linear grouphave featured prominently in some recent work: [Dr, ES] prove noetherian results in thissetting, [BDE] establishes a structural result for certain varieties of this form, and [DLL,ESS1, ESS2] give proofs of Stillman’s conjecture (and some related results in commutativealgebra) using such varieties. These varieties are also closely connected to research in whichGowers norm techniques are used to establish properties of high-rank (tuples of) polynomials[KaZ1, KaZ2, KaZ3, KaZ4], although they do not appear explicitly. This work suggests thatthis class of varieties is well-behaved and can be useful for studying questions of broadinterest. In this paper, we initiate a systematic study of these varieties, and establish anumber of fundamental results about them.

AB was partially supported by JD’s Vici Grant. JD was partially supported by the NWO Vicigrant entitled Stabilisation in Algebra and Geometry, project number 639.033.514. RE was partiallysupported by the NWO Veni grant entitled Stability and structure in infinite-dimensional spaces,project number 016.Veni.192.113. AS was supported by NSF grant DMS-1453893.

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http://arxiv.org/abs/2105.12621v1

2 ARTHUR BIK, JAN DRAISMA, ROB H. EGGERMONT, AND ANDREW SNOWDEN

1.1. GL-varieties. Fix a field K of characteristic 0. Let GL “Ťně1GLnpKq be the

infinite general linear group, and let V “Ťně1K

n be its standard representation. For apartition λ, we let Vλ “ SλpVq be the irreducible polynomial representation of GL withhighest weight λ; here Sλ denotes a Schur functor. For example, Vpdq is the dth symmetricpower of V. We let Aλ be the spectrum of the polynomial ring SympVλq; this is an affinescheme equipped with an action of the group GL. For a tuple of partitions λ “ rλ1, . . . , λrs,we let Aλ “ Aλ1 ˆ ¨ ¨ ¨ ˆ Aλr . The role of the spaces Aλ in our theory is analogous to therole of the usual affine spaces An in classical algebraic geometry.

Main definition. An affine GL-variety is a closed GL-stable subset of Aλ, for some λ.

This paper is a study of affine GL-varieties: our goal is to understand their structure andestablish analogs of classical theorems of algebraic geometry for them. In the process, it willsometimes be convenient to work with the larger class of quasi-affine varieties, which areGL-stable open subsets of affine GL-varieties.

Before proceeding, we give some examples of GL-varieties that are typical of the kindsappearing in applications.

Example 1.1. Let V be a complex vector space. Recall that an element of V bd has (tensor)rank ď r if it can be written as a sum of r pure tensors. Tensor rank has been widely studied;for a general introduction, see [La]. The rank ď r locus in V bd is typically not Zariski closed(see, e.g., [La, §2.4.5]). A point in the closure is said to have border rank ď r.

Now, let X “ pVbdq˚; this space has the form Aλ for an appropriate choice of λ. Onecan make sense of the notions of rank and border rank in X , and the border rank ď r locusis a closed GL-subvariety of X . By studying this GL-variety, one can hope to understandaspects of border rank that are independent of the dimension of the vector space.

Example 1.2. Let R “À

ně0Rn be a graded ring. The strength of a homogeneous elementf P R is the minimal n for which there exists an expression f “

řn

i“1 gihi where gi andhi are homogeneous elements of positive degree. This concept was introduced by Ananyanand Hochster [AH] in their proof of Stillman’s conjecture, though similar ideas had beenpreviously considered. Applying this in the case R “ SympV q, with V a vector space, weobtain a notion of strength for elements of SymdpV q. This is not the most direct analog oftensor rank, but it is a closely related idea. The strength ď k locus in SymdpV q is not closedin general [BBOV], but its closure is an interesting closed subvariety of SymdpV q.

Now let X “ SymdpVq˚ “ Apdq. Then one can make sense of the strength ď k locus inX , and its closure is a closed GL-subvariety of X . As in the previous example, one canhope to understand aspects of strength that are independent of the number of variables bystudying this variety. This strategy has been carried out successfully in [ESS2, DLL] to givenew proofs of Stillman’s conjecture.

1.2. Structural results. We now describe our main structural results about GL-varieties.Let Gpnq be the subgroup of GL consisting of block matrices of the form

îdn 00 ˚

˙.

The group Gpnq is in fact isomorphic to GL. Thus if X is a GL-variety, we can restrict theaction of GL to Gpnq and then identify Gpnq with GL to obtain a new action of GL. We

THE GEOMETRY OF POLYNOMIAL REPRESENTATIONS 3

call this the nth shift of X , and denote it by ShnpXq. This is a very important operation onGL-varieties that appears throughout the paper.

We can now state our first main theorem:

Embedding theorem (Theorem 4.2). Let Y be a GL-variety, let λ be a non-empty parti-tion, and let X be a closed GL-subvariety of Y ˆ Aλ. Then one of the following holds:

(a) We have X “ Y0 ˆ Aλ for some closed GL-subvariety Y0 Ă Y .(b) There is a non-empty open GL-subvariety of ShnpXq, for some n, that embeds into

ShnpY q ˆ Aµ for some tuple µ, each part of which is smaller than λ.

This theorem is important since it yields an inductive approach to studying GL-varieties.Indeed, suppose X is an affine GL-variety. Then, by definition, X is a closed GL-subvarietyof Aκ for some tuple κ “ rκ1, . . . , κrs. Arrange the κ’s so that κr has maximal size, andapply the theorem with Y “ Aκ1 ˆ ¨ ¨ ¨ ˆ Aκr´1 and λ “ κr. In case (a), X has a relativelysimple form. In case (b), we find that an open subset of a shift of X embeds into a space ofthe form ShnpY q ˆ Aµ, where each µi is smaller than κr. In turns out that this space hasthe form Aρ for some tuple ρ which is (in a precise sense) smaller than the original κ.

In fact, the embedding theorem was essentially proved by the second author in the courseof proving his noetherianity theorem [Dr]. However, the theorem was not formally stated inthat paper: it appeared only implicitly within the main proof. We have isolated it here as astandalone theorem since we have found it to be an important tool; indeed, it the input keyto our next theorem.

An important theme in representation stability is that objects can be often made “nice”by shifting. A prototypical result of this kind is Nagpal’s theorem [Na] that a sufficient shiftof a finitely generated FI-module over a field of characteristic 0 is projective. We establisha shift theorem for GL-varieties:

Shift theorem (Theorem 5.1). Let X be a non-empty affine GL-variety. Then there is anon-empty open affine GL-subvariety of ShnpXq, for some n, that is isomorphic to B ˆ Aκ

for some variety B and tuple κ.

Here, and elsewhere in this paper, a variety without further qualifications is a finite-dimensional affine variety over K, i.e., the spectrum of a finitely generated reduced K-algebra.

The shift theorem has a number of important consequences. For one, it shows that con-structing rational points on GL-varieties is no more difficult than on finite dimensionalvarieties. For example, if K is algebraically closed then the K-points of a GL-variety aredense; this follows easily from the definitions if K is uncountable (see [La2] for details), butis not obvious when K is countable. More generally, given a GL-variety X there is some dsuch that the points of X defined over degree d extensions of K are dense in X .

Here is a more interesting result that follows from the shift theorem:

Unirationality theorem (Theorem 5.4). Let X be an irreducible affine GL-variety. Thenthere is a dominant morphism ϕ : B ˆ Aλ Ñ X for some variety B and tuple λ.

Recall that a variety C is unirational if there is a dominant rational morphism An99K C

for some n. Thus the above theorem can be interpreted as saying that X is “unirational inthe GL direction” or “unirational up to a finite dimensional error.” We emphasize that theϕ in the theorem is actually a morphism, and not just a rational map. As an application,we show that the field KpXqGL of invariant rational functions on X is a finitely generated


extension of K. We prove some finer results surrounding the unirationality theorem as well:for instance, we show that one can find ϕ that is surjective (though with B reducible), andshow how one can control λ.

The shift theorem shows that if X is an arbitrary GL-variety then, after shifting, X hasa fairly simple form outside of some closed subset. We now describe how this theorem canbe used to give a more cohesive picture of GL-varieties. An elementary GL-variety is oneof the form B ˆ Aλ, where B is a finite dimensional quasi-affine variety; we apply the sameterminology to Gpnq-varieties. We say that a GL-variety is locally elementary if every pointadmits an open elementary Gpnq-stable neighborhood. With this language, we can now statethe theorem:

Decomposition theorem (Theorem 7.8). Let X be a quasi-affine GL-variety. Then thereis a finite decomposition X “

Ůr

i“1 Yi where each Yi is a locally elementary GL-variety thatis locally closed in X.

We also establish a version of the decomposition theorem for morphisms of GL-varieties;in fact, this is a much more useful statement. The decomposition theorem (for morphisms)allows one to easily prove a variety of results about GL-varieties, such as:

Chevalley’s theorem (Theorem 7.13). Let ϕ : X Ñ Y be a morphism of quasi-affine GL-varieties and let C be a GL-constructible subset of X. Then ϕpCq is a GL-constructiblesubset of Y .

Here, we say that a subset of a quasi-affine GL-variety is GL-constructible if it is a finiteunion of locally closed GL-stable subsets. One consequence of this theorem is that if ϕ issurjective on K-points then it is scheme-theoretically surjective.

1.3. Orbits and types. Given a group action on a variety, it is often of interest to study theorbits. In the case ofGL-varieties, it turns out that orbits in the usual sense are pathological.Indeed, Ap2q can be identified with the space of all symmetric bilinear forms on V; thus itspoints can be represented by infinite symmetric matrices. Since elements of GL differ fromthe identity matrix in only finitely many entries, they cannot change an infinite symmetricmatrix very much. To remedy this, we introduce the notion of generalized orbit : two pointsof a GL-variety belong to the same generalized orbit if and only if they have the same orbitclosure. Generalized orbits are well-behaved. For example, generalized orbits on Ap2q areparametrized by rank. We let Xorb be the space of generalized orbits on a GL-variety X .We are thus interested in the problem of describing Xorb.

The first interesting phenomenon one observes is that many GL-varieties contain a K-point with dense orbit; we call such points GL-generic. For example, the space Ap3q “Sym3pVq˚ of cubic forms in infinitely many variables contains a GL-generic K-point. Thisis rather counterintuitive since the analogous statement in finitely many variables is clearlyfalse: the variety Sym3pKnq˚ of cubic forms in n " 0 variables clearly does not contain aK-point with dense GLn orbit, asGLn has dimension n2 but the space has dimension « 1

6n3.

In fact, we show (Proposition 3.13) that if λ is any pure tuple (meaning it does not containthe empty partition) then Aλ contains a GL-generic K-point. (We note that if λ “ ∅ isthe empty partition then Aλ is the ordinary affine line with trivial GL action, and does notcontain a GL-generic K-point.)

Combining the above observation and the unirationality theorem leads to a strategy forstudying generalized orbits. Let x be a point of X and let Ox be its orbit closure. We assume


for simplicity that x is a K-point and K is algebraically clsoed. Using the unirationalitytheorem, we show that there is a dominant map ϕ : Aλ Ñ Ox for some irreducible varietyB and pure tuple λ. In fact, if we assume that λ is appropriately minimal then ϕ is uniqueup to the action AutpAλq; in this case, we say that x has type λ and that ϕ is a typicalmorphism for x. It then follows that x “ ϕpyq for some GL-generic K-point y P Aλ. Nowsuppose that x1 is a second K-point that happens to have the same typical morphism ϕ; thisis not so improbable as the mapping space Mλ “ MappAλ, Xq is finite dimensional. We canthen write x1 “ ϕpy1q for some GL-generic K-point y1. Since y and y1 are both GL-generic,each belongs to the orbit closure of the other, and so the same is true for x and x1; that is,x and x1 belong to the same generalized orbit. This suggests that we might be able to usetypical morphisms to detect generalized orbits.

To pursue this line of thought in more detail, we introduce some notation. Let Xorbλ be

the subspace of Xorb on points having type ď λ, where ď is the containment partial order.Let Xtype

λ be the quotient of the finite dimensional variety Mλ by the action of the algebraicgroup AutpAλq. We have a well-defined map

ρλ : Xtypeλ Ñ Xorb

λ

given by evaluating a morphism Aλ Ñ X on any GL-generic point and taking the associ-ated generalized orbit. (The well-definedness here is essentially the content of the previousparagraph.) The following is our main theorem about this construction:

Theorem 1.3 (Theorem 8.25). The map ρλ is a continuous bijection.

Let us explain why this is a useful result. The space Xtypeλ is completely algebraic: it is a

quotient of a finite dimensional variety by the action of a finite dimensional algebraic group.The space Xorb

λ , on the other hand, is more analytic in nature. Indeed, to determine if twopoints x and y belong to the same generalized orbit, one must construct a sequence g‚ inGL such that gix converges to y in an appropriate sense. The above theorem connects thesetwo very different looking objects.

We introduced typical morphisms for K-points in the above discussion. In fact, we definethe notion for arbitrary irreducible affine GL-varieties: given such a variety X , a typicalmorphism is a dominant map BÂλ Ñ X that is appropriately minimal. This is unique upto the action of AutpAλq and passing to finite covers and open subsets of B. In particular,the dimension of B is an invariant of X , which we call the typical dimension of X . We givethe following elegant characterization of this quantity:

Theorem 1.4 (Theorem 8.8). Let ϕ : B ˆ Aλ Ñ X be a typical morphism. Then KpBqis a finite extension of the field KpXqGL of invariant rational functions. In particular, thetypical dimension of X is the transcendence degree of KpXqGL over K.

1.4. Further results. Let X be an irreducible GL-variety. In a follow-up paper (in prepa-ration), we prove the following results.

‚ Any two points of X can be joined by an irreducible curve.‚ There exists a surjective morphism B ˆ Aλ Ñ X for some tuple λ and irreduciblevariety B.

‚ The continuous bijection ρλ : Xtypeλ Ñ Xorb

λ is a homeomorphism.‚ The mapping space MappAλ, Xq is irreducible for all λ.‚ Let ϕ : Y Ñ X be a map of GL-varieties. Then any point in the image closure of ϕcan be realized as the limit of a “nice” 1-parameter family in Y .


These statements are closely related to one another; for example, it is easy to see that thefirst is a consequence of the second. The proofs of these results are rather long, which is whywe have deferred them to a separate paper.

1.5. Applications. We discuss a few applications of our theory.

‚ Let R be the inverse limit of the graded rings Krx1, . . . , xns. In [ESS2] (see also[AM]), it is shown that R is isomorphic to a polynomial ring (in uncountably manyvariables), which was an important component in the proofs of Stillman’s conjecturegiven there. We give a new proof of this result using the theory developed in thispaper. When K “ Cpptqq is the field of Laurent series, [ESS2] introduced a subring R5

of R consisting of elements with bounded denominators, and showed that it too is apolynomial ring. In recent work [Sn], the third author showed that R is a polynomialalgebra over R5; this result is significantly more difficult than the polynomiality resultsof [ESS2]. We reprove this result as well. See §9.2.

‚ In [BBOV], Ballico-Oneto-Ventura and one of the the current authors give a noncon-structive proof that the strength ď 3 locus in Sym4pV q is not closed in general. In[BDDE], Danelon and three of the current authors generalize a theorem of Kazhdan-Ziegler [KaZ2] on universality of high-strength polynomials to arbitrary polynomialfunctors. These results build on a weaker variant of the unirationality theorem thatappeared in the first author’s thesis [Bi].

‚ Our version of Chevalley’s theorem shows that the tensor rank r locus in Example 1.1or the strength r locus in Example 1.2 are GL-constructible sets.

‚ Our Chevalley theorem also implies that for any GL-equivariant morphism Aλ Ñ Aµ

there exists a polynomial-time membership test for the image. To illustrate, for fixedk1, k2, k3, there exists such a polynomial-time algorithm that, on input n and a tensorT P Q

nbQ

nbQ

n, tests whether T is the sum of k1 terms of the form v1 b v2 b v3, k2

terms of the form v1 bA23, and k3 terms of the form A12 bv3, where the vi are vectorsin Q

nand the Aij are matrices in Q

nbQ

n. This algorithm just verifies, on this input

tensor, the uniform quantifier-free description of this set promised by Chevalley’stheorem; it does not compute the vi and the Aij . However, using the decompositiontheorem for morphisms, there is a polynomial-time algorithm that computes these,as well.

1.6. On the characteristic. We restrict ourselves to characteristic zero for two relatedreasons. First, in positive characteristic, the Schur functors Sλ are typically not irreducible,and one should replace direct sums of these with objects in the larger category of polynomialfunctors. This category is not semisimple, and this complicates some of the reasoning. Sec-ond, the proof of one of our key tools, the second version of the Embedding Theorem, doesnot work as stated in positive characteristic, even when direct sums of Schur functors arereplaced by arbitrary polynomial functors; see Remark 4.3. However, the proof of Noethe-rianity in [Dr] shows that in positive characteristic, the map that is required to be a closedembedding in Theorem 4.2 is a universal homeomorphism onto its image. This suggeststhat most of our results have analogs in positive characteristic. We will pursue this line ofthought in a follow-up paper.

1.7. Notation. We typically denote ordinary, finite-dimensional varieties by B or C, andGL-varieties by X or Y . Other important notation:


K: the base field, always characteristic 0Ω: a (variable) extension of K, often algebraically closedV: a fixed infinite dimensional K-vector space with basis teiuiě1

GL: the group of automorphisms of V fixing almost all basis vectorsGpnq: the subgroup of GL fixing e1, . . . , en and mapping spanteiuiąn into itself.Shn: the shift functorshn: the shift operation on partitions or tuples

Xr1hs: the open subset of the variety X defined by h ‰ 0.XtV u: the result of taking the affine GL-scheme X , regarding it as functor on vector

spaces, and evaluating on the K-vector space VVλ: the irreducible GL-representation SλpVq, where Sλ is a Schur functorλ: a tuple rλ1, . . . , λrs of partitions

Vλ: the representation Vλ1 ‘ ¨ ¨ ¨ ‘ Vλr

Rλ: the symmetric algebra on Vλ

Aλ: the affine GL-variety with coordinate ring Rλ

MλpXq: the space of maps Aλ Ñ X .

2. GL-varieties

2.1. Partitions and tuples. Recall that a partition is a sequence λ “ pλ1, λ2, . . .q of non-negative integers with λi ě λi`1 for all i and λi “ 0 for i " 0. We write |λ| for the sumλ1 ` λ2 ` ¨ ¨ ¨ , and call it the size of λ. The empty partition ∅ is the partition with all partsequal to 0.

A tuple of partitions (often simply called a tuple) is a finite tuple λ “ rλ1, λ2, . . . , λrswhere each λi is a partition. We use square brackets for the notation since each λi is denotedwith parentheses. We say that a tuple is pure if it does not contain the empty partition.We say that a tuple λ “ rλ1, . . . , λrs contains a tuple µ “ rµ1, . . . , µss if r ě s and, perhapsafter reordering, we have λi “ µi for 1 ď i ď s; we denote this by µ Ă λ. For tuplesλ “ rλ1, . . . , λrs and µ “ rµ1, . . . , µss, we let λ Y µ be the tuple rλ1, . . . , λr, µ1, . . . , µss.

We define themagnitude of a tuple λ, denoted magpλq to be the sequence pn0, n1, . . .q whereni is the number of partitions in λ of size i. We compare magnitudes lexicographically, thatis, magpλq ă magpµq if magpλqi ă magpµqi where i is the largest index where the magnitudesdisagree. This is a well-order. We can therefore argue by induction on magnitude. We definethe degree of a tuple λ, denoted degpλq to be the maximal size of a partition in λ.

2.2. Polynomial representations. We fix a field K of characteristic 0 throughout thispaper. Let GL “

Ťně1GLnpKq be the infinite general linear group, thought of as a discrete

group, and let V “Ťně1K

n be its standard representation. A polynomial representation ofGL is one that occurs as a subquotient of a (possibly infinite) direct sum of tensor powersof V. The category of polynomial representations is a semi-simple abelian category.

For a partition λ, we write Vλ for the polynomial representation SλpVq where Sλ denotesthe Schur functor associated to λ. The Vλ’s are exactly the irreducible polynomial represen-tations. For a tuple λ “ rλ1, . . . , λrs of partitions, we write Vλ for

Àr

i“1Vλi . Every finitelength polynomial representation is isomorphic to some Vλ.

Every polynomial representation V carries a canonical grading. For t P Kˆ, let Anptq P GL

be the diagonal matrix where the first n entries are t and the remaining entries are 1; oneshould think of Anptq as an approximation of the scalar matrix t ¨ id. If v P V then Anptqv isindependent of n for n " 0; we denote this common value by Aptqv. Then v is homogeneous


of degree d if and only if Aptqv “ tdv for all t P Kˆ. Under this grading, the irreduciblerepresentation Vλ is homogeneous of degree |λ|. We note that an element of v is GL-invariant if and only if it has degree 0. We define an action of the multiplicative group Gm

on a polynomial representation V by letting t P Gm act by td on the degree d piece. Werefer to this as the “central Gm.”

The category of polynomial representations is equivalent to the category of polynomialfunctors. Given a polynomial representation V , we let V tKnu be the result of evaluating thecorresponding polynomial functor on Kn. Explicitly, this can be identified with V G whereG is the subgroup of GL fixing each of e1, . . . , en. The operation V ÞÑ V tKnu is compatiblewith tensor products and arbitrary direct sums.

The action of GL on a polynomial representation V extends in a few ways. First theaction of the discrete group GLnpKq Ă GL extends uniquely to an action of the algebraicgroup GLn; that is, V is naturally a comodule over the Hopf algebra KrGLns. And second,the action of GL naturally extends to an action of the monoid EndpVq. These new actionsare completely natural, and compatible with direct sums and tensor products.

2.3. GL-algebras. A GL-algebra is a K-algebra R equipped with an action of the groupGL by algebra automorphisms such that it forms a polynomial representation. Such analgebra is naturally graded, since every polynomial representation is graded, and any GL-equivariant homomorphism of GL-algebras is homogeneous. We let Rλ “ SympVλq; this isa typical, and important, example of a GL-algebra. We say that a GL-algebra R is finitelyGL-generated if it is generated, as a K-algebra, by the GL-oribts of finitely many elements;equivalently, R is a quotient of Rλ for some λ.

Let R be a GL-algebra. A GL-ideal of R is an ideal that is GL-stable. We say thata GL-ideal is finitely GL-generated if it is generated, as an ideal, by the GL-orbits offinitely many elements. Equivalently, a GL-ideal I is finitely GL-generated if there exists aGL-equivariant surjection of R modules R b Vλ Ñ I for some λ.

Let R be a GL-algebra over an ordinary ring A, that is, GL acts trivially on A and wehave a ring homomorphism A Ñ R0. We say that R is pure over A if the map A Ñ R0 isan isomorphism. In this case, there is a natural map R Ñ A that sends all positive degreeelements to 0. We note that Rλ is pure over K if and only if λ is pure.

2.4. GL-varieties. An affine GL-scheme is an affine scheme X equipped with an actionof GL such that ΓpX,OXq is a GL-algebra. An affine GL-variety is a reduced affine GL-scheme such that ΓpX,OXq is finitely GL-generated. A quasi-affine GL-variety is a GL-stable open subscheme of an affine GL-variety. A morphism of GL-schemes is simply aGL-equivariant morphism of schemes.

Let Aλ “ SpecpRλq. This is an affine GL-variety, and serves as the principal example:any affine GL-variety can be realized as a closed GL-subvariety of Aλ for some λ. Thus theAλ take the place of the usual affine spaces in classical algebraic geometry. For a singletonλ “ rλs, we simply write Aλ in place of Aλ. If λ is the empty partition then Aλ “ A1 is theordinary affine line.

Let f : X Ñ S be an affine GL-scheme over the ordinary affine scheme S. We say that Xis pure over S if (locally) the map on coordinate rings is pure In this case, the map f admitsa canonical GL-invariant section S Ñ X that we call the zero section. We note that Aλ ispure over K if and only if the tuple λ is pure.


Let X “ SpecpRq be an affine GL-scheme. Recall that RtKnu is the result of evaluatingR on Kn, where we regard R as a polynomial functor. This is a K-algebra equipped with aGLn-action. We define XtKnu “ SpecpRtKnuq. We note that RtKnu can be realized as asubalgebra of R, and so if R is reduced or integral then so is RtKnu.

2.5. Noetherianity. The following noetherianity theorem was established by the secondauthor, and suggested that the theory pursued in this paper might be within reach:

Theorem 2.1 ([Dr]). For any tuple λ, the closed GL-subsets of Aλ satisfy the descendingchain condition. Equivalently, any closed GL-subvariety of Aλ is the zero locus of a finitelyGL-generated GL-ideal of Rλ.

In fact, [Dr] establishes the natural analog of the above theorem in positive characteristic,and more recently it has been established over arbitrary noetherian base rings [BDD].

2.6. Mapping spaces. Let X and Y be GL-schemes over an affine base K-scheme S. Welet MapSpX, Y q be the set of maps X Ñ Y of GL-schemes over S. The group GmpSq actson this set, via the action of the central Gm on Y (or on X ; the two actions differ by aninverse). If Y is pure over S then we let 0 P MapSpX, Y q be the composition of X Ñ S withthe zero section S Ñ Y . We let MapSpX, Y q be the the functor from affine S-schemes to setsgiven by T ÞÑ MapT pXT , YT q, where p´qT denotes base change to T . This carries an actionof Gm, and if Y is pure over S, then there is a zero section 0 : S Ñ MapSpX, Y q (where weidentify S with the functor it represents).

Example 2.2. Suppose S “ SpecpKq, X “ SpecpAq for some finitely GL-generated GL-algebra A with A0 “ K, and Y “ Aλ for a partition λ. Suppose that Vλ occurs withmultiplicity n in A. Then

MapSpX, Y q “ HomGL-algpSympVλq, Aq “ HomGL-reppVλ, Aq – Kn.

More generally, if R is a K-algebra then MapSpX, Y qpSpecpRqq – Rn. It follows thatMapSpX, Y q is represented by An.

Proposition 2.3. Let S be a noetherian affine scheme over K, let X Ñ S be a GL-schemethat is affine, pure, GL-finite type, and flat over S, and let Y Ñ S be a GL-scheme that isaffine and GL-finite type over S.

(a) MapSpX, Y q is represented by a finite type affine scheme over S.(b) Suppose Y is pure over S. Then the quotient of MapSpX, Y qzt0u by Gm is a closed

subvariety of a weighted projective space over S.(c) Suppose Y “ A

λ

S for a (possibly non-pure) tuple λ. Then MapSpX, Y q is a vectorbundle over S.

Proof. (c) This is essentially the same computation as Example 2.2. Let OS , OX , and OY

be the rings for S, X , and Y , and let OX “À

λOX,λ bK Vλ be the isotypic decompositionof OX . Note that since OX is finitely GL-generated over OS, each multiplicity space OX,λ

is a finitely generated OS-module. Since X is flat over S it follows that OX,λ is flat as anOS-module; since OS is noetherian, it thus follows that OX,λ is projective as an OS-module.

Regarding MapSpX, Y q as a functor on OS-algebras, we have

MapSpX, Y qpRq “HompOS ,GLq-algpOY , R bOSOXq

“HomGL-reppVλ, R bOSOXq “

rà

i“1

R bOSOX,λi


where λ “ rλ1, . . . , λrs. Now, if P is a finitely generated projective OS-module then, puttingP_ “ HomOS-modpP,OSq, we have

R bOSP “ HomOS-modpP_, Rq “ HomOS-algpSympP_q, Rq.

Thus R ÞÑ R bOSP is represented by the vector bundle SpecpSympP_qq over S. We thus

see that if Ei is the vector bundle associated to OX,λi then MapSpX, Y q is represented by thevector bundle E “

Àr

i“1 Ei.Before continuing, we make one additional observation. An element x P Gm acts on Ei via

multiplication by xdi where di “ |λi|. If λ is pure then di ą 0 for all i, and so the quotientof Ezt0u by Gm is a family of weighted projective spaces over S.

(a) Since Y is of finite type over S, it embeds into AλS for some (possibly non-pure) tuple

λ. First suppose the ideal of Y in AλS is equivariantly finitely generated. We can then realize

Y as f´1p0q for some map of GL-schemes f : AλS Ñ A

µS with µ some other tuple (the µi’s

correspond to the generators of the ideal). In other words, we have a cartesian diagram

Y //

AλS

S0

// Aµ

S.

(Note that even if µ is not pure, the space AµS has a natural zero section.) Since Map is

compatible with fiber products in the second variable, we obtain a cartesian diagram

MapSpX, Y q //

MapSpX,AλSq

MapSpX,Sq0

// MapSpX,AµSq.

These functors, other than the top left, are representable by finite type affine schemes by(c), and so it follows that the top left one is as well.

We now treat the case where the ideal for Y is not equivariantly finitely generated. Expressthe ideal for Y as a directed union of equivariantly finitely generated ideals; thus Y “

Şiě1 Yi

for a descending chain tYiuiě1 of closed GL-subschemes of Aλ

S. Since Map is compatible withsuch intersections in its second argument, we see that MapSpX, Y q “

Şiě1MapSpX, Yiq as

subfunctors of MapSpX,Aλ

Sq. Since each MapSpX, Yiq is a closed subscheme of MapSpX,Aλ

Sq,it follows that MapSpX, Y q is too. In fact, since MapSpX,Aλ

Sq is noetherian, this descendingchain stabilizes, and so MapSpX, Y q “ MapSpX, Yiq for i " 0.

(b) Choose a closed embedding Y Ñ AλS for a pure tuple λ. Then MapSpX, Y q is a

Gm-stable closed subscheme of MapSpX,AλSq. We have already seen that the quotient of

MapSpX,Aλ

Sqzt0u by Gm is a weighted projective space, and so the result follows.

Remark 2.4. The hypotheses in the above proposition can be relaxed in various ways. Forexample, instead of X being pure over S, it is enough for X0 Ñ S to be finite.

In this paper, we only apply the proposition when X is an affine space. We introduce aspecial notation for that case:

Definition 2.5. Let Y be an affine GL-variety over K and let λ be a pure tuple. We putMλpY q “ MapKpAλ, Y q, which is a finite type affine scheme over K.


Remark 2.6. Let λ “ rp1qns be the tuple consisting of n copies of the partition p1q. ThenMλpY q is naturally identified with Y tKnu. This is easiest explained on K-points: given aK-point y of Y tKnu, we have a morphism of K-schemes Aλ Ñ Y defined on K-points asfollows. A K-point of the former space is a tuple pv1, . . . , vnq of vectors vi P V˚. This tupledefines a linear map V Ñ Kn, ei ÞÑ pv1i, . . . , vniq and hence, by functoriality, a morphismY tKnu Ñ Y tVu “ Y of K-schemes. Applying this morphism to y yields a K-point of Y .Conversely, if ϕ is a K-morphism Aλ Ñ Y , then we apply ϕ to the K-point pe1, e2, . . . , enqof the former space to get a K-point in Y tKnu; here ei is the linear function on V definedby eipejq “ δij . These two constructions are inverse to each other, and extends to pointswith values in arbitrary K-algebras.

3. Generalized orbits

3.1. Orbit closures. Let X be an affine GL-scheme. We define the orbit closure of a subsetW of X , denoted OW , to be the intersection of all closed GL-subsets of X that contain W .We are most interested in the case where W “ txu is a point, in which case we simply writeOx. The following proposition summarizes the basic properties of orbit closures:

Proposition 3.1. We have the following:

(a) For any subset W of X, we have OW “ GL ¨W .(b) For any subset W of X we have OW “ OW , where W denotes the closure of W .(c) If W is an irreducible subset of X then OW is also irreducible.(d) Every irreducible component (maximal irreducible subset) of X is GL-stable.(e) For every x P X the orbit closure Ox is irreducible and its generic point is GL-fixed.

(f) Every point x P X that is fixed by GL is the generic point of its orbit closure Ox “ txu.

Proof. (a) For Ď observe that the right-hand side is a closedGL-subset ofX that containsW .For Ě observe that each closed GL-subset of X containing W also contains the right-handside.

(b) The inclusion Ď follows from W Ď W . For the inclusion Ě observe that every closedGL-subset of X that contains W also contains W .

(c) By (b), and since the closure of an irreducible set is irreducible, we may assume thatW is a closed subset, hence (the underlying space of) an irreducible affine scheme. For everyn P Zě0, GLn ¨ W is irreducible since it is the closure of the image of the irreducible affinescheme GLn ˆ W under a morphism. (Here we appeal to the comment at the end of §2.2that the action of the discrete group GLnpKq Ă GL on the coordinate ring of X extendsuniquely to an action of the algebraic group GLn.) Hence, if GL ¨ W Ď X1 Y X2 with X1

and X2 closed, then for all n we have GLn ¨W Ď Xin for some in P t1, 2u. Taking the unionof the GLn over a subsequence where the in are constant, we find that GL ¨ W Ď X1 orGL ¨W Ď X2, and hence also GL ¨W Ď X1 or GL ¨ W Ď X2. Now use (a).

(d) Let W be an irreducible component. Then W is contained in the closed GL-subsetOW , which by (c) is irreducible. By maximality of W , we have W “ OW .

(e) The first statement follows from (c). The statement about the generic point says thatthe prime ideal defining Ox is GL-stable.

(f) If x P X is GL-fixed, then Ox “ txu is irreducible and has x as its generic point.

3.2. Generalized orbits. Let X be an affine GL-scheme. We define the generalized orbitof x P X , denoted Ox, to be the set of all points y P X such that Ox “ Oy. The generalized


orbits partition X into disjoint subsets, just as ordinary orbits do. Note that Ox containsthe usual orbit GL ¨ x; in particular, by Proposition 3.1(a), the closure of Ox is Ox, so thenotation is consistent.

Example 3.2. Suppose X “ Ap1q, which is (the scheme associated to) the dual vector spaceV˚. The orbits of GL onV˚ behave pathologically: indeed, a point of V˚ can be representedby infinitely many coordinates but an element of GL only affects finitely many coordinates,and so two points of V˚ can only belong to the same orbit if all but finitely many of theircoordinates are equal. The generalized orbits, on the other hand, behave as expected: thereare two, namely, the origin and everything else.

Example 3.3. Let X “ Ap2q, which one can regard as the space of symmetric bilinear formson V. One can show that two points belong to the same generalized orbit if and only if theyhave the same rank. Thus the generalized orbits are naturally index by the set NY t8u.

Proposition 3.4. The set Ox is the intersection of all non-empty open GL-subsets of Ox.

Proof. LetW be the intersection of all non-empty openGL-subsets ofOx. We showW “ Ox.Let U be a non-empty open GL-subset of Ox, and let Z be its complement. Suppose

y P Ox. If y belonged to Z then Oy “ Ox would be contained in Z, which is not the case.Thus y P U . Since y is arbitrary, we have Ox Ă U , and since U is arbitrary, we find Ox Ă W .

Now suppose that y P OxzOx. Then, by definition, Z “ Oy is not all of Ox. ThusU “ OxzZ is a non-empty open GL-subset of Ox and y R U ; thus y R W . Since y isarbitrary, we have OxzOx Ă OxzW , and so W Ă Ox. This completes the proof.

Proposition 3.5. The generic point of Ox is the unique GL-fixed point in Ox.

Proof. For x P X , the set Ox is irreducible by Proposition 3.1. Let y be its generic point.Then y is GL-fixed since Ox is GL-stable, and we have Oy “ tyu “ Ox, so Ox “ Oy.Suppose that z P Ox were some other GL-fixed point. By Proposition 3.1(f), we see thatz is the generic point of Oz “ Ox, and thus equal to y since Ox has a unique generic point(affine schemes are sober).

3.3. The space of orbits. Let X be an affine GL-scheme. We define the orbit space Xorb

to be the set of generalized orbits in X , and we let π : X Ñ Xorb be the quotient map,defined by πpxq “ Ox. We give Xorb the quotient topology, so that a subset U of Xorb isopen if and only if π´1pUq is an open subset of X . We let XGL denote the subset of Xconsisting of points fixed by the group GL, and we endow it with the subspace topology.

Proposition 3.6. The map XGL Ñ Xorb induced by π is a homeomorphism.

Proof. By Proposition 3.5, the map is a bijection, and it is the restriction of a continuousmap, hence continuous. Next assume that U is an open subset of XGL, say U “ V X XGL

with V open in X . Then V 1 :“ GL ¨ V is also open in X and GL-stable, and V 1 X XGL “V X XGL “ U , so after replacing V by V 1 we may assume that V is a GL-stable opensubset of X . Let Y Ď X be its complement. Then the orbit closure of a point y P Y iscontained in Y and hence Ov “ Oy does not hold for any v P V . It follows that V is a unionof generalized orbits. Consequently, π´1pπpV qq “ V , hence πpV q is open by definition of thequotient topology. Finally, since each generalized orbit contains a GL-stable point, we haveπpV q “ πpUq. So the inverse of π|XGL is also continuous, as desired.


Proposition 3.7. The open (resp. closed) GL-subsets of X are in bijection with the open(resp. closed) subsets of Xorb, via π and π´1.

Proof. A closed GL-suset Z Ď X contains the generalized orbit of each of its points. ByProposition 3.5 we have πpZq “ πpZGLq, which by Proposition 3.6 is closed, and its preimagein XGL is ZGL; it follows that π´1pπpZqq “ Z. Moreover, also by Proposition 3.6, all closedsubsets of Xorb are of the form πpZq with Z Ď X GL-stable and closed, and we findπpπ´1pπpZqqq “ πpZq. Similarly for open GL-sets.

Let U be an open GL-subset of X . We say that U is GL-quasi-compact if any opencover of U by open GL-subsets admits a finite subcover. This is equivalent to πpUq beingquasi-compact by Proposition 3.7.

Proposition 3.8. Let Z be a closed GL-subset of X. Then U “ XzZ is GL-quasi-compactif and only if Z can be defined by a finitely GL-generated ideal.

Proof. Suppose the orbits of f1, . . . , fr define an ideal cutting out Z. Let Dpfq be theusual distinguished open set defined by f ‰ 0. Suppose that tVαuαPI is an open cover ofU by open GL-subsets. Since Dpfiq Ă U , the Vα’s cover Dpfiq. But Dpfiq is an affinescheme, and thus quasi-compact, and so there exists a finite subset Ji of I such that Dpfiq ĂŤαPJi

Vα. Since the Vα’s are GL-stable, we thus haveŤgPGL

gDpfiq ĂŤαPJi

Vα. Since

U “Ťr

i“1

ŤgPGL

gDpfiq, we see that U “ŤαPJ Vα, where J “

Ťr

i“1 Ji is finite.

Now suppose that U is GL-quasi-compact. Let tfαuαPI define an ideal cutting out Z.Then U “

ŤαPI

ŤgPGL

gDpfαq is a cover by open GL-sets, and so there is a finite subset J

of I such that U “ŤαPJ

ŤgPGL

gDpfαq. We thus see that Z “ŞαPJ

ŞgPGL

gV pfαq. Thus

tfαuαPJ is a finite set of elements that generated a GL-ideal cutting out Z.

Proposition 3.9. The space Xorb is a spectral topological space.

Proof. Let W be an irreducible closed subset of Xorb. By Proposition 3.6, its preimage Uin XGL is closed and irreducible. Then U “ OU is a closed irreducible GL-set in the soberspace X , hence has a unique generic point x. Then x P XGL, and x lies in the closure ofU in XGL, which is U itself. So W has a generic point, namely, πpxq. The pre-image in Uof any other generic point of W would also be a generic point of U , hence equal to x. Thisshows that X is sober.

Let U be an open GL-stable subset of X , and let Z “ XzU . Let Z “ V paq for some GL-ideal a of ΓpX,OXq. Write a “

řαPI aα where each aα is finitely GL-generated. Let Zα “

V paαq, so that Z “ŞαPI Zα. Let Uα “ XzZα, so that U “

ŤαPI Uα. By Propositions 3.7

and 3.8, we see that πpUq “ŤαPI πpUαq is a cover of πpUq by quasi-compact open subsets.

Thus the quasi-compact open subsets of Xorb form a basis for the topology. It followsimmediately from the same propositions that the intersection of two quasi-compact opensubsets is again quasi-compact. Thus X is spectral.

Proposition 3.10. If X is an affine GL-variety then Xorb is a noetherian topological space.

Proof. This follows from Theorem 2.1 and Proposition 3.7.

3.4. GL-generic points. We say that a point x P X is GL-generic if Ox “ X . ByProposition 3.7, this is equivalent to πpxq being a generic point of Xorb. An interestingfeature of GL-varieties is that there can be closed points that are GL-generic.


Example 3.11. Let λ “ rp1qds, so that Vλ “ V‘d. One can identify a point of Aλ witha d-tuple pv1, . . . , vdq where vi P V˚. Such a point is GL-generic if and only if the vi’s arelinearly independent.

Example 3.12. A point ofAp2q isGL-generic if and only if it has infinite rank, when thoughtof as a symmetric bilinear form on V. This follows immediately from the description of thegeneralized orbits given in Example 3.3.

In fact, it is not difficult to generalize the above examples:

Proposition 3.13. The space Aλ admits a GL-generic K-point if and only if λ is pure.

Proof. First suppose that λ is not pure. We then have a GL-invariant projection mapAλ Ñ A1, under which any GL-generic point of Aλ maps to a generic point of A1. SinceA1 does not have a generic K-point, it follows that Aλ does not have a GL-generic K-point.

We now prove the converse direction. We start by considering a special case. Let Td “ Vbd

be the dth tensor power representation and let Td “ SpecpSympTdqq be the correspondingGL-variety. Assume that d ą 0, so that Td is pure. Let teiuiě1 be a basis of V, so that wehave a natural basis of Td given by elements of the form ei1,...,id “ ei1 b ¨ ¨ ¨ b eid. A K-pointv of Td is an element of the dual space T ˚

d , and can be expressed as a (possibly infinite) sumři1,...,id

ci1,...,ide˚i1,...,id

where the c’s belong to K.

Let f : rdsˆN Ñ N be an injective function, and put v “řně0 e

˚fp1,nq,...,fpd,nq. For example,

if d “ 2 one could take v “ e˚1,2 ` e˚

3,4 ` ¨ ¨ ¨ . This will be our GL-generic point of Td. Beforeproving this, we establish a property of its orbit closure. Let Td,ďn be the subspace of Tdspanned by the vectors ei1,...,id with i1, . . . , id ď n. We claim that T ˚

d,ďn is contained in Ov.Indeed, let w P T ˚

d,ďn be given. Write w “ w1 ` ¨ ¨ ¨ `wr where each wk is a pure tensor. Wecan then find an endomorphism m of V that satisfies mpe˚

fp1,kq,...,fpd,kqq “ wk for 1 ď k ď r

and mpe˚fp1,kq,...,fpd,kqq “ 0 for k ą r. We thus see that mpvq “ w. As remarked in §2.2, the

monoid EndpVq naturally acts on any polynomial representation. Thus m acts on SympTdq,and leaves any subrepresentation stable; in particular, the ideal of Ov is stable. It followsthat the action m induces on Td carries Ov into itself. Since mpvq “ w, we see that w P Ov.

It now follows that Ov “ Td. Indeed, suppose that h is some non-zero element of SympTdq.Let n be such that h belongs to SympTd,ďnq. Since h is non-zero, it does not vanish on all ofT ˚d,ďn, and so we can find some w P T ˚

d,ďn such that hpwq ‰ 0. Since w P Ov, it follows that

h does not vanish on Ov. Thus no element of SympTdq vanishes on Ov, and so Ov is Zariskidense in Td. Since it is also closed, it must be the entire space.

Now consider the space Td1 ˆ ¨ ¨ ¨ˆTdr , where each di is positive. Let vi P TdipKq be as inthe previous paragraph, but choose the vi’s so that they use distinct basis vectors. In otherwords, if vi is associated to the function fi, then the images of the fi’s should be disjointfrom one another. This ensures that we can move the vi’s independently under the actionEndpVq. Put v “ pv1, . . . , vrq. Then from our previous argument, we see that Ov containsT ˚d1,ďn

ˆ ¨ ¨ ¨ ˆ T ˚dr ,ďn

for all n. Thus v is GL-generic just as before.

Finally, suppose that λ is an arbtirary pure tuple. We have Td1 ˆ ¨ ¨ ¨ ˆ Tdr “ Aλ ˆ Aµ

for appropriate d1, . . . , dr ą 0 and µ. Taking a GL-generic K-point of Td1 ˆ ¨ ¨ ¨ ˆ Tdr , itsprojection to Aλ is a GL-generic K-point.

We have the following mild generalization of the above proposition:


Proposition 3.14. Let B be an irreducible variety and let λ be a pure tuple. Suppose that Badmits a generic point over the extension ΩK. Then BÂλ admits a GL-generic Ω-point.

Proof. Let x be a generic Ω-point of B and let v be a GL-generic K-point of Aλ. Then oneeasily sees that px, vq is a GL-generic Ω-point of B ˆ Aλ. (For instance, base change alongx : SpecpΩq Ñ B and then apply the previous proposition over the base field Ω.)

We observe that, in general, one does need to pass to an extension of K to obtain GL-generic points:

Example 3.15. Let X be the closed GL-subvariety of Arp1q,p1qs consisting of pairs of linearlydependent vectors. We have a surjective map of GL-varieties ϕ : A2 ˆ Ap1q Ñ X defined bypα, β, vq ÞÑ pαv, βvq. Moreover, ϕ is surjective on K-points, and so the orbit closure of anynon-zero K-point is a copy of Ap1q inside of X . Thus X does not admit a GL-generic K-point. The restriction of ϕ to A1ˆt1uÂp1q Ă A2Âp1q (the line y “ 1 in the first factor) isdominant, and so, by Proposition 3.14, we see that X admits a GL-generic Kptq-point.

4. The embedding theorem

The proof of the noetherianity theorem in [Dr] relies on a fundamental embedding theorem.Roughly speaking, this theorem states that if X is a proper closed GL-subvariety of Aλ thenan open subset of some shift of X embeds into Aµ where magpµq ă magpλq. This providesan effective tool for performing inductive arguments on GL-varieties. We now formulate twoprecise versions of this theorem.

4.1. The shift operation. For any partition λ and any n P Zě0, we define ShnpVλq tobe the polynomial GL-representation SλpKn ‘ Vq, where GL acts trivially on Kn. Thefunctoriality of Sλ implies that for m,n P Zě0 we have Shm`npVλq – ShmpShnpVλqq. Weextend Shn (additively) to arbitrary polynomial GL-representations, and as such it is afunctor from the category of polynomial representations to itself that commutes with tensorproducts, symmetric powers, and indeed general Schur functors.

As a consequence, if R is a GL-algebra, then the polynomial GL-representation ShnpRqis naturally a GL-algebra, with the product coming from the image of the multiplicationhomomorphism R b R Ñ R under the shift functor. Note that Shn commutes with tensorproducts of GL-algebras.

If X is an affine GL-scheme with coordinate ring R, then we define ShnpXq to be thespectrum of the GL-algebra ShnpRq. In particular, ShnpAλq is the spectrum of ShnpRλq –SympShnpSλqq. Note that Shn commutes with products of affine GL-schemes.

Since we take V and coordinate rings as the primary objects, the shift operation is dual tothat in [Dr]. In particular, the natural inclusionV Ñ Kn‘V yields injective homomorphismsR Ñ ShnpRq of GL-algebras and surjective morphisms ShnpXq Ñ X of affine GL-schemes.Indeed, the natural projection Kn ‘ V Ñ V yields a left inverse ShnpRq Ñ R and a rightinverse X Ñ ShnpXq, respectively. Similarly, note that the natural inclusion Kn Ñ Kn ‘ V

yields the inclusion ShnpRqGL Ñ ShnpRq of the subalgebra of GL-invariant functions onShnpXq into ShnpRq.

Alternatively, the shift operation Shn may and will be understood as follows. The linearshift isomorphism σn : Kn ‘ V Ñ V that sends the standard basis of Kn to e1, . . . , en andthe basis vector ei of V to ei`n is GL-equivariant if we let GL act on the right-hand sidevia the isomorphism GL Ñ Gpnq that shifts an infinite matrix to the south-east and inserts


an nˆ n-identity matrix in the top left corner. Consequently, ShnpVλq is isomorphic to therestriction of the GL-representation SλpVq to the subgroup Gpnq, regarded as a polynomialGL-representation via the isomorphism GL Ñ Gpnq. We write σn : ShnpVλq Ñ Vλ forthis isomorphism, but stress that the GL-action on Vλ is non-standard. Note that thelinear inclusion Kn Ñ V sending the standard basis vectors to e1, . . . , en yields the inclusionRGpnq Ñ R of the subalgebra of Gpnq-invariant functions on X into R.

Similarly, for a GL-algebra R, σn induces an isomorphism σn : ShnpRq Ñ R if we letGL act on R via its isomorphism into Gpnq, and for an affine GL-scheme X , σn induces anisomorphism σn : X Ñ ShnpXq if we let GL act on X via that isomorphism.

For a tuple λ, we write shnpλq for the tuple such that ShnpAλq “ Ashnpλq. The tuple shnpλqalways contains λ; we let shn,0pλq be the complement. Thus ShnpAλq – Aλ ˆ Ashn,0pλq. Fora partition λ, we write shnpλq and shn,0pλq in place of shnprλsq and shn,0prλsq. We notethat every partition appearing in shn,0pλq has size strictly smaller than that of λ, and thatshn,0pλq is the empty tuple if λ is the empty partition.

4.2. First version. Let Y be an affine GL-variety, and let λ be a non-empty partition. Letf be a regular function on Y ˆ Aλ that does not factor through Y and let X Ă Y ˆ Aλ bethe closed GL-subvariety defined by the orbit of f . Let h be a non-zero partial derivativeof f with respect to some coordinate on Aλ. Let n be such that f and the coordinate areGpnq-invariant; then so is h. Recall that ShnpAλq “ Aλ ˆ Aµ, where µ “ shn,0pλq is atuple whose constituents are smaller than λ, and also that ShnpY ˆ Aλq can be regarded asY Âλ but with GL acting via the isomorphism GL Ñ Gpnq. Via this latter interpretation,h defines a GL-invariant function on ShnpY ˆ Aλq “ ShnpY q ˆ Aλ Âµ. In particular, thisinvariant function does not involve the coordinates on Aλ (which have positive degree sinceλ is not the empty partition), and therefore factors through the projection

π : ShnpY ˆ Aλq Ñ ShnpY q ˆ Aµ.

We can thus regard h as a function on either the source or target. The embedding theoremis the following statement:

Theorem 4.1. The projection map π restricts to a closed immersion

π : ShnpXqr1hs Ñ pShnpY q ˆ Aµqr1hs.

Proof. This is proved in [Dr, §2.9]; we only clarify some minor differences in the set-up. Theproof there, for the purpose of an induction, assumes that X is a closed GL-subvariety ofAν ˆ Aλ with |λ| ě |νi| for all i, but this inequality is not needed in the proof. There, Yis just the closure of the image of X in Aν , and the requirement is that X is not equal toY ˆ Aλ; here we have made sure of this fact by requiring that f does not factor through Y .Also, there, X might be cut out from Y ˆ Aλ by further equations in addition to the orbitof f , but of course the proof in particular applies to the case where X is defined by thissingle orbit. Finally, in [Dr] it is required that h does not vanish identically on X . But if itdoes (e.g., if f is a square), then the conclusion of the theorem is trivial since ShnpXqr1hsis empty.

4.3. Second version. The above formulation of the embedding theorem is very precise, butrather cumbersome to apply (ensuring h is non-zero can be subtle). We now formulate asimpler version which is usually sufficient:


Theorem 4.2. Let Y be an affine GL-variety, let λ be a non-empty partition, and let X bea closed GL-subvariety of Y ˆ Aλ. Then one of the following two cases hold:

(a) There exists a closed GL-subvariety Y0 Ă Y such that X “ Y0 ˆ Aλ.(b) There exists an integer n ě 0 and a GL-invariant function h on ShnpY q ˆ Aµ that

does not vanish identically on ShnpXq such that the projection map

π : ShnpXqr1hs Ñ pShnpY q ˆ Aµqr1hs

is a closed immersion. (Here π and µ “ shn,0pλq are defined as before.)

Proof. Suppose case (a) does not hold. Let ϕ : Y ˆ Aλ Ñ Y be the projection map and

let Y0 “ ϕpXq. Since (a) does not hold, it follows that X is a proper closed subset ofϕ´1pY0q “ Y0 ˆ Aλ. There is therefore some function on Y ˆ Aλ that vanishes identicallyon X but not on Y0 ˆ Aλ. Of all such functions, choose one f of minimal degree. Let txiube coordinates on Aλ, and write f “

řαPS gαx

α where S is a finite set of exponent vectors,xα denotes the monomial xα1

1 xα2

2 ¨ ¨ ¨ , and gα is a function on Y . We can assume that no gαvanishes identically on Y0; indeed, we can simply subtract off such terms from f . Let h bethe partial derivative of f with respect to some xi that appears in f . Then h does not vanishidentically on Y0 ˆ Aλ, since its coefficients are among the gα’s. Since h is non-vanishing onY0 ˆ Aλ and has smaller degree than f , it does not vanish on X by the minimality of the

degree of f . Now, let rX be the GL-subvariety of Y Âλ defined by the orbit of f ; note thatX is a closed subvariety of rX . Again take n such that f and xi are Gpnq-invariant. Then,by the previous version of the theorem, the map

π : Shnp rXqr1hs Ñ pShnpY q ˆ Aµqr1hs

is a closed immersion. Since ShnpXqr1hs is a closed subvariety of Shnp rXqr1hs, we obtainthe desired result.

Remark 4.3. In the proof of Theorem 4.2, characteristic zero is used in an essential manner:indeed, in positive characteristic, it is possible for all derivatives of all possible f ’s to vanishidentically. We do not know if Theorem 4.2 remains valid in positive characteristic; we hopeto return to this topic in the future.

4.4. A companion result. The following result is often used in conjunction with the em-bedding theorem:

Proposition 4.4. Let X be an affine GL-variety over a field K and let h be a non-zeroinvariant function on ShnpXq.

(a) The natural inclusion ι : V Ñ Kn ‘ V induces a GL-morphism ϕ : ShnpXqr1hs ÑX.

(b) The image of ϕ contains a non-empty open GL-subset U of X.(c) Every point x P U admits an open neighborhood V such that the map ϕ´1pV q Ñ V

has a section. (The open set and section are not necessarily GL-stable.)(d) Let ΩK be an extension field. Given any Ω-point x of U , there is an Ω-point y of

ShnpXqr1hs such that ϕpyq “ x.

Proof. (a) By the setup in §4.1, that inclusion yields an inclusion at the level of coordinaterings and a surjection ShnX Ñ X ; ϕ is the restriction to ShnpXqr1hs.


(b) Consider the linear map τ : Kn ‘ V Ñ V that is the identity on V and sends thestandard basis of Kn to e1, . . . , en. We have τ ˝ ι “ 1V , and therefore τ induces a (non-GL-equivariant) section τ˚ of the GL-equivariant surjective morphism ShnpXq Ñ X comingfrom ι. Let Kn Ñ Kn ‘ V be the natural inclusion and let Kn Ñ V be the linear inclusionsending the standard basis to e1, . . . , en. Then, since h is GL-invariant and nonzero and thediagram

Kn //

$$

Kn ‘ V

τ

V

commutes, the pull-back τh of h along τ˚ is a nonzero Gpnq-invariant function on X . Itfollows that Xr1τhs is in the image of ϕ : ShnpXqr1hs Ñ X . Since imϕ is GL-stable, theopen GL-set U :“ GL ¨Xr1τhs is also contained in imϕ.

(c) On Xr1τhs we have constructed the section τ˚. On g ¨ Xr1hs for g P GL we havethe section gτ˚g´1.

(d) Use the section from (c).

5. The shift theorem and consequences

5.1. The shift theorem. We now establish the important shift theorem. We first introducea piece of terminology. Suppose that X is a non-empty closed GL-subvariety of Aλ for sometuple λ. Let d be the maximal size of a partition appearing in λ, and decompose λ as µY ν,where ν consists of the partitions in λ of size d and µ consists of the remaining partitions. Wesay that X is imprimitive if d ą 0 and X has the form Y Âν for some closed GL-subvarietyY of Aµ, and primitive otherwise. It is not difficult to see that this notion is independentof coordinates, i.e., invariant under the action of AutpAλq. Given any X , one can find adecomposition λ “ µY ν (not necessarily the same as the decomposition above) with ν puresuch that X “ Y ˆ Aν for some primitive closed GL-subvariety Y of Aµ.

Theorem 5.1. Let X be an affine GL-variety.

(a) There exists an integer n ě 0 and a non-zero invariant function h on ShnpXq suchthat ShnpXqr1hs is isomorphic, as a GL-variety, to B ˆ Aκ for some variety BKand some pure tuple κ.

(b) Suppose that X is a primitive closed GL-subvariety of Aλ, and λ contains a non-empty partition. Then one can realize the situation in (a) with magpκq ă magpλq.In fact, if d is the maximal size of a partition in λ, then one can take κ so thatmagpκqi “ 0 for i ą d and magpκqd ă magpλqd.

Proof. (a) Let λ be a pure tuple and let d be the maximum size of a partition appearing inλ. For a pure tuple κ, we write κ Æ λ if magpκqi “ 0 for i ą d and magpκqd ď magpλqd.Consider the following statement for a tuple λ:

Spλq: Given a closed GL-subvariety X of Aλ there exists n ě 0 and a non-zero invariantfunction h on ShnpXq such that ShnpXqr1hs is isomorphic to BÂκ for some varietyB and pure tuple κ Æ λ.

We prove the statement Spλq for all λ by induction on the magnitude of λ. This will establishthe theorem.

Let λ be given, and suppose that Spµq holds for all µ with magnitude strictly less thanthat of λ. We prove Spλq. If λ consists of empty partitions the statement is obvious, so


assume this is not the case. Let X Ď Aλ be given. Let d ą 0 be the maximal size of apartition in λ and let ν be a partition in λ of size d. Let µ be the tuple obtained by removingν from λ, so that λ “ µ Y ν. Let Y “ Aµ, so that X is a subvariety of Y ˆ Aν . We nowapply Theorem 4.2; we consider the two possible cases in turn.

In the first case, there is a closed GL-subvariety Y0 of Y such that X “ Y0 ˆ Aν . Sincemagpµq ă magpλq, the statement Spµq holds, and so ShnpY0qr1hs – BÂτ for some varietyB, tuple τ Æ µ, integer n, and non-zero invariant function h on ShnpY0q. We thus find

ShnpXqr1hs “ ShnpY0qr1hs ˆ ShnpAνq – B ˆ AτYshnpνq.

We note that τ Y shnpνq Æ λ, since all partitions in shnpνq other than ν have size ă d. Hencethe conclusion of the theorem holds for X .

In the second case, there exists an integer n ě 0 and a non-zero invariant function h onShnpY q ˆ Ashn,0pνq that does not vanish identically on ShnpXq such that the map

π : ShnpXqr1hs Ñ pShnpY q ˆ Ashn,0pνqqr1hs

is a closed immersion. Now, ShnpY q ˆ Ashn,0pνq “ Aσ where σ “ shnpµq Y shn,0pνq. Sinceh is a GL-invariant function on Aσ, we can realize Aσr1hs as a closed GL-subvariety ofA1 Âσ “ Aτ , where τ “ r∅s Yσ. We have magpτ q ă magpλq; in fact, magpτ qd ă magpλqd.Thus the statement Spτq holds by the induction hypothesis. Hence, there exists an integerm and a non-zero invariant function h1 on ShmpShnpXqr1hsq “ Shn`mpXqr1hs such thatShn`mpXqr1hh1s is isomorphic to B ˆ Aκ with κ Æ τ . Note that κ Æ λ; in fact, we havemagpκqd ă magpλqd. Thus Spλq holds.

(b) We maintain the notation from part (a). After possibly making a linear change ofcoordinates, we can choose the decomposition λ “ µ Y ν such that X does not have theform Y0 ˆ Aν ; this follows from the assumption that X is primitive. Following the proof inpart (a), we find ourselves in the second case, and so, as noted, we can realize the conclusionwith magpκqd ă magpλqd.

Remark 5.2. The variety B in Theorem 5.1 is in fact pXtKnuqr1hs, as one sees by eval-uating on K0. We can therefore deduce certain properties about B from those of X . Forexample, if X is irreducible then so is B.

The shift theorem has the following consequence for GL-algebras:

Corollary 5.3. Let R be a reduced GL-algebra that is finitely GL-generated over a field K.Then there is a non-zero element h P R such that, ignoring the GL-actions, Rr1hs isisomorphic to a polynomial algebra over a finitely generated K-algebra.

5.2. Unirationality. We now obtain the important unirationality theorem:

Theorem 5.4. Let X be an irreducible affine GL-variety. Then there exists a dominantGL-morphism ϕ : BÂµ Ñ X for some irreducible variety BK and pure tuple λ. Moreover:

(a) The image of ϕ contains a non-empty open GL-subset U of X.(b) Every point x P U admits an open neighborhood V such that the map ϕ´1pV q Ñ V

admits a section.(c) Let ΩK be an extension. Then any Ω-point of U lifts to an Ω-point of B ˆ Aλ.(d) Suppose X is a closed GL-subvariety of Ar ˆ Aλ, for some r ě 0 and pure tuple λ.

Then either X “ C ˆ Aλ for some closed subvariety C of Ar, or else one can find ϕas above with magpµq ă magpλq.


Proof. By the shift theorem (Theorem 5.1), we have an isomorphism ShnpXqr1hs – BÂλ

for some n, h, B, and λ. Parts (a), (b) and (c) now follow from Proposition 4.4 applied toShnpXqr1hs Ñ X . This morphism is dominant by (a) since X is irreducible.

We now handle part (d). Write X “ Y ˆ Aτ where λ “ σ Y τ and Y is a primitivesubvariety of Ar ˆ Aσ. If σ is the empty tuple then C “ Y is a subvariety of Ar, andX “ CÂλ. Thus assume this is not the case. By the shift theorem, we have an isomorphismShnpY qr1hs – BÂκ for some n, h, B, and κ with magpκq ă magpσq. As noted in the firstparagraph, ψ : ShnpY qr1hs Ñ Y satisfies the analogues of (a)–(c) and is dominant. Nowconsider the map Aτ ˆ ShnpY qr1hs Ñ Aτ ˆ Y given by id ˆ ψ; call this ϕ. The domainof ϕ is identified with B ˆ AκYτ and the target with X . Clearly, ϕ satisfies (a)–(c) and isdominant. Moreover, µ “ κ Y τ satisfies magpµq ă magpλq. Thus (d) holds.

Corollary 5.5. Let X be an irreducible affine GL-variety. Then X admits a GL-genericpoint defined over a finitely generated extension of K.

Proof. Let ϕ : B ˆ Aλ Ñ X be a dominant map as in Theorem 5.4. Then B ˆ Aλ admitsa GL-generic point x defined over a finitely generated extension of K by Proposition 3.14,and ϕpxq is a GL-generic point of x.

Combining the above result with a noetherian induction argument, we can obtain a finitecollection of maps from varieties of the form B ˆ Aµ that are jointly surjective. Precisely:

Proposition 5.6. Let X be a closed GL-subvariety of Aλ for some tuple λ. Then we canfind a finite family tϕi : Bi ˆ Aκi Ñ Xu1ďiďn of morphisms GL-varieties such that:

(a) Bi is an irreducible variety and magpκiq ď magpλq.(b) For any extension ΩK, the ϕi’s are jointly surjective on Ω-points.

Proof. By noetherian induction, we can assume that the result holds for every proper closedGL-subvariety of X . Applying Theorem 5.4 to an irreducible component of X , we have amap ϕ1 : B1 ˆ Aκ

1 Ñ X for an irreducible variety B1 such that:

‚ the image of ϕ1 contains a non-empty open GL-subset U of X ,‚ every Ω-point of U lifts to an Ω-point under ϕ1, and‚ magpκ1q ď magpλq.

To see the third point, we apply Theorem 5.4(d); in the first case we take κ1 “ λ, and in thesecond we actually get magpκ1q ă magpλq. Let Z be the complement of U . By the inductivehypothesis, we can find maps tϕi : Bi ˆ Aκi Ñ Zu2ďiďn such that magpκiq ď magpλq andthe ϕi’s are jointly surjective on Ω-points for all Ω. We thus see that tϕiu1ďiďn satisfies therequisite conditions.

Note that by taking B “ B1>¨ ¨ ¨>Bn and κ “ κ1Y¨ ¨ Ÿκn, one can create a single morphismϕ : B ˆ Aκ Ñ X that is surjective. However, even if X is irreducible, the B produced bythis method may well be reducible. In our follow-up (see §1.4), with a considerable amountof extra work, we show that one can actually find an irreducible B in this case.

5.3. Invariant function fields. Let X be an irreducible affine GL-variety. We define theinvariant function field of X to be KpXqGL, i.e., the fixed field of GL acting on the usualfunction field KpXq.

Proposition 5.7. Let BK be an irreducible variety with function field E and let λ be apure tuple. Then the invariant function field of B ˆ Aλ is E.


Proof. The function field EpAλq is the field of rational functions in the coordinate variables onAλ. No non-constant rational function is invariant under GL; in fact, none is invariant underthe infinite symmetric group, which is a subgroup of GL. We thus see that EpAλqGL “ E,which establishes the result.

Proposition 5.8. Let X be an irreducible GL-variety over a field K. Then the invariantfunction field KpXqGL is a finitely generated extension of K.

Proof. By the unirationality theorem (Theorem 5.4), we have a dominant GL-morphismB ˆ Aλ Ñ X with B an irreducible variety over K and λ a pure tuple. We thus haveKpXqGL Ă KpB ˆ AλqGL “ E, where E is the function field of B. Thus KpXqGL iscontained in a finitely generated extension of K, and is therefore itself finitely generated.

Remark 5.9. We can also consider the invariants of KpXq under the subgroups Gpnq. Thisleads to a tower of fields

KpXqGL “ KpXqGp0q Ď KpXqGp1q Ď KpXqGp2q Ď ¨ ¨ ¨

each of which is a finitely generated extension of K. The shift theorem implies that form " 0, KpXqGpnq is a rational function field over KpXqGpmq for each n ě m.

5.4. Dimension functions. Let X be an affine GL-variety over K. We define the dimen-sion function of X to be the function δX : N Ñ N given by δXpdq “ dimXtQdu.

Proposition 5.10. The dimension function δX is eventually polynomial. That is, there isa polynomial p with rational coefficients such that δXpdq “ ppdq for all d " 0.

Proof. First suppose X is irreducible. By the shift theorem, we have ShnpXqr1hs – BÂλ

for some integer n. We have ShnpXqtQdu “ XtQn`du, and so δShnpXqpdq “ δXpd ` nq.Of course, inverting h does not affect the dimension since X is irreducible. We thus seethat δXpd ` nq “ dimB ` δAλpnq. Now, δAλpdq is just the dimension of the representationÀr

i“1 SλipQdq, where λ “ rλ1, . . . , λrs, which is known to be polynomial in d of degree the

maximum among the sizes of the λi. Thus δAλ is polynomial. The result follows; in fact, wesee that d ÞÑ δXpdq is polynomial for d ě n.

We now treat the general case. Let Y1, . . . , Yr be the irreducible components of X . ThenδXpdq “ maxpδY1pdq, . . . , δYrpdqq. Since each δYi is eventually polynomial by the previousparagraph, it follows that δX is too.

6. Some properties of affine spaces

6.1. Dominant maps to Aλ. It is very easy to write down many examples of dominantmaps An Ñ Am in standard algebraic geometry. By contrast, we now show that essentiallythe only dominant maps Aµ Ñ Aλ are projection maps when λ and µ are pure.

Proposition 6.1. Let B and C be irreducible varieties, let µ and λ be pure tuples, and letϕ : B ˆ Aµ Ñ C ˆ Aλ be a dominant morphism of GL-varieties. Then λ Ă µ. Moreover,letting ϕ0 : B Ñ C be the map induced by ϕ, there exist non-empty open affine subvarietiesU Ă B and V Ă C such that ϕ0pUq Ă V and the restriction of ϕ to U ˆ Aµ factors as

U ˆ Aµ σ// U ˆ Aµ idˆπ

// U ˆ Aλ ϕ0îd// V ˆ Aλ

where σ is an U-automorphism of U Âµ and π is the projection map. In fact, it is possibleto choose U and V such that ϕ maps U ˆ Aµ surjectively onto V ˆ Aλ.


We require a few lemmas before giving the proof. Recall that Rλ “ SympVλq is thecoordinate ring of Aλ. We write Rλ,` for the ideal of positive degree elements of Rλ.

Lemma 6.2. Let E Ă F be field extensions of K, let λ and µ be pure tuples, and let f : EbRλ Ñ F b Rµ be an injection of GL-algebras over E. Then dimSλpKnq ď c ` dimSµpKnqfor some c P N and all n ě 0.

Proof. The restriction of f to Rλ is still an injection of GL-algebras, so we may as wellassume E “ K. Then f induces a map of representations Vλ Ñ F b Rµ. The image ofthis map is necessarily contained in U b Rµ for some finite dimensional K-subspace U ofF . It follows that the image of f is contained in B b Rµ where B is the K-subalgebra of Fgenerated by U .

Evaluating the polynomial functors on Kn, we see that the map RλtKnu Ñ B bRµtKnuis an injection of finitely generated K-algebras. It follows that the Krull dimension of thesource, which is dimSλpKnq, is at most that of the target, which is c ` dimSµpKnq, wherec “ dimpBq. The result follows.

Lemma 6.3. Let E Ă F be field extensions of K, let λ and µ be pure tuples, and letf : E b Rλ Ñ F b Rµ be an injection of GL-algebras over E. Then the induced map

f : F b Rλ,`R2λ,` Ñ F b Rµ,`R2

µ,`

is an injection. (Note: it is F on both the left and right sides above.)

Proof. Suppose that f fails to be injective on the Vν isotypic piece. Let n be the multiplicityof ν in λ, and letm ă n be the multiplicity ofVν in the image of f . We thus see that the n “νvariables” in EbRλ are mapped to elements that make use of only m “ν variables” in FbRµ

and lower degree elements. In other words, f induces an injection E b Rνn Ñ F b Rµ1Yνm

where µ1 consists of the partitions in µ of size ă |ν|, and νn denotes the n-tuple rν, . . . , νs.This contradicts the Lemma 6.2

Proof of Proposition 6.1. Let B “ SpecpRq and C “ SpecpSq where R and S are finitelygenerated integral K-algebras. Since ϕ is dominant, it follows that ϕ0 : B Ñ C is dominant,and that the corresponding K-algebra homomorphism S Ñ R is injective. Let E “ FracpSqand F “ FracpRq; we regard E as a subfield of F . Then ϕ induces an injection f : EbRλ ÑF b Rµ of GL-algebras over E. Lemma 6.3 implies that the multiplicity of Vν in Vλ is atmost the multiplicity of Vν in Vµ, for any partition ν; note that Rλ,`R2

λ,` – Vλ. Thusλ Ă µ. Write µ “ λ Y ν.

Let f be as in Lemma 6.3. The image of f is isomorphic to F b Vλ, and thus hasa complementary representation isomorphic to F b Vν . Choose an F -linear injection of

representations g : F bVν Ñ F bRµ such that the images of f and g in F bRµ,`R2µ,` form

complementary subrepresentations. Let g : F bRν Ñ F bRµ be the algebra homomorphisminduced by g. Consider the following sequence of maps

E b Rλ// F b Rλ

// F b Rλ b Rν

fbg// // F b Rµ

where the first two maps are the inclusions. Since the right map is an isomorphism modulothe maximal ideal, Nakayama’s lemma implies that it is an isomorphism. We identify thethird algebra with F b Rµ. The second map is then the standard inclusion coming fromλ Ă µ, and the right map is an automorphism.


Now, consider the automorphism fbg of F bRµ,`R2µ,`. For each partition ν in µ, let Aν

be the matrix of f b g on the ν-multiplicity space in some basis. Each Aν is a finite matrix,and there are only finitely many of them. It follows that we can find a non-zero h P R

such that each of our bases is defined over Rr1hs, the determinants of all these matrices areunits in Rr1hs, and f and g are defined over Rr1hs. We thus see that f b g induces anautomorphism of Rr1hs b Rµ. We thus get maps

S b Rλ// Rr1hs b Rλ

// Rr1hs b Rµ

fbg//// Rr1hs b Rµ

where, again, the first map comes from the inclusion S Ă Rr1hs and the second from theinclusion λ Ă µ. Let U “ SpecpRr1hsq, an open affine of B. The above maps translate tomaps

U ˆ Aµ σ// U ˆ Aµ idˆπ

// U ˆ Aλϕ0îd

// C ˆ Aλ

where σ is a U -automorphism and π is the projection map. This gives the main statementof the proposition, with V “ C.

To prove the final part, about surjectivity, simply take V to be a non-empty open subsetof C contained in ϕ0pUq (which exists by Chevalley’s theorem) and replace U with U Xϕ´10 pV q.

6.2. A lifting result. Using Proposition 6.1, we obtain a very useful lifting result:

Proposition 6.4. Let ϕ : B Âλ Ñ X and ψ : Y Ñ X be morphisms of GL-varieties, withB an irreducible variety and λ a pure tuple. Let x be the generic point of B ˆ Aλ, andsuppose that ϕpxq P impψq. We can then find a commutative diagram

C ˆ Aλ αîd//

B ˆ Aλ

ϕ

Yψ

// X

where C is an irreducible variety and α : C Ñ B is a quasi-finite dominant morphism.

Proof. Let Z be the reduced subscheme of the fiber product of pB ˆ Aλq ˆX Y . This is aGL-variety, and the condition ϕpxq P impψq ensures that the projection map Z Ñ B ˆ Aλ

is dominant. Let Z 1 be an irreducible component of Z mapping dominantly to BÂλ, and,applying Theorem 5.4, choose a dominant morphism C ˆ Aµ Ñ Z 1 with C an irreduciblevariety. We thus have a commutative diagram

C ˆ Aµ //

B ˆ Aλ

ϕ

Yψ

// X

where the top morphism is dominant. We now apply Proposition 6.1. We find that λ Ă µ.Furthemore, replacing C with a dense open and applying an automorphism of C ˆ Aµ,we can assume that the top map has the form α ˆ π where α : C Ñ B is a morphism,necessarily dominant, and π : Aµ Ñ Aλ is the projection map. We now simply restrict toC ˆ Aλ Ă C ˆ Aµ. This yields the stated result, except for quasi-finiteness. For this, wereplace C with an appropriate closed subvariety.

We note an elegant corollary:


Corollary 6.5. Assume K is algebraically closed. Suppose that ϕ : Y Ñ Aλ is a dominantmorphism of GL-varieties, with Y irreducible and λ pure. Then ϕ admits a GL-equivariantsection. In particular, ϕ is surjective.

Proof. Apply the proposition with B “ SpecpKq and X “ Aλ. Then C is necessarilySpecpKq (since K is algebraically closed), and the map C ˆ Aλ Ñ Y is the sought aftersection.

6.3. Extending maps. Let Z be a closed subscheme of an affine scheme X . Then anymorphism Z Ñ An can be extended to a morphism X Ñ An. We now show that thisproperty also holds for the GL-analogs of affine spaces.

Proposition 6.6. Let X be an affine GL-scheme, let Z be a closed GL-subscheme of X,and let ϕ : Z Ñ Aλ be a morphism of GL-schemes, with λ an arbitrary tuple. Then thereexists a morphism of GL-schemes ψ : X Ñ Aλ extending ϕ.

Proof. Write X “ SpecpAq and Z “ SpecpAIq, where A is a GL-algebra and I is a GL-idealof A. Write Aλ “ SpecpRλq as usual, with Rλ “ SympVλq. Consider the following diagram

Rλ

ψ˚

//

ϕ˚!!

A

AI

We must find a morphism ψ˚ of GL-algebras that makes the diagram commute. The re-striction of ϕ˚ to Vλ Ă Rλ is a map Vλ Ñ AI of representations. It lifts to a map ofrepresentations Vλ Ñ A, since polynomial representations of GL are semi-simple. Thismap induces a map of GL-algebras ψ˚ : Rλ Ñ A, by the universal property of Sym, whichsatisfies the necessary properties.

Remark 6.7. This proposition does not remain valid in positive characteristic: for instance,over a fieldK of characteristic 3, the space of infinite cubics contains as a proper closed subsetthe linear space spanned by third powers of linear forms, which is stable under the infinitegeneral linear group. The identity map from that space to itself does not extend to anequivariant morphism defined on the entire space of infinite cubics.

7. The decomposition theorem

Throughout this section, “GL-variety” means “quasi-affine GL-variety” by default.

7.1. Elementary varieties and morphisms. We say that a GL-variety X is elementaryif it is isomorphic to a GL-variety of the form B ˆ Aλ for some irreducible affine variety Band tuple λ; this implies that X is irreducible and affine. Let ϕ : X Ñ Y be a morphism ofGL-varieties. We say that ϕ is elementary if there exists a commutative diagram

(7.1)

Xϕ

//

i

Y

j

B ˆ Aλψˆπ

// C ˆ Aµ

where i and j are isomorphisms of GL-varieties, B and C are irreducible affine varieties,ψ : B Ñ C is a surjective morphism of varieties, λ and µ are pure tuples with µ Ă λ, and


π : Aλ Ñ Aµ is the projection map. This implies that X and Y are elementary and that ϕis surjective. Note that a GL-variety is elementary if and only if its identity map is. Wesay that a Gpnq-variety, or morphism of Gpnq-varieties, is Gpnq-elementary if it is so afteridentifying Gpnq with GL.

We say that a quasi-affine GL-variety X is locally elementary at x P X if there exists mand a Gpmq-stable open neighborhood of x that is Gpmq-elementary; we say that X is locallyelementary if it is so at all points. Let ϕ : X Ñ Y be a morphism of GL-varieties. We saythat ϕ is locally elementary at x P X if there exists m and Gpmq-stable open neighborhoodsU of x and V of ϕpxq such that ϕ induces a Gpmq-elementary morphism U Ñ V . We say thatϕ is locally elementary if it is surjective and locally elementary at all x P X . This impliesthat both X and Y are locally elementary. We note that a GL-variety is locally elementaryif and only if its identity morphism is. We again define a Gpnq-variety, or a morphism ofGpnq-varieties, to be locally Gpnq-elementary if it is so after identifying Gpnq with GL.

Example 7.2. Let λ be a non-empty pure tuple. Then X “ Aλzt0u is an example of alocally elementary variety that is not elementary. The fact that X is locally elementaryfollows from Corollary 7.6 below. It is clear that X is not elementary, as it is not affine. Theidentity morphism of X is a (somewhat boring) example of a locally elementary morphismthat is not elementary.

Proposition 7.3. Let ϕ : X Ñ Y be a morphism of Gpnq-varieties, and let m ě n.

(a) If ϕ is Gpnq-elementary then it is Gpmq-elementary.(b) ϕ is locally Gpnq-elementary (at x P X) if and only if it is locally Gpmq-elementary

(at x).

Proof. (a) There is a diagram (7.1) after identifying Gpnq with GL. Now apply Shmń tothat diagram. Recall that ShmńX – X where the action of GL on the right-hand side isvia its isomorphism to Gpmńq, which under the identification of GL with Gpnq is mappedto Gpmq. So the top row remains unchanged (except that we remember only that ϕ is Gpmq-equivariant). In the bottom row, B and C are replaced by products with ordinary affinespaces Ab and Ac where b ě c are the multiplicities of the empty partition in shmń λ andshmń µ, respectively, ψ is replaced by its product with a linear surjection Ab Ñ Ac, and πis replaced by a GL-equivariant linear surjection Aλ1

Ñ Aµ1where λ1 and µ1 are the pure

parts of shmń λ and shmń µ1, respectively. We thus see that ϕ is Gpmq-elementary.

(b) Assume that ϕ is locally Gpnq-elementary at x P X . Then there exist a k ě n andGpkq-stable open neigborhoods U of x and V of ϕpxq such that ϕ induces a Gpkq-elementarymorphism U Ñ V . By part (a), ϕ : U Ñ V is also Gpmaxtk,muq-elementary, so ϕ is locallyGpmq-elementary at x. The converse is similar (but does not require the use of part (a),since an integer ě m is automatically ě n).

Due to part (b) above, we simply say “locally elementary” in place of “locally Gpnq-elementary,” since the condition is independent of n.

Proposition 7.4. Let ϕ : X Ñ Y be a morphism of GL-varieties. The following are equiv-alent:

(a) ϕ is locally elementary.(b) There exist m and non-empty Gpmq-stable open subvarieties U1, . . . , Un of X and

V1, . . . , Vn of Y such that X is the union of the GL-translates of the Ui’s, Y is the


union of the GL-translates of the Vi’s, and ϕ induces a Gpmq-elementary map Ui Ñ Vifor all i.

Proof. It is clear that (b) implies (a). Conversely, suppose (a) holds. For each x P X ,choose open Gpmxq-stable neighborhoods Ux of x and Vx of ϕpxq such that ϕ induces aGpmxq-elementary map Ux Ñ Vx. Let U 1

x “ŤgPGL

gUx. The sets U 1x form an open cover

of X by GL-stable open subsets. Since Xorb is quasi-compact, it follows that there is afinite subcover; thus we can find points x1, . . . , xn such that X “

Ťn

i“1 U1xi. Put Ui “ Uxi

and Vi “ Vxi; these are open and Gpmq-stable, where m “ maxpmx1 , . . . , mxnq. Note thatϕ : Ui Ñ Vi is Gpmq-elementary by Proposition 7.3(a). It is thus clear that (b) holds.

Proposition 7.5. Locally elementary maps are stable under base change along open immer-sions. More precisely, consider a pullback diagram

X 1 //

ϕ1

X

ϕ

Y 1 // Y

where ϕ is a locally elementary map of GL-varieties and Y 1 Ñ Y is an open immersion ofGL-varieties. Then ϕ1 is locally elementary.

Proof. We proceed in three steps. In what follows, we simply identify Y 1 and X 1 with openGL-subvarieties of Y and X .

Step 1: Suppose ϕ is elementary and Y 1 “ Y r1hs for non-zero GL-invariant functionh on Y . Identify X with B ˆ Aλ and Y with C ˆ Aµ and ϕ with ψ ˆ π, where ψ : B Ñ C

is a surjective morphism of varieties and π : Aλ Ñ Aµ is the projection map. We findthat Y 1 “ Cr1hs ˆ Aµ and X 1 “ Br1hs ˆ Aλ. The induced map ϕ1 : X 1 Ñ Y 1 is simplyψ|Br1hs ˆ π, and is thus elementary.

Step 2: Suppose ϕ is elementary. Let x P X 1 and y “ fpxq. We can then find a functionh on Y such that y P Y r1hs Ă Y 1. Let m be such that h is Gpmq-invariant. Regard ϕ

as a morphism of Gpmq-varieties—as such, it is still elementary by Proposition 7.3(a)—andapply Step 1 with the open set Y r1hs. We thus see that ϕ´1pY r1hsq Ñ Y r1hs is Gpmq-elementary. Since x P ϕ´1pY r1hsq Ă X 1, it follows that ϕ1 is locally elementary at x. Sincex is arbitrary and ϕ1 is surjective, it follows that ϕ1 is locally elementary.

Step 3: the general case. Let x P X 1 and let y “ ϕpxq. Since ϕ is locally elementary, wecan find Gpmq-stable open neighborhoods U of x and V of y such that ϕ induces a Gpmq-elementary morphism ϕ : U Ñ V . Let V 1 “ V X Y 1 and let U 1 “ U X X 1 “ ϕ´1pV 1q. ByStep 2, we see that the induced morphism U 1 Ñ V 1 is locally elementary. Since x P U 1 Ă X 1,it follows that ϕ1 is locally elementary at x. Since x is arbitrary and ϕ1 surjective, it followsthat ϕ1 is locally elementary.

Applying the proposition to the identity map, we find:

Corollary 7.6. Let X be a locally elementary GL-variety and let U be an open GL-subvariety. Then U is locally elementary.

7.2. Locally elementary decompositions. Let X be a GL-variety. A locally elementarydecomposition of X is a finite decomposition X “

ŮiPI Xi of X such that each Xi is GL-

stable, locally closed (i.e., open in its closure), and locally elementary. Let ϕ : X Ñ Y bea morphism of GL-varieties. A locally elementary decomposition of ϕ consists of locally


elementary decompositions X “ŮiPI Xi and Y “

ŮjPJ Yj such that for each i P I there is

some j P J such that ϕpXiq Ă Yj and the map ϕ : Xi Ñ Yj is locally elementary. In whatfollows, we sometimes abbreviate “locally elementary decomposition” to LED.

Proposition 7.7. Consider a pullback diagram

X 1 //

ϕ1

X

ϕ

Y 1 // Y

where ϕ is a map of GL-varieties and Y 1 Ñ Y is an open immersion of GL-varieties.Suppose ϕ admits an LED given by X “

ŮiPI Xi and Y “

ŮjPJ Yj. Put X 1

i “ Xi X X 1 and

Y 1j “ Yj X Y 1. Then ϕ1 admits the LED X 1 “

ŮiPI X

1i and Y

1 “ŮjPJ Y

1j .

Proof. This follows immediately from Proposition 7.5.

7.3. The decomposition theorem. The following is our main structure theorem for mor-phisms of GL-varieties.

Theorem 7.8. Every morphism of GL-varieties admits a locally elementary decomposition.

Applying the theorem to the identity map, we find:

Corollary 7.9. Every GL-variety admits a locally elementary decomposition.

We require a number of lemmas before proving the theorem.

Lemma 7.10. Let X be a non-empty GL-variety. Then X contains an irreducible (and, inparticular, non-empty) locally elementary open GL-subvariety.

Proof. First suppose that X is affine. By the shift theorem (Theorem 5.1), there is aninteger n ě 0 and a non-zero GL-invariant function h on ShnpXq such that ShnpXqr1hs iselementary. Under the identification pGL, ShnpXqq “ pGpnq, Xq, the open GL-subvarietyShnpXqr1hs of ShnpXq corresponds to an open Gpnq-subvariety U0 of X . Now, let U “ŤgPGL

gU0, an irreducible open GL-subvariety of X . Since U0 is Gpnq-elementary, it followsthat U is locally elementary.

We now treat the general case. Let U be a non-empty open affine Gpnq-subvariety of X .By the previous paragraph, U contains a non-empty open locally elementary Gpnq-subvarietyV0. Let V “

ŤgPGL

gV0. Then V is an irreducible open locally elementary GL-subvariety ofX .

Lemma 7.11. Let ϕ : X Ñ Y be a dominant morphism of irreducible locally elementaryGL-varieties. Then there exist non-empty open GL-subvarieties U Ă X and V Ă Y suchthat ϕpUq Ă V and the induced map ϕ : U Ñ V is locally elementary.

Proof. First suppose that X and Y are elementary, say X “ B ˆ Aλ and Y “ C ˆ Aµ. ByProposition 6.1, we have µ Ă λ and there are non-empty open subschemes U0 of B and V0of C such that ϕ0pU0q “ V0 and the restriction of ϕ to U0 ˆ Aµ factors as

U0 ˆ Aλ σ// U0 ˆ Aλ idˆπ

// U0 ˆ Aµϕ0îd

// V0 ˆ Aµ


where σ is some U0-automorphism of U0 ˆ Aλ and π is the projection map. Thus, puttingU “ U0 ˆ Aλ and V “ V0 ˆ Aµ, we see that the diagram

Uϕ

//

i

V

U0 ˆ Aλϕ0ˆπ

// V0 ˆ Aµ

commutes, where i “ σ. Thus ϕ : U Ñ V is elementary.We now treat the general case. Since Y is non-empty and locally elementary, we can

find a non-empty open elementary Gpnq-subvariety Y 1 of Y . Since ϕ´1pY 1q is a locallyelementary Gpnq-variety (Proposition 7.5), by the same reasoning we can find a non-emptyopen elementary affine Gpmq-subvariety X 1 of ϕ´1pY 1q, for some m ě n. Note that Y 1 is anelementary Gpmq-variety (Proposition 7.3(a)). Thus ϕ : X 1 Ñ Y 1 is a dominant morphism ofirreducible elementary affine Gpmq-varieties. By the previous paragraph, we can find non-empty open Gpmq-subvarieties U 1 Ă X 1 and V 1 Ă Y 1 such that ϕ induces an elementary mapU 1 Ñ V 1. Put U “

ŤgPGL

gU 1 and V “ŤgPGL

gV 1. Then ϕ induces a locally elementarymap U Ñ V , which completes the proof.

Lemma 7.12. Let ϕ : X Ñ Y be a dominant morphism of non-empty GL-varieties. Thenthere exist non-empty open GL-subvarieties U Ă X and V Ă Y such that ϕpUq Ă V and themap ϕ : U Ñ V is locally elementary.

Proof. By Lemma 7.10, we can find an irreducible open locally elementary GL-subvarietyY 1 of Y . By Lemma 7.10 again, we can find an irreducible open locally elementary GL-subvariety X 1 of ϕ´1pY 1q. By Lemma 7.11, we can find non-empty open GL-subvarietiesU Ă X 1 and V Ă Y 1 such that ϕpUq Ă V and the map ϕ : U Ñ V is locally elementary.

Proof of Theorem 7.8. Let ϕ : X Ñ Y be a given morphism of GL-varieties. By noetherianinduction on Y , we can assume that for any proper closed GL-subvariety Z of Y , the mapϕ´1pZq Ñ Z admits an LED.

First suppose that Y is reducible. Write Y “ Y 1 YY 2 where Y 1 and Y 2 are proper closedGL-subvarieties. Let Y3 “ Y 1 XY 2 and Y1 “ Y 1zY 2 and Y2 “ Y 2zY 1; thus Y “ Y1 \Y2 \Y3is a decomposition of Y into locally closed GL-subvarieties. For i P t1, 2u, the morphismϕ´1pY iq Ñ Y i admits an LED by the inductive hypothesis, and so ϕ´1pYiq Ñ Yi admitsone by Proposition 7.7. The morphism ϕ´1pY3q Ñ Y3 also admits an LED by the inductivehypothesis. Putting together the LED’s for ϕ´1pYiq Ñ Yi for i P t1, 2, 3u, we obtain one forX Ñ Y .

Now suppose that Y is irreducible. By noetherian induction on X , we can assume thatfor any proper closed GL-subvariety Z of X , the map ϕ : Z Ñ Y admits an LED. If ϕ is notdominant then its image is contained in a proper closed GL-subvariety Z of Y . We obtainan LED for ϕ by taking one for ϕ´1pZq Ñ Z (which exists by the first inductive hypothesis)and simply throwing in the open set Y zZ into the decomposition of Y . We can thus assumein what follows that ϕ is dominant.

By Lemma 7.12, we can find non-empty open GL-subvarieties W Ă X and V1 Ă Y suchthat ϕpW q Ă V1 and the map ϕ : W Ñ V1 is locally elementary. Let Z “ XzW . By thesecond inductive hypothesis, the map ϕ : Z Ñ Y admits an LED, say Z “

ŮiPI Zi and

Y “ŮjPJ Yj. Let j0 be the unique element of J such that V2 “ Yj0 is open, and let I0 Ă I

be the set of indices i such that Zi maps into V2. Let V “ V1 X V2 and let U “ ϕ´1pV q. We


have U “ pW X Uq \ŮiPI0

pZi X Uq. The maps W X U Ñ V and Zi X U Ñ V are locallyelementary by Proposition 7.5. We thus see that this decomposition of U , together with thetrivial decomposition V “ V of V , is a LED for ϕ : U Ñ V . Combining this LED with theone for ϕ´1pY zV q Ñ Y zV (which exists by the first inductive hypothesis), we obtain onefor ϕ.

7.4. Chevalley’s theorem. Let X be a GL-variety. We say that a subset of X is GL-constructible if it is a finite union of sets of locally closed GL-stable subsets. The followingis an analog of Chevalley’s theorem for GL-varieties:

Theorem 7.13. Let ϕ : X Ñ Y be a morphism of quasi-affine GL-varieties, and let C be aGL-constructible subset of X. Then ϕpCq is a GL-constructible subset of Y .

Proof. First suppose that C “ X . Let X “ŮiPI Xi and Y “

ŮjPJ Yj be a LED for ϕ. Thus

for each i P I there is j P J such that ϕ induces a locally elementary map Xi Ñ Yj. Since alocally elementary map is surjective, it follows that the image of ϕ is a union of some subsetof tYjujPJ , and thus GL-constructible.

Now if C is a general GL-constructible set in X , then it is a finite union of locally closedGL-stable subsets of X , and we apply the previous paragraph to the restrictions of ϕ tothese subsets.

Remark 7.14. In EGA and the Stacks project (see [Stacks, Tag 04ZC]), constructible setsare defined for arbitrary schemes (in fact, arbitrary topological spaces). However, only theso-called retrocompact open subsets are considered constructible. The basic example of anon-retrocompact open subset is A8zt0u. In particular, the open GL-subset Aλzt0u of Aλ

is not retrocompact, and therefore is not constructible in the usual sense. This is why weuse the term GL-constructible. We note, however, that GL-constructible subsets of X arein bijective correspondence with constructible subsets (in the usual sense) of Xorb.

7.5. Lifting points. Suppose that ϕ : X Ñ Y is a morphism of GL-varieties and y is aK-point in the image of ϕ. Finding a point of X lifting y typically involves solving infinitelymany equations, so it is not immediately clear how to control the extension of K requiredto find a solution. Using the decomposition theorem, we find that the nicest reasonablestatement is true:

Proposition 7.15. Let ϕ : X Ñ Y be a morphism of quasi-affine GL-varieties. Let Ω be anextension of K and let y be an Ω-point of the image of ϕ. Then there exists a finite extensionΩ1Ω and an Ω1-point x of X mapping to y. In fact, there exists an integer d (dependingonly on ϕ) such that one can always take rΩ1 : Ωs ď d.

Proof. The statement is clear for maps of ordinary varieties, thus for elementary maps ofGL-varieties, thus for locally elementary maps (use Proposition 7.4 to get uniformity ofrΩ1 : Ωs), and thus for all maps by Theorem 7.8.

7.6. A variant. Suppose that P is some property of varieties and morphisms. We definea GL-variety to be P-elementary if it has the form B ˆ Aλ where B is a variety satisfyingP. We define a morphism of GL-varieties to be P-elementary if there is a diagram like(7.1) where ψ satisfies P. Our definitions of elementary correspond to taking P to be theproperty “irreducible affine” on varieties and the property “surjective” on morphisms. Onecan prove an analog of the decomposition theorem for many different choices of P; essentially,one just needs to be able to decompose a variety or morphism of varieties into pieces that

http://stacks.math.columbia.edu/tag/04ZC


satisfy P. For instance, there is a version of the decomposition theorem where the strata aresmooth and the morphisms on strata are flat. The particular P we worked with is somewhatarbitrary, but was chosen to make the proof of Chevalley’s theorem as simple as possible.

8. Theory of types

8.1. Types. We fix an irreducible GL-variety X for this section.

Definition 8.1. A typical morphism for X is a dominant morphism ϕ : B ˆ Aλ Ñ X , withB an affine variety and λ a pure tuple, with the following property: if Z is a proper closedGL-subvariety of B ˆ Aλ then ϕ|Z is not dominant.

We note that in a typical morphism, the variety B is necessarily irreducible.

Proposition 8.2. A typical morphism exists.

Proof. Let Λ be the set of all pure tuples λ for which a dominant morphism B ˆ Aλ Ñ X

exists. This set is non-empty by Theorem 5.4. Let λ P Λ be any element of minimalmagnitude; such a minimal element exists since magnitudes are well-ordered. Now let d P N

be minimal such that a dominant morphism B ˆ Aλ Ñ X exists with dimpBq “ d.Let ϕ : B ˆ Aλ Ñ X be any dominant morphism with B irreducible of dimension d.

We claim that ϕ is typical. Indeed, suppose that Z is a proper closed subset of B ˆ Aλ

such that ϕ|Z is dominant. We will obtain a contradiction. We may as well suppose thatZ is irreducible. We now apply Theorem 5.4(d). There are two cases. In the first case,Z “ C ˆ Aλ for some (necessarily proper) subvariety C of B (to apply Theorem 5.4 aswritten, embed B into some Ar). This gives a contradiction since dimpCq ă d. In thesecond case, there is a dominant morphism BÂµ Ñ Z for some µ with magpµq ă magpλq;thus µ P Λ. This contradicts the minimality of λ.

Proposition 8.3. Let ϕ : B ˆ Aλ Ñ X be a typical morphism and let ψ : Y Ñ B ˆ Aλ beanother morphism of GL-varieties. If ϕ ˝ ψ is dominant then ψ is dominant.

Proof. Let Z “ impψq. Since ϕ ˝ ψ is dominant, it follows that ϕ|Z is dominant. Since ϕ istypical, we have Z “ B ˆ Aλ, and thus ψ is dominant.

Proposition 8.4. Let ϕ : B ˆ Aλ Ñ X be a typical morphism, and let ψ : C ˆ Aµ Ñ X

be a dominant morphism with C irreducible and µ pure. Then there exists a commutativediagram

D ˆ Aµ αîd//

θ

C ˆ Aµ

ψ

B ˆ Aλϕ

// X

where D is an irreducible variety, α : D Ñ C is dominant and quasi-finite, and θ is dominant.In particular, λ Ă µ and dimpBq ď dimpCq.

Proof. Let x be be the generic point of C ˆ Aµ and let y be the generic point of B ˆ Aλ.Then ψpxq “ ϕpyq is the generic point of X since both ϕ and ψ are dominant; in particular,ψpxq P impϕq. Applying Proposition 6.4 (with Y “ BÂλ), we obtain a commutative squarehaving the stated properties except perhaps for the dominance of θ. However, since ϕ ˝ θ isdominant, Proposition 8.3 shows that θ is dominant. Proposition 6.1 now shows that λ Ă µ.Since θ induces a dominant morphism D Ñ B, we have dimpBq ď dimpDq “ dimpCq.


Corollary 8.5. Let ϕ : B ˆ Aλ Ñ X and ψ : C ˆ Aµ Ñ X be typical morphisms. Thenλ “ µ and dimpBq “ dimpCq.

Definition 8.6. Let ϕ : B ˆ Aλ Ñ X be a typical morphism. We define the type of X ,denoted typepXq, to be the pure tuple λ. We define the typical dimension of X , denotedtdimpXq, to be dimpBq. These are well-defined by the above corollary.

Proposition 8.7. Let ϕ : B ˆ Aλ Ñ X be a dominant morphism with B irreducible and λpure. Then typepXq Ă λ and tdimpXq ď dimpBq, with equalities if and only if ϕ is typical.

Proof. The inequalities typepXq Ă µ and tdimpXq ď dimpCq follow from Proposition 8.4.If ϕ is typical, then both are equalities by definition. Conversely, suppose that both areequalities. We follow the proof of Proposition 8.2. By the inequalities proven here, the set Λdefined there consists of all tuples µ such that typepXq Ă µ and the numer d defined thereis just tdimpXq. Thus the second paragraph of the proof shows that ϕ is typical.

8.2. Characterization of typical dimension. Let X be an irreducible GL-variety. Thefollowing is the main theorem of this section:

Theorem 8.8. Let ϕ : BÂλ Ñ X be a typical morphism. Then KpBq is a finite extensionof KpXqGL.

We note that ϕ˚ induces a field extension KpXq Ă KpB ˆ Aλq, and thus an extensionKpXqGL Ă KpBÂλqGL “ KpBq, where we have used Proposition 5.7. The content of thetheorem is that this extension has finite degree.

Corollary 8.9. The typical dimension of X is equal to the transcendence degree over K ofthe invariant function field KpXqGL.

Proof. Let ϕ be as in Theorem 8.8. Then we have

tr. degpKpXqGLKq “ tr. degpKpBqKq “ dimpBq “ tdimpXq.

where the first equality follows from Theorem 8.8, the second is standard, and the third isthe definition of typical dimension.

The proof of the theorem occupies the remainder of §8.2. It is inspired by Rosenlicht’stheorem on separating general orbits by rational invariants [Ro, Theorem 2]. It needs thefollowing lemma, as well as its proof.

Lemma 8.10. Let V be a vector space over a field K, F Ą K a field extension, G a groupof K-automorphisms of F , and W an F -subspace of F bK V that is stable under the actionof G on the first tensor factor. Then W is spanned over F by elements in WG Ă FG bK V .

Proof. Choose a well-ordered K-basis peiqiPI of V , so that each element w of W has a uniqueexpression w “

řiPI ci b ei where only finitely many of the ci P F are nonzero; the largest ei

with ci ‰ 0 is called the leading term of w, and ci its leading coefficient.Call w minimal if the support ti : ci ‰ 0u is inclusion-wise minimal among all supports of

nonzero vectors of W and if moreover the leading coefficient equals 1. Then for each g P G,gw ´ w P W has support strictly contained in that of w, and must therefore be zero. Henceany minimal w has coefficients in FG.

Furthermore, every u P W is an F -linear combination of minimal elements: find a minimalw whose leading term equals that of u, subtract w times the leading coefficient of u from u

so that the leading term becomes smaller, and use well-orderedness to conclude.


Proof of Theorem 8.8. Let ϕ : BÂλ Ñ X be a typical morphism. We may assume that Kis algebraically closed; we then may and will work exclusively with K-points in this proof.After passing to an affine open dense subset of B, we may further assume that for all K-points b P B, the map ϕpb, .q : Aλ Ñ impϕpb, .qq is typical, i.e., does not factor through Aµ

for any µ Ă λ.Let Γ Ď X ˆ B ˆ Aλ be the graph of ϕ, and let π : Γ Ñ X ˆ B be the restriction of the

projection morphism; this is a morphism of GL-varieties. Note that the points of impπq arethe pairs px, bq with x P impϕpb, .qq.

Let I Ď KrX ˆBs be an ideal whose vanishing set is impπq and which is generated by theGL-orbits of finitely many elements h1, . . . , hm; I exists by Noetherianity (Theorem 2.1). LetJ be the ideal generated by I in KpXq bKrBs. Then J is stable under the action of GL onKpXq and hence by Lemma 8.10, J is spanned overKpXq by elements inKpXqGLbKrBs. Inparticular, we can find GL-invariant elements f1, . . . , fk P J that are minimal relative to anarbitrary well-ordered K-basis of KrBs and such that each hi is a KpXq-linear combinationof the fj. Write fj “

řl gjl b fjl where the fjl are elements of the chosen basis of KrBs and

gjl P KpXqGL. The proof of Lemma 8.10 shows that we can arrange things such that eachhi is in fact a linear combination of the fj with coefficients in the subalgebra KrXsrtgjlusof KpXq generated by KrXs and the gjl. Note that then, by GL-invariance of the gjl, eachelement in GL ¨ hi is also a KrXsrtgjlus-linear combination of the fj .

By Chevalley’s theorem (Theorem 7.13), impπq contains a dense, GL-stable, open subset

U1 of impπq. We then have U1 “ impπq X U2 for some GL-stable, dense open subset U2 ofXˆB. Now intersect U2 with the preimage in XˆB of the domains of definition in X of thefinitely many rational functions in KpXq needed to express each fj as a linear combinationof elements in I. This yields a (not necessarily GL-stable) open, dense subset U3 of X ˆB,

which, since impπq Ñ X is dominant, still intersects impπq in an open dense subset containedin impπq and satisfies fjpx, bq “ 0 for all px, bq P impπq X U3. Next intersect U3 with thepreimage in X ˆ B of the domains of definition in X of the finitely many gjl P KpXqGL.

This yields an open, dense subset U4 of X ˆB, which still intersects impπq in an open denseset and such that, for px, bq P U4, we have px, bq P impπq if and only if fjpx, bq “ 0 forall j—this follows since, over the open subset U4, every element of each GL ¨ hi is a linearcombination of the fj . Finally, let U5 Ď BÂλˆB be the preimage of U4 under the morphismψ : BÂλˆB Ñ XˆB, pb1, a, b2q ÞÑ pϕpb1, aq, b2q. Note that ψpb, a, bq “ pϕpb, aq, bq P impπqand that this precisely parameterises impπq, so U6 :“ tpb, aq P BÂλ | pb, a, bq P U5u is opendense in B ˆ Aλ and for pa, bq P U6 we have fjpψpb, a, bqq “ 0 for all j.

By the first paragraph of this proof, the gjl P KpXqGL may be regarded as elements ofKpBq. After shrinking B to a dense open affine subset (and adapting the open subsets aboveaccordingly), we may assume that they are in KrBs. For pb1, a, b2q P U5 Ď B ˆ Aλ ˆ B wehave pϕpb1, aq, b2q P impπq if and only if, for all j,

0 “ fjpϕpb1, aq, b2q “ÿ

l

gjlpb1qfjlpb2q.

Note that this condition is, in fact, independent of a.Now pick a GL-generic K-point a P Aλ such that pb, aq P U6 for some, and hence most,

b P B; such a point exists by Proposition 3.13. Then V :“ U5 X pB ˆ tau ˆ Bq is nonemptyand therefore open dense in Bˆ tau ˆB, and contains pb, a, bq for b in an open dense subset


of B. After replacing B by an open dense affine subset (and updating all open subsetsaccordingly), we may and will assume that pb, a, bq P V for all b P B.

Then, for pb1, a, b2q P V , we have

ϕpb1, aq P impϕpb2, .qq ôÿ

l

gjlpb1qfjlpb2q “ 0 for all j “ 1, . . . , k.

Since, by assumption, ϕpb1, .q and ϕpb2, .q are typical morphisms and since a P Aλ isGL-generic, we have ϕpb1, aq P impϕpb2, .qq if and only if the more symmetric propertyimpϕpb1, .qq “ impϕpb2, .qq holds.

Now the functions gjl P KpXqGL X KrBs define a dominant morphism γ from B to avariety C whose coordinate ring is generated by the gjl. Let D be an irreducible closedsubvariety of B such that γ|D : D Ñ C is dominant and quasi-finite. After further shrinkingB,C,D we may assume that γ|D and γ are surjective. Since pb, a, bq P V for any b P D,V X pD ˆ tau ˆ Bq is nonempty and hence dense in D ˆ tau ˆ B.

For every b1 P D, the set of b2 P B with pb1, a, b2q P V and γpb2q “ b1 is an openneighborhood of b1 in the fiber γ´1pγpb1qq. The union of these open neighborhoods in fibersis a constructible set in B which is easily seen to be dense. Hence it contains an open densesubset of B, which, after shrinking B,C,D appropriately, we may assume to be all of B.Then, for any b2 P B, there exists a b1 P D such that γpb1q “ γpb2q and pb1, a, b2q P V .

We claim that DÂλ Ñ X is already dominant. Indeed, consider a point x P impϕpb2, .qqfor some b2 P B and let b1 P D be such that pb1, a, b2q P V and γpb1q “ γpb2q. Thengjlpb1q “ gjlpb2q holds for all j, l and we find that, for each j,

ÿ

l

gjlpb1qfjlpb2q “ÿ

l

gjlpb2qfjlpb2q “ 0

so that impϕpb1, .qq “ impϕpb2, .qq Q x. Since B was assumed to have minimal dimension, wehave D “ B. We thus see that KpBqKpCq is a finite extension. Since KpCq Ă KpXqGL,the result follows.

Remark 8.11. Rosenlicht’s theorem that inspired the proof above says that when an alge-braic group G acts on an ordinary algebraic variety Y , there is a G-stable open subset of Y inwhich the orbits are separated by rational invariants in KpY qG. The proof above shows thata similar statement holds for orbit closures of certain points in X . Indeed, after shrinking Bas we did, suppose that x, y are K-points in impϕq of type λ (see the next section for typesof points), and that gjlpxq “ gjlpx

1q for all j, l. Then we claim that Ox “ Ox1. Indeed, writex “ ϕpb1, a1q and y “ ϕpb2, a2q, where a1, a2 P Aλ are GL-generic. Then the orbit closures

of x and y equal impϕpb1, .qq and impϕpb2, .qq and these are equal since, in the notation ofthe proof above, γpb1q “ γpb2q.

8.3. Types of points. We now extend the definitions we made for irreducible GL-varietiesto points by taking orbit closures:

Definition 8.12. Let X be an arbitrary GL-variety and let x P X .

‚ The type of x, denoted typepxq, is the type of Ox.‚ The typical dimension of x, denoted tdimpxq, is the typical dimension of Ox.‚ A typical morphism for x is one for Ox.


We note that if X is an irreducible GL-variety, then type, typical dimension, and typicalmorphisms for X coincide with those for the generic point of X . We also note that the typeof a point depends only on its generalized orbit, and thus makes sense for points of Xorb.

Proposition 8.13. Let ϕ : X Ñ Y be a morphism of GL-varieties and let x P X. Thentypepϕpxqq Ă typepxq and tdimpϕpxqq ď tdimpxq.

Proof. Let ψ : B ˆ Aλ Ñ Ox be a typical morphism. The morphism ϕ induces a dominantmorphism ϕ1 : Ox Ñ Oϕpxq. The composition ϕ1 ˝ ψ : B ˆ Aλ Ñ Oϕpxq is dominant, and sothe result follows from Proposition 8.7.

Proposition 8.14. Let λ be a pure tuple and let x P Ar ˆ Aλ. Then either typepxq “ λ

or magptypepxqq ă magpλq, with the former occurring if and only if Ox “ C ˆ Aλ for someclosed subvariety C Ă Ar.

Proof. This follows from Theorem 5.4(d).

8.4. A stratification of the orbit space. Let X be a GL-variety. For a pure tuple λ, letXorbλ be the subset of Xorb consisting of all points x with typepxq Ă λ. Endow this subset

with the subspace topology. We note that if λ Ă µ then Xorbλ Ă Xorb

µ ; also Xorb “ŤλX

orbλ .

We can thus regard tXorbλ uλ as a stratification of the space Xorb.

Proposition 8.15. Suppose that the points of type λ are dense in X. Then each genericpoint of X has type λ.

Proof. Without loss of generality, we assume X is irreducible. Let ϕ : B ˆ Aµ Ñ X be atypical morphism. The natural map MλpXq ˆ Aλ Ñ X contains all points of type λ in itsimage, and it is therefore dominant. It follows that there is some irreducible component Cof MλpXq such that the map ψ : C ˆ Aλ Ñ X is dominant. We thus see that µ Ă λ. ByChevalley’s theorem, there is an open dense GL-subset U of X contained in both impϕq.Since the type λ points are dense in X , there is a point x P U of type λ. Let y P ϕ´1pxq. Thenλ “ typepxq Ă typepyq (Proposition 8.13) and magptypepyqq ď magpµq (Proposition 8.14).Thus, putting ν “ typepyq, we have

µ Ă λ Ă ν, magpνq ď magpµq.

It follows that λ “ ν “ µ. Thus X has type λ, as required.

Proposition 8.16. The space Xorbλ is a noetherian spectral space.

Proof. Let Z be an irreducible closed subset of Xorbλ . For µ Ă λ, let Zµ be the set of points

of Z of type µ. Since Z is the union of the Zµ’s and there are only finitely many choices for

µ, it follows that there is some Zµ that is dense in Z. Let Z be the Zariski closure of Z in

Xorb, which is irreducible. Then Zµ is dense in Z, and thus the generic point z of Z has type

µ by Proposition 8.15. It follows that z P Z X Xorbλ “ Z. Clearly, z is a generic point of Z.

It is also unique: if z1 is a generic point of Z then it is also one for Z, and thus z “ z1 sinceXorb is sober. We have thus shown that Xorb

λ is sober. It is noetherian since it is a subspace

of the noetherian space Xorb. It is therefore also spectral.


8.5. Self-maps of affine spaces. Let λ be a pure tuple. We let Γ`λ be the endomorphism

monoid of Aλ and we let Γλ be the automorphism group of Aλ. We regard both as algebraicvarieties. More precisely, Γ`

λ is the mapping space MapKpAλ,Aλq and Γλ is the closedsubvariety of Γ`

λ ˆ Γ`λ consisting of pairs px, yq such that xy “ 1.

Proposition 8.17. Both Γλ and Γ`λ are irreducible varieties. Moreover, Γλ is an open dense

subvariety of Γ`λ .

Proof. Let µ1, . . . , µn be the distinct partitions appearing in λ, arranged so that #µi ď#µi`1, and let mi be the multiplicity of µi in λ. Let Yi “ Aµi b Kmi , which is simply aproduct of mi copies of µi, and let Xi “ Y1 ˆ ¨ ¨ ¨ ˆ Yi. Thus Xn “ Aλ. We have Γ`

λ “śn

i“1MapKpX, Yiq. Now, any map X Ñ Yi must factor through the projection X Ñ Xi.Any map Xi Ñ Yi decomposes as f ` g, where f is induced by a linear endomorphism ofKmi and g factors through the projection Xi Ñ Xi´1. We thus have

Γ`λ “

nź

i“1

pMmiˆ MappXi´1, Yiqq,

where Md denotes the space of d ˆ d matrices. By Proposition 2.3(c), the mapping spacesappearing above are affine spaces. We thus see that Γ`

λ is an affine space, and, in particular,irreducible.

It is not difficult to see that, under the above identification, we have

Γλ “nź

i“1

pGLmiˆ MappXi´1, Xiqq,

from which it easily follows that Γλ is irreducible, and open and dense in Γ`λ .

Corollary 8.18. Any irreducible component of MλpXq is stable by Γλ.

8.6. The principal component of the mapping space. We now isolate an importantirreducible component of the mapping space MλpXq, in certain cases.

Lemma 8.19. Let X be an irreducible variety of type µ. Let λ be a pure tuple containingµ and let π : Aλ Ñ Aµ be the projection map. Let ϕ : B ˆ Aµ Ñ X be a typical morphismand let ψ : C ˆ Aλ Ñ X be a dominant morphism, with C irreducible. Then there existnon-empty open subsets U Ă B and V Ă C such that, for any algebraically closed field Ωcontaining K, we have:

(a) Given x P UpΩq there exists z P CpΩq and σ P ΓλpΩq such that π˚pϕxq “ ψz ˝ σ.(b) Given z P V pΩq there exists x P BpΩq and σ P ΓλpΩq such that ψz “ π˚pϕxq ˝ σ.

Proof. Applying Proposition 8.4, we obtain a commutative diagram

D ˆ Aµ αîd//

θ

C ˆ Aµ

ψ

B ˆ Aλϕ

// X

with α and θ dominant. Applying Proposition 6.1, after possibly replacing D with a denseopen, we see that θ “ pβ ˆ πq ˝ σ where β : D Ñ B is a dominant morphism and σ is aD-automorphism of D ˆ Aµ. Let U be an open subset of B contained in impβq and V bean open subset of C contained in impαq; these exist by Chevalley’s theorem (for ordinary


varieties). Suppose that x P UpΩq. Let y P DpΩq satisfy βpyq “ x, and let z “ αpyq P CpΩq.We then have π˚pϕxq ˝ σy “ ψz. This proves (a). The proof of (b) is similar.

Proposition 8.20. Let X be an irreducible GL-variety of type µ and let λ be a pure tuplewith µ Ă λ.

(a) There exists a unique irreducible component M of MλpXq for which the natural mapM ˆ Aλ Ñ X is dominant.

(b) Suppose that CÂλ Ñ X is dominant with C irreducible. Then the map C Ñ MλpXqfactors through M , and the induced map Γλ ˆ C Ñ M is dominant.

Proof. Fix a typical morphism ϕ : BÂµ Ñ X , and let π : Aλ Ñ Aµ be the projection map.Composing ϕ with π yields a dominant map BÂλ Ñ X . It follows that the universal mapMλpXq ˆ Aλ Ñ X is dominant, and so there is some component M of MλpXq such thatM ˆ Aλ Ñ X is dominant. This proves the existence statement in (a).

Let α : B Ñ MλpXq be the map b ÞÑ π˚pϕbq, and let M be any irreducible component ofMλpXq such that the map ψ : M ˆ Aλ Ñ X is dominant. We claim that M is the closureof Γλ impαq, which will establish the uniqueness in (a). Let U Ă B and V Ă M be as inLemma 8.19. Let Ω be an algebraically closed field containing K and let Γ “ ΓλpΩq. Thelemma shows that

αpUpΩqq Ă Γ ¨ MpΩq, V pΩq Ă Γ ¨ αpUpΩqq

as subsets of MλpXqpΩq. Since M is Γλ-stable, we see that αpUq Ă M , and thus αpBq Ă M .Since V is dense in M , we see that Γλ ¨ αpBq is dense in M . This verifies the claim, andcompletes the proof of (a).

We now prove (b). Let θ : C ˆ Aλ Ñ X be dominant with C irreducible, and let β : C ÑMλpXq be the induced map. Let U Ă B and V Ă C be as in Lemma 8.19. Then, as above,we find

αpUpΩqq Ă Γ ¨ βpCpΩqq, V pΩq Ă Γ ¨ αpUpΩqq.

We thus see that βpV q Ă M , and thus βpCq Ă M . Since Γ ¨ αpUpΩqq is dense in M , we seethat Γλ ¨ βpV q is also dense in M , which completes the proof.

Definition 8.21. Let X be an irreducible GL-variety and let λ be a pure tuple withtypepXq Ă λ. We define the principal component of MλpXq, denoted Mprin

λ pXq, to bethe irreducible component M of MλpXq from Proposition 8.20.

In fact, in our subsequent paper, we show that MλpXq is irreducible in the situation above(see §1.4).

8.7. The space of types. Let λ be a pure tuple. Recall that Mλ “ MλpXq is the mappingspace MapKpAλ, Xq. The group Γλ acts on this space. We define Xtype

λ to be the set of Γλfixed points of Mλ, equipped with the subspace topology. We note that here we are thinkingof Mλ as a scheme, and most points of Xtype

λ will not be closed points. One should think of

a point of Xtypeλ as corresponding to an irreducible closed Γλ-stable subvariety of Mλ. We

have a natural map Mλ Ñ Xtypeλ taking a point to the generic point of the closure of its

Γλ-orbit, and this realizes Xtypeλ as a quotient space of Mλ; the proof is similar to that of

Proposition 3.6.

Proposition 8.22. The space Xtypeλ is a noetherian spectral space.


Proof. Sobriety follows from the same argument as in the first paragraph of the proof ofProposition 3.9. Since Xtype

λ is a subspace of the noetherian space Mλ, it is also noetherian,and therefore spectral.

Now suppose λ Ă µ. Choose a projection map π : Aµ Ñ Aλ, and let π˚ : Mλ Ñ Mµ be theinduced map. Let x P Xtype

λ , and let Z Ă Mλ be the corresponding Γλ-stable closed subset,i.e., the Zariski closure of txu Ă Mλ. Then the Zariski closure W of Γµπ

˚pZq is a Γµ-stableirreducible closed subset of Mµ, and thus defines a point of Xtype

µ . We have thus defined a

function Xtypeλ Ñ Xtype

µ . This construction is independent of the choice of π, since any twoprojection maps differ by an element of Γµ.

Lemma 8.23. Let Z and W be as above. Then Z “ pπ˚q´1pW q.

Proof. Let Z 1 “ pπ˚q´1pW q; we show Z “ Z 1. The inclusion Z Ă Z 1 is clear. Let i : Aλ Ñ Aµ

be the inclusion, so that π ˝ i “ id. We thus see that Z 1 “ i˚pπ˚pZ 1qq Ă i˚pW q. Now, i˚pW qis the closure of the set i˚pΓµπ

˚pZqq. An element of this set has the form ϕ ˝ π ˝ γ ˝ i, whereϕ : Aλ Ñ X belongs to Z and γ P Γµ. Note that π˝γ ˝ i is a self-map of Γλ. We thus see thati˚pW q is contained in the closure of the set Γ`

λZ. But Z is Γ`λ -stable since it is closed and

Γλ-stable and Γλ is dense in Γ`λ . We thus see that i˚pW q Ă Z, which shows that Z 1 Ă Z.

Proposition 8.24. The map Xtypeλ Ñ Xtype

µ is a homeomorphism onto its image.

Proof. Call the map in question f . Let x, y P Xtypeλ . Let Zx Ă Mλ be the Zariski closure of

txu, and let Wx Ă Mµ be the Zariski closure of Γµπ˚pZxq, so that fpxq is the generic point

of Wx. Analogously define Zy and Wy. Note that y is a specialization of x if and only ifZy Ă Zx, and fpyq is a specialization of fpxq if and only if Wy Ă Wx. Of course, Zy Ă Zximplies Wy Ă Wx. The reverse implication holds by Lemma 8.23. We thus see that y is aspecialization of x if and only if fpyq is a specialization of fpxq. Since the spaces in questionare noetherian spectral spaces, the result follows [Stacks, Tag 09XU].

8.8. The map ρ. Let x be a point of Xtypeλ and let B Ă Mλ be its Zariski closure. We have

a natural map B ˆ Aλ Ñ X . We define ρλpxq P Xorb to be the image of the generic pointof B ˆ Aλ. We note that typepρλpxqq Ă λ by Proposition 8.7, and so ρλpxq P Xorb

λ . Thefollowing is result is one of the main points of type theory:

Theorem 8.25. The function ρλ : Xtypeλ Ñ Xorb

λ is a continuous bijection.

Proof. Let x be a GL-generic K-point of Aλ. Consider the maps

Xtypeλ

// Mλidˆx

//// Mλ ˆ Aλ // X // Xorb.

The first map is the inclusion, the second is as labeled, the third is the evaluation map,and the last is the quotient map. The composition of these maps is ρλ. Since each map iscontinuous, so is ρλ.

We now show that ρλ is injective. Let x P Xtypeλ , let M Ă Mλ be its Zariski closure, and

let Z be the Zariski closure of ρλpxq. We thus have a dominant map ϕ : M ˆ Aλ Ñ Z. Wethus see that M is contained in MλpZq Ă MλpXq. Since M is Γλ stable, it follows fromPropsoition 8.20 that M must be the principal component of MλpZq. Thus x is uniquelydetermined from ρλpxq.

Finally, we show ρλ is surjective. Thus let y P Xorbλ be given. Let W Ă X be the

irreducible closed GL-subvariety with generic point y. Since typepyq Ă λ, we can find a

http://stacks.math.columbia.edu/tag/09XU


dominant morphism ψ : B ˆ Aλ Ñ W with B irreducible. (If typepyq is smaller than λ thenψ will factor through a projection map.) Let C be the closure of the Γλ-orbit of the imageof the natural map B Ñ Mλ. Then the natural map C ˆ Aλ Ñ X also maps dominantly toW . It follows that if x P Xtype

λ is the generic point of C then ρλpxq “ y. This completes theproof.

Proposition 8.26. Let λ Ă µ be pure tuples. Then the restriction of ρµ to Xtypeλ agrees with

ρλ.

Proof. Let x P Xtypeλ and let Z Ă Mλ be its Zariski closure. Let π : A

µ Ñ Aλ be a projectionmap, let W0 Ă Mµ be the set Γµ ¨ π˚pZq, and let W be the Zariski closure of W0. The mapsZÂλ Ñ X andW0 Âµ Ñ X have the same image, since applying π˚ and automorphismsof Aµ does not change the image. It follows that the maps Z ˆ Aλ Ñ X and W ˆ Aµ Ñ X

have the same image closures. Thus ρλpxq “ ρµpxq.

8.9. Putting the pieces together. As we have seen, the spaces Xtypeλ form a directed

system. We define Xtype to be their direct limit. This spaces carries a direct limit topology,and there is a natural bijection

ρ : Xtype Ñ Xorb

that is continuous with respect to the direct limit topology on Xtype. However, ρ is typicallynot a homeomorphism when Xtype is given the direct limit topology; this is due to the factthat Xorb is typically not the direct limit of its subspaces Xorb

λ . There is a refined topology

one can give Xtype for which we expect ρ to be a homeomorphism. We will return to thistopic in the future.

8.10. An example. Let κ “ p2q and consider X “ Aκ. One can think of X as the space ofsymmetric bilinear forms on V. For r P N, let Zr be the locus of forms of rank ď r. Thesesets, together with ∅ and X , account for all of the closed GL-subvarieties. We thus see thatXorb is identified with NY t8u. The topology on Xorb is the order topology: the closed setsare the finite intervals t0, . . . , ru, together with ∅ and Xorb.

Let λ “ rp1qn, p2qs. Then Xorbλ “ t0, . . . , n,8u with the subspace topology. It is not

difficult to see in this case that Xtypeλ Ñ Xorb

λ is a homeomorphism. The space Xtype can beidentified with N Y t8u equipped with the direct limit topology with respect to the Xtype

λ .

Since N X Xtypeλ is closed for all λ, it follows that N is closed in Xtype. However, it is not

closed in Xorb. We thus see that ρ is not a homeomorphism in this case.

9. Examples and applications

9.1. Systems of variables. Let λ be a pure tuple. We write AλpKq for the set of K-pointsof Aλ. We say that x P AλpKq is degenerate if it is not GL-generic. We have the followingcharacterization of these points:

Proposition 9.1. Let x P AλpKq. Then x is degenerate if and only if x has the form ϕpyqwhere ϕ : Aµ Ñ Aλ is a morphism where µ is a pure tuple with magpµq ă magpλq andy P AµpKq.

Proof. Suppose x is degenerate. Then Ox is a proper closed GL-subvariety of Aλ. Letϕ : B ˆ Aµ Ñ Ox be the morphism produced by Theorem 5.4(d). Thus magpµq ă magpλq,the image of ϕ contains a non-empty GL-open subset U of Ox, and every K-point of U lifts


to a K-point of Aµ. Since U contains Ox (Proposition 3.4), it follows that x “ ϕpb, yq forsome b P BpKq and y P AµpKq. Letting ϕb be the restriction of ϕ to tbu ˆ Aµ, we havex “ ϕbpyq, and so x has the stated form.

Now suppose that x “ ϕpyq for some ϕ : Aµ Ñ Aλ with magpµq ă magpλq. Since ϕ is notdominant (by, e.g., Proposition 6.1) it follows that Ox is a proper subset of Aλ, and so x isdegenerate.

Proposition 9.2. Let λ be a non-empty partition. Then the set of degenerate points inAλpKq forms a K-vector subspace of AλpKq.

Proof. The subset of degenerate points clearly contains 0 and is closed under scaling. Wenow show that it is closed under addition. Let x, y P Aλ be degenerate. Applying theprevious proposition, write x “ ϕpx1q where ϕ : Aµ Ñ Aλ is a morphism with magpµq ămagpλq and x1 P AµpKq; similarly, write y “ ψpy1q where ψ : Aν Ñ Aλ is a morphism withmagpνq ă magpλq and y1 P AνpKq. Then x ` y is the image of px1, y1q under the map

AµYν Ñ Aλ, pu, vq ÞÑ ϕpuq ` ψpvq.

Since magpµ Y νq ă magpλq, the previous proposition shows that x ` y is degenerate.

The above proposition is not true for tuples. Indeed, if px, yq is a non-degenerate point ofArλ,µs “ Aλ ˆ Aµ then px, yq “ px, 0q ` p0, yq and both px, 0q and p0, yq are degenerate. Wesay that a set ofK-points ofAλ is collectively non-degenerate if they are linearly independentmodulo the subspace of degenerate points, and collectively degenerate otherwise.

Proposition 9.3. Let λ1, . . . , λn be distinct non-empty partitions, let xi,1, . . . , xi,mpiq be K-

points of Aλi, and let x “ pxi,jq, which we regard as a K-pointśn

i“1pAλiqmpiq. Then x

is non-degenerate if and only if for each i the points xi,1, . . . , xi,mpiq are collectively non-degenerate.

Proof. Suppose that x is degenerate, and write x “ ϕpyq where ϕ : Aµ Ñ Aλ is a morphism

with magpµq ă magpλq and λ “ rλmp1q1 , . . . , λ

mpnqn s. Now, if κ is a partition then any mor-

phism ψ : Aµ Ñ Aκ has the form ψ “ ψ1 ` ψ2, where ψ1 is a linear map defined on factorsof Aµ with µi “ κ and ψ2 depends only on the factors of Aµ with #µi ă #κ. In particular,we see that ψpyq is equal to a linear combination of the yi’s with µi “ κ modulo degenerateelements. Now, since magpµq ă magpλq there must be some λi that appears r ă mpiq timesin µ. We thus see that xi,1, . . . , xi,mpiq is a linear combination of r components of y modulodegenerate elements, and so collectively degenerate.

Now suppose that xi,1, . . . , xi,mpiq is collectively degenerate for some i. Applying an elementof GLmpiq, we can assume that xi,mpiq is itself degenerate. Since the projection xi,mpiq of x toAλi is not GL-generic, it follows that x is not GL-generic.

We now come to the central concept of §9.1.

Definition 9.4. For a non-empty partition λ, a system of λ-variables is a subset Ξλ ofAλpKq

that forms a basis modulo the subspace of degenerate elements. A system of variables is acollection Ξ “ tΞλuλ where Ξλ is a system of λ-variables for each non-empty partition λ.

We fix a system of variables Ξ in what follows. Given a K-point x of Aκ, an expressionof the form x “ ϕpξ1, . . . , ξnq means that ϕ is a morphism Aλ Ñ Aκ with ξi P Ξλi . Thefollowing theorem, which is our main result on systems of variables, essentially says thatevery element of Aκ can be expressed uniquely in terms of Ξ.


Theorem 9.5. Let x be a K-point of Aκ. Then we have x “ ϕpξ1, . . . , ξnq for distinctξ1, . . . , ξn. Fix such an expression with n minimal, and consider a second such expressionx “ ψpξ1

1, . . . , ξ1mq with ξ1

1, . . . , ξ1m distinct. Then, after applying a permutation, we have

ξ1i “ ξi for 1 ď i ď n and ψpy1, . . . , ymq “ ϕpy1, . . . , ynq identically.

Proof. For the first statement, it suffices to treat the case where κ is a single partition κ.Write x “

řr

i“1 ciξi ` y where the ci’s are scalars, the ξi belong to Ξκ, and y is degenerate;we can do this since Ξκ is a basis modulo degenerate elements. Since y is degenerate, wecan write y “ ψpzq for some morphism ψ : Aµ Ñ Aκ with magpµq ă magpκq. By induction,we have an expression for z in terms of ξ’s. Combining all of this, we obtain the requiredexpression for x.

Now, fix an expression x “ ϕpξ1, . . . , ξnq with ξ1, . . . , ξn with n minimal; this impliesthat the ξi are distinct. Consider a second expression x “ ψpξ1

1, . . . , ξ1mq with ξ1

1, . . . , ξ1m

distinct. Apply a permutation so that ξi “ ξ1i for 1 ď i ď r with r maximal; thus

the elements ξ1, . . . , ξn, ξ1r`1, . . . , ξ

1m is distinct. Let ξi P Ξλi and ξ1

i P Ξλ1i, and put λ “

rλ1, . . . , λn, λ1r`1, . . . , λ

1ms. Let ϕ : Aλ Ñ Aκ be the composition of ϕ with the appropriate

projection, and similarly define ψ. Then both ϕ and ψ map pξ1, . . . , ξn, ξ1r`1, . . . , ξ

1mq to x.

Since this point is GL-generic by Proposition 9.3, it follows that ϕ “ ψ. Evaluating bothsides on the point pξ1, . . . , ξr, 0, . . . , 0, ξ

1r`1, . . . , ξ

1mq, we find ϕpξ1, . . . , ξr, 0, . . . , 0q “ x. By the

minimality of n, we conclude that r “ n. It now follows that ϕpy1, . . . , ynq “ ψpy1, . . . , ymqfor all y1, . . . , ym, which completes the proof.

9.2. Big polynomial rings. Let R be the inverse limit of the standard-graded polynomialrings Krx1, . . . , xns in the category of graded rings. Thus R is a graded ring, and a degreed element of R is a formal K-linear combination of degree d monomials in the variablesx1, x2, . . .; in particular, Rd is identified withApdqpKq. The following theorem was establishedin [ESS2] (and previously in [AM]), and used to give two new proofs of Stillman’s conjecture:

Theorem 9.6. The ring R is (isomorphic to) a polynomial ring.

In fact, this result is essentially a special case of the material in the previous section, aswe now show:

Proof of Theorem 9.6. It follows from the Littlewood–Richardson rule that if a symmetricpower Vpdq appears in a tensor product Vλ b Vµ then λ and µ each have a single row,i.e., Vλ and Vµ are themselves symmetric powers; moreover, in this case, Vpdq occurs withmultiplicity one, and so, up to scalaing, the only map Vλ b Vµ Ñ Vpdq is the multiplication

map. It follows that any morphism Aλ Ñ Apdq only depends on the coordinates of Aλ thatare symmetric powers, and is simply given by some polynomial in those coordinates.

Now, fix a system of variables Ξ. By Theorem 9.5 and the above discussion, we see thatevery element of Rd can be expressed as a polynomial in the elements S “

Ťně1 Ξpnq, and

that this expression is unique up to permutations. Thus R is a polynomial ring on thegenerators S.

Now, suppose that K is the field Cpptqq of Laurent series. Let R5 be the subring of Rconsisting of elements f with bounded denominators, i.e., f has coefficients in tńCJtK forsome n. In [ESS2], it was also shown that R5 is also a polynomial ring over K. The followingcomplementary (but much deeper) result was established in [Sn]:

Theorem 9.7. The ring R is a polynomial algebra over R5.


We now show how this more difficult theorem can be deduced from our theory in a fairlysimple manner. We say that a K-point of Aλ is bounded if there is some n such that itscoordinates all belong to tńCJtK. Thus R5

d can be identified with the set of bounded K-points of Apdq. More generally, suppose that x is a K-point of a quasi-affine GL-variety X .Let R “ ΓpX,OXq and let ϕ : R Ñ K be the homomorphism corresponding to x. We saythat x is bounded if for every finite length subrepresentation V of R there is some n such thatϕpV q Ă tńCJtK; in fact, it suffices to verify this for any subrepresentation V that generatesR as an algebra. The key result we need is the following:

Proposition 9.8. Let ϕ : X Ñ Y be a morphism of GL-varieties and let x be a boundedK-point of X belonging to impϕq. Then there is a bounded Kpt1nq-point y of Y , for somen, such that x “ ϕpyq.

Proof. This is clear for elementary morphisms, and thus the result follows from the de-composition theorem. (We note that there is a notion of boundedness for K-points in aGpnq-variety, and a point is Gpnq-bounded if and only if it is Gpmq-bounded for m ě n.This theory will be better developed in our subsequent paper.)

Recall from Example 1.2 the notion of strength. As explained in the introduction to [Sn],Theorem 9.7 follows easily from the following:

Proposition 9.9. Suppose that f is an element of R5 that has finite strength in R. Then fhas finite strength in R5.

Proof. Say f has degree d, and we have f “řr

i“1 gihi where gi, hi P R are homogeneous ofdegrees ă d. Put degpgiq “ ei degphiq “ e1

i and let λ “ rpe1q, pe11q, . . . , perq, pe

1rqs. Con-

sider the map π : Aλ Ñ Apdq given by πpa1, b1, . . . , ar, brq “řr

i“1 aibi. We have f “πpg1, h1, . . . , gr, hrq, and so f is in the image of π. By Proposition 9.8, f is the imageof a bounded Kpt1nq-point for some n ě 1. This means we can write f “

řr

i“1 g1ih

1i where

g1i, h

1i P R5

Kpt1nqare homogeneous of degrees ă d. Write g1

i “řn´1

k“0 g1i,kt

kn with g1i,k P R5, and

similarly express h1i. Multiplying g1

ih1i out and taking the piece that belongs to R5, we see

that f has strength ď rn in R5.

9.3. An extended example. Suppose that K is algebraically closed field and let X ĂArp1q,p1qs be the rank ď 1 locus. Precisely, we identify Arp1q,p1qs with the spectrum of S “Krxi, yisiě1, with each set of variables generating a copy of the standard representation ofGL, and X is the spectrum of R “ Spxiyj ´ xjyiq. We look at several of our definitionsand results in this case.

‚ For each ra : bs P P1pKq we have the line La,b Ă X where bxi “ ayi. This is a closedGL-subset. Besides these, there are only two other closed GL-subsets, namely t0uand X . We thus see that Xorb can be identified with P1pKq plus two additionalpoints. In fact, the generalized orbit of the generic point of X corresponds to thegeneric point of P1

K . Thus Xorb is the topological space underlying the scheme P1K

plus one extra point ˚ corresponding to t0u. The point ˚ is closed, and the closure ofany point contains ˚; this completely describes the topology on Xorb.

‚ Let t “ xiyi P FracpRq; note that this is independent of i. Then KpXqGL “ Kptq.Thus X gives an example of a pure GL-variety with a non-trivial invariant functionfield.


‚ We now consider the shift theorem for X . Identify Sh1pSq with Srξ, ηs, where ξ the“shifted off” x variable, and η is the “shifted off” y variable. Then Sh1pRq is thequotient of Srξ, ηs by the relations

xiyj ´ xjyi, ξyi ´ ηxi

for all i and j. We thus see that in Sh1pRqr1ηs we have xi “ pξηqyi for all i. Wethus find that Sh1pRqr1ηs “ Krξ, η˘1, y1, y2, . . .s. Hence Sh1pXq – B ˆ Ap1q whereB “ A1 ˆpA1zt0uq. From this, we see that the invariant function field of any ShnpXqis a rational function field over K.

‚ Besides 0, every point in X has type rp1qs.‚ A typical morphism for X is given by the map A1 ˆ Ap1q Ñ X corresponding to thering homomorphism R Ñ Krt, y1, y2, . . .s mapping yi to yi and xi to tyi.

‚ Consider the mapping spaceM “ MapKpAp1qK , Xq. Let A be aK-algebra. An A-point

of M corresponds to a GL-algebra homomorphism f : R Ñ Arz1, z2, . . .s. Under sucha homomorphism x1 must map to a Gp1q-invariant element, and thus a scalar multipleof z1; similarly, y1 must also map to a scalar multiple of z1. Say fpx1q “ az1 andfpy1q “ bz1. Then fpxiq “ azi and fpyiq “ bzi, and so f is completely determined.Conversely, for any a and b, we see that these formulas define a GL-algebra homo-morphism, i.e., the relations in R hold. We thus see that M “ SpecpKra, bsq “ A2.

9.4. Examples of invariant function fields. Let λ be the tuple consisting of n`1 copiesof p1q, and let X be the rank ď 1 locus in Aλ. We can think of Xzt0u as the space of rank 1maps V Ñ Kn`1. We thus have a natural map π : Xzt0u Ñ Pn by associating to a rank 1map its image. The map π is clearly GL-equivariant, using the trivial action on the target.Let W be an irreducible subvariety of Pn, and let Z Ă X be the closure of π´1pW q, which isa GL-variety. The map π induces a field homomorphism π˚ : KpW q Ñ KpZqGL, which onecan show is an isomorphism. We thus see that any finitely generated extension of K can berealized as the invariant function field of a GL-variety that is pure over K.

9.5. Morphisms of affine spaces need not have closed image. Suppose that ϕ : Aλ ÑAµ is a morphism of GL-varieties with λ and µ pure. From Examples 1.1 and 1.2, we knowthat ϕ need not have closed image. We now give a simple example that directly exhibits thisphenomenon.

Consider the GL-morphism

ϕ : Arp2q,p2q,p2qs Ñ Ap4q

pf, g, hq ÞÑ fg ´ h2

and the closure X of its image. We show that the image of ϕ is not closed by finding aGL-morphism ψ to X such that ψ ‰ ϕ ˝ γ for every GL-morphism γ.

Note that for all closed points px, y, f, g, hq P Arp1q,p1q,p2q,p2q,p2qs and t P Kzt0u, we have

t´1ϕ`y2 ` tf, x2 ` tg, xy ´ 1

2th

˘“ t´1

`py2 ` tfqpx2 ` tgq ´ pxy ´ 1

2thq2

˘

“ x2f ` y2g ` xyh` tp. . . q P X

Hence the GL-morphism

ψ : Arp1q,p1q,p2q,p2q,p2qs Ñ Ap4q

px, y, f, g, hq ÞÑ x2f ` y2g ` xyh


maps into X . If impψq Ď impϕq, then ψ “ ϕ ˝ γ for some GL-morphism

γ : Arp1q,p1q,p2q,p2q,p2qs Ñ Arp2q,p2q,p2qs

by Propositions 3.13 and 6.4. Such a GL-morphism γ must be of the form

γpx, y, f, g, hq “

¨˝

c11x2 ` c12xy ` c13y

2 ` c14f ` c15g ` c16h

c21x2 ` c22xy ` c23y

2 ` c24f ` c25g ` c26h

c31x2 ` c32xy ` c33y

2 ` c34f ` c35g ` c36h

˛‚

for some constants cij P K. This turns the condition ψ “ ϕ ˝ γ into polynomial equationsin the cij. Now, one can check using a Grobner basis calculation that this system has nosolutions. Hence impψq Ę impϕq. So, in particular, we see that impϕq ‰ X is not closed.

Question 9.10. Is impϕq Y impψq closed?

Question 9.11. Is impψq closed?

9.6. The typical locus need not be open. We consider the set of all GL-morphismsArp1q,p1q,p2q,p2q,p2qs Ñ Ap4q. Such a GL-morphism is typical (as a map to its image closure)precisely when it does not factor via Aλ for any λ Ĺ rp1q, p1q, p2q, p2q, p2qs. Note that for allt P Kzt0u, the GL-morphism

ϕt : Arp1q,p1q,p2q,p2q,p2qs Ñ Ap4q

px, y, f, g, hq ÞÑ t´1`px2 ` tfqpy2 ` tgq ´ pxy ` thq2

˘

factors viaArp2q,p2q,p2qs and is hence not typical. If the set of typicalGL-morphisms were open,then the limit ϕ0 “ limtÑ0 ϕt would also have to be not typical. However the GL-morphism

ϕ0 : Arp1q,p1q,p2q,p2q,p2qs Ñ Ap4q

px, y, f, g, hq ÞÑ x2g ` y2f ´ 2xyh

cannot factor through Arp1q,p1q,p2q,p2qs for dimension reasons and it cannot factor throughArp1q,p2q,p2q,p2qs since the coefficients x2, y2 of g, f in ϕ0 are linearly independent. So ϕ0 istypical. Hence the set of typical GL-morphisms Arp1q,p1q,p2q,p2q,p2qs Ñ Ap4q is not open.

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University of Bern, Switzerland, and MPI for Mathematics in the Sciences, Germany

Email address : [email protected]

URL: http://arthurbik.nl

University of Bern, Switzerland, and Eindhoven University of Technology, The Nether-

lands


URL: https://mathsites.unibe.ch/jdraisma/

Eindhoven University of Technology, The Netherlands


URL: https://www.tue.nl/en/research/researchers/rob-eggermont/

Department of Mathematics, University of Michigan, Ann Arbor, MI


URL: http://www-personal.umich.edu/~asnowden/







http://arxiv.org/abs/1505.04294v1


http://stacks.math.columbia.edu

mailto:[email protected]

http://arthurbik.nl


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https://www.tue.nl/en/research/researchers/rob-eggermont/


http://www-personal.umich.edu/~asnowden/

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