CALYXES AND COROLLAS - College of LSA | U-M LSA U-M ... · monoids by Mlat. Note that, while the monoid Mmay not be commutative in general, joins and meets in Mlat are still commutative

CALYXES AND COROLLAS

E. DEAN YOUNG, H. DERKSEN

Abstract. A calyx (multiplicative lattice) is a complete lattice endowed with the struc-ture of a monoid such that multiplication by an element is a left adjoint functor of completelattices (equivalently, a left adjoint functor which preserves colimits). Calyxes are a gener-alization of the set of ideals of a ring, which form a complete lattice under intersection andsummation; in conjunction with the natural multiplication operation (ideal multiplication),this lattice forms a calyx. One can axiomatize the submodules of a module similarly toarrive at the definition of a corolla (module lattice). We present here a notion of principalelement equivalent to Dilworth’s original definition of principal on a multiplicative lattice,and show that this is equivalent to weak principal. The new concept of principal elementbegins with a notion of morphism in the category of corollas over a given calyx. We char-acterize principal elements in the calyx induced by a Noetherian ring as those ideals in thelattice which are principal ideals (in the ordinary sense) after localizing at any given primeideal. We include here w

Contents

1. Intrinsic Definitions 32. Intrinsic Definitions 32.1. Preliminaries 32.2. Lattices 52.3. Calyxes 62.4. Corollas 113. Universal Constructions in Lat, Cal, and E-Cor 143.1. The Category Lat 143.2. The Category Cal 183.3. The Category E-Cor 203.4. The Category E-Cor 214. Commutative Calyxes and their Corollas 254.1. Maximal, Minimal, Irreducible, Co-Irreducible, and Prime Elements [-1] 254.2. The Isomorphism Theorems [0] 274.3. Duality [1] 274.4. Principal Elements [2] 284.5. Exact Sequences [3] 314.6. Hom and Tensor [10, 11] 324.7. Deciduous, Coniferous, and Cupulate Calyxes [4] 324.8. Modularity 334.9. Perianths 334.10. Finitely Generated Perianths, Finitely Generated Elements 344.11. Finite Corollas, Finite Elements, and Finitely Presented Corollas [5] 344.12. Calyx Maps of Finite Type and Finite Presentation 34

1

2 E. DEAN YOUNG, H. DERKSEN

4.13. Finite Calyx Maps 344.14. Compact Elements and Cupulate Calyxes 344.15. Localization [9] 354.16. Base Change [13] 434.17. The Chinese Remainder Theorem [14] 434.18. The Spectrum of a Calyx [16] 444.19. Completions 454.20. Graded Calyxes and Corollas [55, 57]!!! 454.21. Noetherian Calyxes [30]!!! 464.22. Local Calyxes [17] 474.23. The Nilradical and the Jacobson Radical [18] 494.24. Nakayama’s Lemma [19] 494.25. Zerodivisors, the Corolla Quotient [24] 504.26. Supports and Annihilators [39] 514.27. Valuation Rings [49] 514.28. Primary Decomposition 524.29. Dimension [59] 554.30. Associated Primes [62] 554.31. Completion [95] 554.32. Regular Local Rings [105] 565. Schemes 565.1. Intrinsic Definitions 565.2. Some Examples 605.3. Projective Calical Schemes 605.4. First Properties of Schemes 605.5. Smooth Schemes 615.6. Separated and Proper Morphisms 615.7. Sheaves of Corollas 615.8. Divisors 615.9. Projective Morphisms 615.10. Differentials 615.11. Formal Calical Schemes 616. Cohomology of Calical Sheaves 616.1. 61References 62

In this section we define multiplicative lattices and module lattices from a categoricalpoint of view, providing the necessary insight in establishing a new definition of morphism.This leads to a concept of ‘perennial’ morphism between module lattices, which is a notionmuch closer to an actual map of modules. For instance, take a multiplicative lattice E.Principal elements of an E-module lattice M are in direct correspondence with perennialmaps φ : E →M , where we view the multiplicative lattice E as a module lattice over itself.

In this document, multiplicative lattices are referred to as calyxes and module latticesare referred to as corollas. This is partially to emphasize the distinction between morphisms

CALYXES AND COROLLAS 3

in the category of multiplicative lattices and morphisms in the category of calyxes. Ourterminology is taken from the subject of flower anatomy.

TODOfinitely generated corollas and finitely generated elements. the notation E[a1, ..., an].

finite as in (a1, ..., an). need definition of quotient notation. the free calyx on n generators,free algebra calyx. etc. we write ΩX for the structure sheaf of X. domain calyx open sets Uin X change name to calical schemes change abstract tensor universal property finite moduleis different than a finite element. section on modularity move annihilators from Zerodivisors,the Corolla Quotient to section on associated primes fix graded corollas. remember finitelygenerated. check ”the hom functor” fix primary decomposition localization at a principalelement notation. note that it is possible to fully recover a spectrum from its caylx. compactelements

1. Intrinsic Definitions

Let C be athe category set and the category of sup-lattices-there is a left adjoint functor set to sup-lattices, injective on objects -a set can be

recovered from its sup-latticethe category monoid and the category of calyxes-ab = xy : x in a, y in b sumd : ad ¡= c ¡- c b -¿ ab -xy in c for all x in a and y in b iff x

in b implies ax ¡= c thus every monoid induces a calyx -the functor monoid to calyx is leftadjoint.

the category of group actions and the category of corollasthe category of rings and the category of perianthsthe category of modules and the category of sepals

2. Intrinsic Definitions

2.1. Preliminaries.

Definition 0. A lattice is a poset E for which every two elements x, y ∈ E have a supremumand an infimum. A bounded lattice is a lattice which has two elements 0, 1 such that0 ≤ x ≤ 1 ∀x ∈ E. A complete lattice is a lattice such that any arbitrary collection ofelements has a unique infimum and supremum. Any complete lattice is bounded. Meets andjoins in a lattice are automatically associative, commutative, and idempotent.

Lemma 1. Suppose a poset P has arbitrary suprema. Then P has arbitrary infima, so thatP is a complete lattice.

Proof. Take a subset X ⊆ P and consider the set Y = y ∈ P : y ≤ x ∀x ∈ X of lowerbounds of X. This set has a supremum p ∈ P . Take x ∈ X. y ≤ x ∀y ∈ Y , so p ≤ x. Sincex was chosen arbitrarily, p ∈ Y . But by definition p ≥ y ∀y ∈ Y , so p is an infimum forX.

Lemma 2. Let f : P → Q be a morphism of complete lattices which preserves suprema(colimits). Then f has a right adjoint.


Proof. Define a function g : Q → P where g 7→ supx ∈ P : f(x) ≤ y. If x ∈ P, y ∈ Q aresuch that f(x) ≤ y, then x ≤ supx ∈ P : f(x) ≤ y = g(y). Conversely, if x ≤ g(y), then

f(x) ≤ f(g(y)) = f(supx ∈ P : f(x) ≤ y) = sup f(x ∈ P : f(x) ≤ y) ≤ y

Lemma 3. It follows from the proof above that, for an adjoint pair f a g of completelattice morphisms f : P → Q and g : Q → P , f(x) = lim←−y ∈ Q : x ≤ g(y) andg(y) = lim−→x ∈ P : f(x) ≤ y.

Lemma 4. Let F : C → D be a left adjoint functor with right adjoint G : D → C. LetAX,Y : HomD(F (X), Y ) → HomC(X,G(Y )) be the isomorphisms testifying to the adjointrelationship F a G for each X ∈ C and Y ∈ D. We show that F preserves colimits.

Proof. Let Λ be a small category and Φ : Λ → C a functor such that lim−→Φ exists. Letαλ : Φ(λ)→ lim−→Φ be the canonical maps. We show that F (lim−→Φ), F (αλ) forms a colimitfor F Φ. Take an object P ∈ D with morphisms βλ : F Φ(λ)→ P such that the followingdiagram commutes for each φ : λ→ µ in Λ:

F Φ(λ) F Φ(µ)

P P

βλ

F (φ)

βµ

1P

Since F a G the following diagram commutes:

Φ(λ) Φ(µ)

G(P ) G(P )

A(βλ)

φ

A(βµ)

1G(P )

By the universal property of colimit, there is β : lim←−Φ → G(P ) such that the followingdiagram commutes:

Φ(λ) Φ(µ) lim−→Φ

G(P ) G(P ) G(P )

A(βλ)

φ

A(βµ)

αµ

β

1G(P ) 1G(P )

Since F a G the following diagram commutes:

F Φ(λ) F Φ(µ) F (lim−→Φ)

P P P

βλ

φ

βµ

αµ

A−1(β)

1G(P ) 1P

We can apply this argument in reverse to get uniqueness.

Analogously, right adjoints preserve limits.

Corollary 5. Colimits commute and limits commute.


Proof. Follows since colimits are left adjoint and limits are right adjoint. A proof can befound in ‘A Term of Commutative Algebra’ By Allen Altman and Steven Kleiman. It occursas 6.6.

Definition 6 (Composition of Adjoints). Let f = (f∗, f∗) : C → D be an adjoint pair with

f∗ : C → D, f∗ : D → C. Suppose g = (g∗ : g∗) : D → E is another adjoint pair withg∗ : D → E and g∗ : E → D. We can form the composition g f = (g∗ f∗, f∗ g∗) of the twoadjoints, which is an adjoint in its own right. We check this:

HomE(g∗ f∗(X), Y ) ∼= HomD(f∗(X), g∗(Y )) ∼= HomC(X, f∗(g∗(Y )))

Moreover the identity functor on a category C composes with any adjoint pair f to makef again. In this way, one can replace the functors in Cat with adjoint pairs to obtain acategory. This same process works with the category of posets viewed as a full subcategoryof Cat.

2.2. Lattices. To motivate the definition of a calyx we start with the notion of a latticeinduced by a monoid. We form the corresponding category of complete lattices with mor-phisms adjoint pairs of poset functors. From complete lattices we construct calyxes just asrings are constructed from abelian groups.

Definition 7. Let M be a (not necessarily commutative) monoid and consider the latticeE = (E,+,∩) of its submonoids, where + is the join and ∩ is the meet. The lattice E iscomplete, with M and 0 the bounds. To each monoid M we assign the lattice of its sub-monoids by M lat. Note that, while the monoid M may not be commutative in general, joinsand meets in M lat are still commutative.

lat can be made into a functor in a natural way, which is desirable as it vital to theother definitions to come, but this cannot happen on the ordinary category of lattices andlattice morphisms. To make E functorial, let us examine the properties of extension andcontraction of submonoids under a morphism f : M → N of monoids. Write Ke ∈ E(N)for the image (extension) of K ∈ E(M) and Lc ∈ E(M) for the preimage (contraction) ofL ∈ E(N). Take K,K ′, Kii∈I ∈ E(M) and L,L′, Lii∈I ∈ E(N). Then

K ⊆ Kec Lce ⊆ LK ⊆ K ′ ⇒ Ke ⊆ K ′e L ⊆ L′ ⇒ Lc ⊆ L′c(∑

i∈I Ki

)e=∑

i∈I Kei

⋂i∈I L

ci =

(⋂i∈I Li

)c(⋂i∈I Ki)

e ⊆⋂i∈I K

ei

∑i∈I L

ci ⊆

(∑i∈I Li

)c0e = 0 1c = 1

These properties can be seen to follow from the fact that extension and contraction are apair of adjoint poset functors, where the lattices in question are viewed as poset categories.We denote by Lat the category of complete (bounded) lattices which has as its morphismsadjoint pairs φ = (φ∗, φ

∗), φ∗ a φ∗ with φ∗ : M → N and φ∗ : N → M , written φ : M → N .An unfortunate consequence of this is that the left adjoint φ∗ is written left to right. Notethe distinction between Lat and the ordinary category of lattices and lattice morphisms.In the second, morphisms are not adjoint pairs but maps f : P → Q such that x ≤ y ⇒f(x) ≤ f(y). The composition of an adjoint pair is an adjoint pair and there is the identityadjoint pair on each complete lattice, so Lat indeed forms a category. We write φ∗ for the


left adjoint and φ∗ for the right adjoint of a morphism pair φ : M → N in Lat, with φ∗written left to right and φ∗ right to left.

Example 8. Let E be a complete lattice. The set EndLat(E) of complete lattice morphismsforms a monoid under composition. EndLat(E) also has a canonically induced partial orderwhere φ ≤ ψ when φ∗(x) ≤ ψ∗(x) ∀x ∈ E and φ∗(x) ≥ ψ∗(x) ∀x ∈ E. Under this orderEndLat(E) is bounded and complete. To show this it suffices to show that EndLat(E) hasarbitrary joins, by lemma 1.

Let φii∈I be elements of EndLat(E). Define α∗(x) =∑

i∈I(φi)∗(x) and α∗(x) =⋂i∈I(φi)

∗(x). α∗ and α∗ are poset functors E → E; since colimit is a functor,

x ≤ y ⇒∑i∈I

(φi)∗(x) ≤∑i∈I

(φi)∗(y)

since limit is a functor,

x ≤ y ⇒⋂i∈I

(φi)∗(x) ≤

⋂i∈I

(φi)∗(y)

To show α∗ a α∗ we must show∑

i∈I(φi)∗(x) ≤ y ⇔ x ≤⋂i∈I(φi)

∗(y). Observe that∑i∈I

(φi)∗(x) ≤ y

⇔ (φi)∗(x) ≤ y ∀i ∈ I⇔ x ≤ (φi)

∗(y) ∀i ∈ I

⇔ x ≤⋂i∈I

(φi)∗(y)

A 0 element for F is (φ∗, φ∗) where φ∗(x) = 0 for each x ∈ E and φ∗(x) = 1 for each

x ∈ E. A 1 element for F is (ψ∗, ψ∗) where ψ∗(x) = x for each x ∈ E and ψ∗(x) = x for

each x ∈ E.

2.3. Calyxes.

Remark. Notice that a ring A is an abelian group G with a group homomomorphism φ :G→ EndAb(G) such that φ(1) = 1 and φ(φ(x)(y))(z) = φ(x)(φ(y)(z)) ∀x, y, z ∈ G. We canuse this as motivation in constructing the category of calyxes from the category of completelattices.

Definition 9. A calyx is a complete lattice E along with a complete lattice morphismφ = (φ∗, φ

∗) : E → EndLat(E) such that

ψ(φ∗(a)(b))(c) = ψ(a)(ψ(b)(c))

where ψ = φ∗ or φ∗ (called associativity of φ) and φ(1) = 1. We denote the join in E withsummation notation and the meet with intersection notation. We denote φ∗(a)(b) ∈ E byab for a, b ∈ E (called product) and φ∗(a)(b) by (b : a) (called quotient), to match thenotation for product and ideal quotient. It follows from the axioms that φ : E → EndLat(E)is a morphism of monoids (with the product in E being (a, b) 7→ ab and the product inEndLat(E) being composition of adjoint pairs). A calyx is called commutative if φ∗(a)(b) =φ∗(b)(a) ∀a, b ∈ E.


In light of the following motivating example, the theory of calyxes can be viewed as ageneralization of ring theory:

Example 10. The set of ideals of any commutative ring A forms an example of a completelattice Acal, whose meet is intersection and whose join is ideal sum. We define the left adjointφ : (φ∗, φ

∗) : Acal → End(Acal) where φ∗(a)(b) = ab (the ideal product) and φ∗(a)(b) = (b :a) (the ideal quotient), and this makes E into a calyx.

Proof. Take a ∈ Acal. For each b, c ∈ Acal, b ≤ (c : a) ⇔ ab ≤ c, so φ∗(a) a φ∗(a).

The map φ forms the left adjoint of a lattice morphism pair. We check this as follows: ifa ≤ b and c ∈ Acal then ac ≤ bc and (c : b) ≤ (c : a). Suppose aii∈I and b are elements ofAcal. Then

φ∗

(∑i∈I

ai

)(b) =

(∑i∈I

ai

)b =

∑i∈I

(aib) =(lim−→φ∗(ai)

)(b)

φ∗(ai)(b) =

(b :∑i∈I

ai

)=⋂i∈I

(b : ai) =(lim−→φ∗(ai)

)(b)

So that φ preserves colimits. By lemma 2, φ is indeed left adjoint.

Lemma 11. ab =⋂

b≤(c:a) c and (a : b) =∑

cb≤a c for elements a, b, c of any calyx.

Proof. ab ≤ x ⇔ b ≤ (x : a). So

y = ab

⇔ y ≤ x ⇔ ab ≤ x

⇔ y = infx ∈ E : ab ≤ x = infx ∈ E : b ≤ (x : a)

The other claim follows similarly.

Definition 12. We often write µa∗(b) for ab in a calyx E. We write µ∗a(b) for (b : a). Thisis to emphasize that µa is an element of EndLat(E).

Lemma 13. The following familiar properties of ideals in ring theory hold in a generalfor an arbitrary calyx E with left adjoint structure map φ = (φ∗, φ

∗) : E → End(E). Leta, b, c, aii∈I be elements of E.


Reason Property Dual property

φ(1) = 1 1a = a (a : 1) = aAssociativity of φ (ab)c = a(bc) (a : b) : c) = (a : bc)φ is a functor a ≤ b⇒ ac ≤ bc a ≤ b⇒ (c : a) ≥ (c : b)φ∗(a) and φ∗(a) are functors. b ≤ c⇒ ab ≤ ac b ≤ c⇒ (b : a) ≤ (c : a)φ∗(a) and φ∗(a) are functorsand b ≤ 1

ab ≤ a a ≤ (a : b)

Since φ∗ is left adjoint to φ∗,a ≤ φ∗φ∗(a) and φ∗φ∗(a) ≤a

(a : b)b ≤ a a ≤ (ab : b)

φ(a) is an adjoint pair soφ∗(a) (resp. φ∗(a)) dis-tributes over coproducts(resp. products).

∑i∈I(aai) = a

(∑i∈I ai

) ⋂i∈I(ai : a) =

(⋂i∈I ai : a

)

φ is an adjoint pair so φ∗(resp. φ∗) preserves initial ob-jects (resp. terminal objects)

a0 = 0 (a : 1) = a

Canonical morphism fromuniversal property of product(resp. coproduct)

a(⋂

i∈I ai)≤⋂i∈I aai

⋂i∈I(a : ai) ≤

(a :∑

i∈I ai)

φ is itself left adjoint, and sodistributes over colimits.

(∑i∈I ai

)a =

∑i∈I(aia)

(a :∑

i∈I ai)

=⋂i∈I(a : ai)

φ is left adjoint and so sendsinitial objects to initial ob-jects.

0a = 0 (a : 0) = 1

The following identities follow from the properties listed above:

(a + b)(a ∩ b) ≤ ab a + b ≤ (ab : a ∩ b)(a : a + b) = (a : b)

ab ≤ a ∩ ba + b = 1⇒ a ∩ b = aba ≤ b ⇔ (b : a) = 1

a + c = 1, a + b = 1⇒ a + (b ∩ c) = b + c(a ∩ b : b) = (a : b)

a + b = 1, a + c = 1⇒ (ab) + c = 1

The list of properties tabulated above determines the definition of a calyx, but it is clearlynot minimal. To satisfy those who would like a minimal set of axioms, we show that this isequivalent to a multiplicative lattice.

Lemma 14. Let E be a complete lattice on which there is a binary operation called product,written (a, b) 7→ ab, such that the following hold for each a, aii∈I ∈ E:

(1) Product forms an abelian monoid.(2) a

∑i∈I ai =

∑i∈I aai

(3) a0 = 0

We call E a multiplicative lattice. Obviously each calyx is a multiplicative lattice. We showthat each multiplicative lattice is a calyx.


Proof. For each a ∈ E define µ(a) : E → E to be the adjoint pair whose left adjoint µ(a)∗ ismultiplication by a. Indeed, multiplication by a is left adjoint by the adjoint functor theoremfor posets since it distributes over sums and preserves the initial object. This defines a setmap µ : E → EndLat(E) which again distributes over the initial object and sums. Thus µis itself a left adjoint. Since multiplication is associative, the left adjoint of the maps belowmatch, so that they are equal:

µ(µ(a)∗(b))(c) = µ(a)(µ(b)(c)) ∀a, b, c ∈ E

Also µ(a)∗(b) = µ(b)∗(a), so that µ(a)∗(b) = µ(b)∗(a) for each a, b ∈ E. Thus E is acalyx.

Example 15. Let E = Zcal, the calyx over the integers. Then + and ∩ distribute over eachother, so that E is a distributive lattice. This is not true in general. The best one obtainsin this direction in general is the modular law, which holds for any calyx induced by a ring:

If b ≤ a or c ≤ a then a ∩ (b + c) = a ∩ b + a ∩ c

Definition 16. A morphism of calyxes E and F is a set map φ : E → F which is both amorphism of complete lattices on the underlying complete lattice structure and a morphismof monoids on the underlying monoid structure. We denote the category of calyxes and calyxmorphisms by Cal.

Lemma 17. There is a functor Cal → Mon. For a calyx E define a monoid M withthe same underlying set as E, where multiplication µ : M ×M → M in M is defined by(a, b) 7→ ab. Indeed, M has 1 ∈ E as an identity element and is associative by requirementon E. There is also a forgetful functor Cal→ Lat which forgets all but the lattice structure.

Definition 18. We call a calyx E representable if E ∼= Acal for some ring A. Not everycalyx is representable. For example, take a semiring S over the group D8 of symmetries of asquare. The lattice structure of D8 is nonmodular, and a semiring over D8 exists whose twosided ideal structure is nonmodular. One can construct a calyx from S in the same way aswith rings, forming in particular a lattice with non-modular ideal structure. Wheras everycalyx arising from a ring has modular ideal structure.

One might ask if the representable calyxes are exactly the modular ones.

Lemma 19. The following familiar properties of extension and contraction of ideals hold ingeneral for an arbitrary calyx morphism f : E → F . Let a, a′, a′′, aii∈I be elements of thecalyx E and let b, b′, b′′, bjj∈J be elements of the calyx F .


Reason Property Dual Property

f is a monoid morphism andso sends 1 to 1

f∗(1) = 1

f is a monoid morphism andso distributes over product.

f∗(aa′) = f∗(a)(a′)

Lemma 20 f∗(a)f∗(a′) ≤ f∗(aa′) f∗(b)f∗(b′) ≤ f∗(bb′)

Lemma 20 f∗((a : a′)) ≤ (f∗(a) : f∗(a′)) f∗((b : b′)) ≤ (f∗(b) : f∗(b′))

f∗ (resp. f∗) is left adjoint(resp. right adjoint) and sodistributes over colimits (resplimits)

f∗(∑

i∈I ai)

=∑

i∈I f∗(ai) f∗(⋂

i∈I bi)

=⋂i∈I f

∗(bi)

f∗ (resp. f∗) is left adjoint(resp. right adjoint) and sopreserves initial objects (resp.terminal objects).

f∗(0) = 0 f∗(1) = 1

Canonical morphism from theuniversal property of product(resp. coproduct)

f∗(⋂

i∈I ai)≤⋂i∈I f∗(ai)

∑i∈I f∗(bi) ≤ f∗

(∑i∈I bi

)Unitor and counitor from ad-joint relationship f∗ a f∗

a ≤ f∗(f∗(a)) f∗(f∗(b)) ≤ b

f∗(a) = f∗(f∗(f∗(a))) f∗(f∗(f

∗(b))) = f∗(b)

Lemma 20. Let f : E → F be a morphism of calyxes E and F , and take elements a, a′ ∈ E,and b, b′ ∈ F . Then we have the following:

(1) f∗(a)f∗(a′) ≤ f∗(aa

′)(2) f∗(b)f∗(b′) ≤ f∗(bb′)(3) f∗((a : a′)) ≤ (f∗(a) : f∗(a

′))(4) f∗((b : b′)) ≤ (f∗(b) : f∗(b

′))

Notice that in one case we have equality by requirement: f∗(a)f∗(a′) = f∗(aa

′).

Proof. As stated above, (1) holds automatically. To show (2), note that

f∗(b)f∗(b′) ≤ f∗(bb′) ⇔ f∗(f∗(b)f∗(b′)) ≤ bb′

Andf∗(f

∗(b)f∗(b′)) = f∗(f∗(b))f∗(f

∗(b′))) ≤ bb′

To show (3), note that

f∗((a : a′)) ≤ (f∗(a) : f∗(a′)) ⇔ f∗(a

′)f∗((a : a′)) ≤ f∗(a) ⇔ f∗(a′(a : a′)) ≤ f∗(a)

To show (4), note that

f∗((b : b′)) ≤ (f∗(b) : f∗(b′)) ⇔ f∗(b′)f∗((b : b′)) ≤ f∗(b)

andf∗(b′)f∗((b : b′)) ≤ f∗(b′(b : b′)) ≤ f∗(b)

Lemma 21. Every ring morphism f : A → B induces a calyx morphism f cal : Acal → Bcal

where f cal∗ is extension of ideals under f and f cal∗ is contraction of ideals under f . cal :Rng→ Cal is a functor sending A to Acal and f : A→ B to f cal.


Proof. We already know there is a functor Rng → Lat which comes as a restriction of thefunctor Grp → Lat. Take rings A and B and let f : A → B be a ring-homomorphism.Then f(ab) = f(a)f(b) for ideals a, b ∈ Acal. And since f(1) = 1, the extension of f(A) is B.Thus f cal is a morphism of the monoids induced by ideal product in Acal and Bcal. Clearly(idA)cal = idAcal and (g f)cal = gcal f cal for ring maps f : A→ B and g : B → C.

2.4. Corollas.

Definition 22. For a complete lattice M , the complete lattice E = EndLat(M) is in fact acalyx. To see this, define a map Comp : E → EndLat(E) where φ = (φ∗, φ

∗) is sent to theadjoint pair whose left adjoint is composition by φ. Explicitly, we define Comp(φ)∗ to sendan adjoint pair ψ to the adjoint pair whose left adjoint is φ∗ ψ∗ and whose right adjoint isφ∗ ψ∗. To see that Comp(φ)∗ is left adjoint, take ψii∈I in E. Then

φ∗

((lim−→i∈I

(ψi)∗

)(x)

)= φ∗

(lim−→i∈I

((ψi)∗(x))

)= lim−→

i∈Iφ∗ ((ψi)∗(x)) =

(lim−→i∈I

(φ∗ (ψi)∗)

)(x)

since φ∗ is left adjoint, so that Comp(φ)∗ distributes over colimits and is left adjoint by theadjoint functor theorem for posets, lemma 2. Define Comp(φ) to be this adjoint pair.

Next let φii∈I be a collection of adjoint pairs in E and take ψ ∈ E.

Comp(lim−→i∈I

φi)∗(ψ) = (lim−→i∈I

φi)∗ ψ = lim−→i∈I

(φi∗ ψ) = lim−→i∈I

Comp(φi∗)∗(ψ)

It follows that Comp(lim−→i∈I φi)∗ = lim−→i∈I Comp(φi)∗ so that Comp(lim−→i∈I φi) = lim−→i∈I Comp(φi).

Thus Comp distributes over colimits, so that by lemma 2, Comp is left adjoint. Lastly, wecheck associativity and unity as follows:

Comp(φ ψ)(ρ) = φ ψ ρ = Comp(φ)(Comp(ψ)(ρ))

Comp(1)(φ) = 1 φ = φ

Definition 23. Let E be a calyx. An E-corolla is a complete lattice M equipped with acalyx morphism E → EndLat(M). We often write M for 1 in the lattice M , an abuse ofnotation.

Lemma 24. An A-module M induces an Acal-corolla, called M cor.

Proof. Define Φ : E(A)→ EndLat(E(M)) by taking

Φ(a)∗(x) = ax =

n∑i=1

aiyi : ai ∈ a, yi ∈ x

and

Φ(a)∗(x) = (x : a) = y ∈M : ya ≤ xΦ(a)∗ and Φ(a)∗ are adjoint for each a ∈ E(A). Indeed, ax ≤ y ⇔ x ≤ (y : a). Φ distributesover colimits, so that it is left adjoint by the adjoint functor theorem.

Moreover Φ(ab)∗(x) = abx = Φ(a)∗(Φ(b)∗(x)) and

Φ(ab)∗(x) = (x : ab) = ((x : a) : b) = Φ(a)∗(Φ(b)∗(x))


Remark. A calyx E can be viewed as a corolla over itself. By definition there is a Latmorphism E → EndLat(E). This morphism becomes a morphism of calyxes under theapparent calyx structure of EndLat(E).

Lemma 25. The following properties hold in a general E-corolla. Notice in particular thatthey hold for M cor for an A-module M .

Reason Property Dual Property

φ∗ is a functor. a ≤ b⇒ ax ≤ bx a ≤ b⇒ (x : b) ≤ (x : a)φ∗(a)∗ and φ∗(a)∗ are func-tors.

x ≤ y ⇒ ax ≤ ay x ≤ y ⇒ (x : a) ≤ (y : a).

φ∗(1) = 1. 1x = x (x : 1) = xφ∗ is left adjoint and there-fore preserves initial objects.

0x = 0 (x : 0) = 1

φ∗ is left adjoint and there-fore preserves colimits.

∑i∈I(aix) =

(∑i∈I ai

)x

⋂i∈I(x : ai) =

(x :∑

i∈I ai)

φ∗(a)∗ (resp. φ∗(a)∗ is leftadjoint (resp. right adjoint)and therefore preserves col-imits (resp. limits).

a∑

i∈I xi =∑

i∈I axi(⋂

i∈I xi : a)

=⋂i∈I (xi : a)

Canonical morphism fromuniversal property of colimit(resp. limit)

a⋂i∈I xi ≤

⋂i∈I axi

∑i∈I (xi : a) ≤

(∑i∈I xi : a

)Unitor and counitor fromadjoint relationship φ∗(a)∗ aφ∗(a)∗

x ≤ (ax : a) a(x : a) ≤ x.

φ∗(ab) = φ∗(a) φ∗(b) a(bx) = (ab)x ((x : a) : b) = (x : ab)φ∗(a) ≤ 1 ax ≤ x (x : a) ≥ x.φ∗(a)∗ (resp. φ∗(a)∗) is leftadjoint (resp. right adjoint)and so preserves initial ob-jects (resp. terminal ob-jects).

a0 = 0 (1 : a) = 1

φ∗(a)∗ a φ∗(a)∗ ax ≤ y ⇔ x ≤ (y : a)φ∗(a)∗ is a limit, φ∗(b)∗ is aright adjoint.

a(x : b) = (ax : b)

Definition 26. Let M and N be E-corollas with structure maps φ : E → EndLat(M) andψ : E → EndLat(N). An E-lattice morphism is a morphism f : M → N in Lat such that,for each a ∈ E, the following diagram commutes:

M N

M N

f

φ(a) ψ(a)

f

Lemma 27. Every A-module morphism f : M → N induces an Acal-corolla morphism f cor :M cor → N cor by extension and contraction. The resulting function cor : A-mod→ Acal-Coris a functor.


Proof. Take a ∈ M cor, b ∈ N cor. f(a) ⊆ b ⇔ a ⊆ f−1(b), so extension and contraction ofsubmodules is indeed an adjoint relationship. Also extension and contraction by an identitymap of modules does nothing, and

(f g)cor∗ (a) = f g(a) = f cor∗ gcor∗ (a)

(f g)cor∗(a) = g−1(f−1(a)) = gcor∗ f cor∗(a)

Lemma 28. Let f : M → N be a morphism of E-corollas. Take elements a, aii∈I in E,x, xii∈I in M , and y, yii∈I in N . Then

Reason Property Dual Propertyf∗ (resp. f∗) is left adjoint(resp. right adjoint) and sodistributes over colimits (resplimits)

f∗(∑

i∈I ai)

=∑

i∈I f∗(ai) f∗(⋂

i∈I bi)

=⋂i∈I f

∗(bi)

f∗ (resp. f∗) is left adjoint(resp. right adjoint) and sopreserves initial objects (resp.terminal objects).

f∗(0) = 0 f∗(1) = 1

Canonical morphism from theuniversal property of product(resp. coproduct)

f∗(⋂

i∈I ai)≤⋂i∈I f∗(ai)

∑i∈I f∗(bi) ≤ f∗

(∑i∈I bi

)Unitor and counitor from ad-joint relationship f∗ a f∗

a ≤ f∗(f∗(a)) f∗(f∗(b)) ≤ b

f∗(a) = f∗(f∗(f∗(a))) f∗(f∗(f

∗(b))) = f∗(b)

Lemma 29. Let E be a calyx and M an E-corolla. Note that, for a ∈ E and x, y ∈ M ,ax ≤ y ⇔ x ≤ (y : a). This leads to a characterization of ax.

ax = lim←−ax≤y

y = lim←−x≤(y:a)

y

Likewise,

(x : a) = lim−→y≤(x:a)

y = lim−→ay≤x

y

Lemma 30. Let f : M → N be a morphism of E-corollas. Then

f∗(ax) = af∗(x) af∗(x) ≤ f∗(ax)f∗((x : a)) ≤ (f∗(x) : a) f∗((x : a)) ≤ (f∗(x) : a)

Proof. The top left follows by definition. For the top right,

f∗(ax) = f∗( lim←−ax≤y

y) = lim←−ax≤y

f∗(y) ≤ lim←−f∗(ax)≤y

y = f∗(ax)

The others follow similarly.


3. Universal Constructions in Lat, Cal, and E-Cor

In this section we calculate many of the limits and colimits in Lat, Cal, and E-Cor.

Lat The categry of complete lattices whose morphisms are adjoint pairs.

Cal The category of calyxes and calyx morphisms.

E-Cor The category of E-corollas with E-corolla morphisms.

For the case of corollas we also establish a notion of perennial morphism f : M → N inE-Cor, meaning that f∗ : M → N is injective on f∗(N) and f∗ is injective on f∗(M). If fis not perennial we call it annual. Perennial morphisms more closely resemble morphismsof modules, and indeed every morphism of modules induces a perennial morphism of E-corollas. The category E-Cor is complete with a zero object. However, only the perennialmonomorphisms are normal and only the perennial epimorphisms are conormal, so E-Corlacks the crucial property of being abelian. Restricting the category to perennial morphisms,the category loses existence of products and coproducts.

3.1. The Category Lat.

Lemma 31. Let f : M → N be a morphism in Lat. Then f∗(f∗(f∗(x))) = f∗(x) ∀x ∈M and

f∗(f∗(f∗(x))) = f∗(x) ∀x ∈ N .

Proof. f∗(f∗(x)) ≤ x ∀x ∈ N , so, applying f∗, f∗(f∗(f

∗(x))) ≤ f∗(x). And f∗(f∗(y)) ≥ y ∀y ∈M ,so taking y = f∗(x), f∗(f∗(f

∗(x))) ≥ f∗(x). The other claim follows similarly.

Theorem. Let f : M → N be a morphism in Lat. The following are equivalent:

(1) f∗(f∗(x)) = x ∀x ∈M(2) f∗ is injective.(3) f∗ is surjective.(4) f is a monomorphism.

Proof. We show (1) ⇔ (2), (1) ⇔ (3), (2) ⇔ (4).

(1) ⇒ (2). Suppose f∗(f∗(x)) = x ∀x ∈ M . Then f∗(x) = f∗(y) ⇒ x = f∗(f∗(x)) =f∗(f∗(y)) = y.

(2) ⇒ (1). Suppose f∗ is injective. By 31, f∗(f∗(f∗(x))) = f∗(x) ∀x ∈ M , so f∗(f∗(x)) =

x ∀x ∈M .

(1)⇒ (3). Suppose f∗(f∗(x)) = x ∀x ∈M . It follows immediately that f∗ is surjective.

(3)⇒ (1). Suppose f∗ is surjective. By 31, f∗(f∗(f∗(y))) = f∗(y) ∀y ∈M . Taking x ∈M ,

we can write x = f∗(y) for some x ∈M , so that

f∗(f∗(x)) = f∗(f∗(f∗(y))) = f∗(y) = x

Clearly (2)⇒ (4) using the uniqueness of right adjoints for a given left adjoint. Suppose¬(2). Then take x, y ∈ M such that f∗(x) 6= f∗(y). There are left adjoint morphismsg, h : I = 0, 1 →M where g(1) = x and h(1) = y. f g = f h but g 6= h.

Theorem. Let f : M → N be a morphism in Lat. The following are equivalent:


(1) f∗(f∗(x)) = x ∀x ∈ N

(2) f∗ is injective.(3) f∗ is surjective.(4) f is an epimorphism.

Proof. Follows in the same way as before.

Example 32. Not all monomorphisms in Lat are kernels. For example take the lattice ZLat.There is a morphism f : I → ZLat such that f∗(0) = 0 and f∗(1) = 1. But clearly this is nota kernel.

Lemma 33. Take a complete lattice E ∈ Lat. The set

C = X ⊆ E : X is closed under sums and contains 0

forms a poset category. Note that elements X ∈ C form complete lattices with an intersectionand terminal object possibly distinct from the ones in E (a poset with arbitrary upper boundsand an initial object is complete). Morphisms f : X → Y are lattice morphisms f : X → Ysuch that the following diagram commutes:

X Y

E

f

The category Sub(E) of subobjects of E is categorically equivalent to C.

Proof. Define a functor Φ : Sub(E)→ E where a subobject f : X → E is sent to a = f∗(X).The resulting set a is closed under sums and contains the initial object 0 of the lattice E, sincef∗ is left adjoint. To define an inverse map Ψ : E → Sub(E), send a to the monomophisma→ E whose left adjoint is the inclusion map a→ E.

Lemma 34. Take a complete lattice E ∈ Lat. The set

D = X ⊆ E : X is closed under intersections and contains 1

forms a poset category. Note that elements X ∈ C form complete lattices with a sum andinitial object possibly distinct from the ones in E (a poset with arbitrary lower bounds and aterminal object is complete). Morphisms f : X → Y are morphisms f : X → Y such that thefollowing diagram commutes:

E

X Yf

The category Quot(E) of subobjects of E is categorically equivalent to D.

Proof. Similar to the previous proof.

Lemma 35. There is a unique element 0 in Lat which has one element, which forms a zeroobject for Lat.

Definition 36. Let M be a complete lattice. We define (x) = y ∈ M : y ≤ x and[x] = y ∈ M : y ≥ x. There are canonical adjunctions i = (i∗, i

∗) : (x) → M where


i∗(y) = y and i∗(y) = y ∩ x, and π : M → [x] where π∗(y) = y + x and π∗(y) = y. To checkthat i∗ a i∗, take y ∈ (x) and z ∈M . Then

y ≤ i∗(z) ⇔ y ≤ z ∩ x ⇔ y ≤ z ⇔ i∗(y) ≤ z

To check that π∗ a π∗, take y ∈M and z ∈ [x]. Then

y ≤ π∗(z) ⇔ y ≤ z ⇔ y + x ≤ z ⇔ π∗(y) ≤ z

Lemma 37. Lat has kernels.

Proof. Let f : M → N be a morphism in Lat. Let g∗ : (f∗(0)) → M be the embedding andlet g∗ : M → (f∗(0)) send x to x ∩ f∗(0). These form an adjoint pair: take x ≤ f∗(0) andy ∈M . Then, since x ≤ f∗,

x ≤ g∗(y) ⇔ x ≤ y ∩ f∗(0) ⇔ x ≤ y

g = (g∗, g∗) is a kernel for f in Lat. To show this, take a complete lattice P and a morphism

h = (h∗, h∗) : P →M such that fh factors through the 0 object. Then f∗(h∗(x)) = 0 ∀x ∈ P ,

so that h∗(x) ≤ f∗(f∗(h∗(x))) = f∗(0). Thus h∗ factors through (f∗(0)) by a morphismk∗ : P → (f∗(0)). Define a lattice morphism k∗ : (f∗(0))→ P where k∗(x) = h∗(x). k∗ and k∗

form an adjoint pair: take x ∈ P and y ≤ f∗(0). Then

x ≤ k∗(y) ⇔ x ≤ h∗(y) ⇔ h∗(x) ≤ y ⇔ k∗(x) ≤ y

Thus g = ker(f) in Lat.

P

(f∗(0)) M N

hk

g f

Lemma 38. Lat has cokernels.

Proof. Let f : M → N be a morphism in Lat. Let g∗ : N → [f∗(1)] be the quotient mapsending x to x+ f∗(1) and let g∗ : [f∗(1)]→ N be the restriction sending x to x. These forman adjoint pair: take x ∈ N and y ≥ f∗(1). Then, since y ≥ f∗(1),

g∗(x) ≤ y ⇔ x+ g∗(0) ≤ y ⇔ x ≤ y

g = (g∗, g∗) is a cokernel for f in Lat. To show this, take a complete lattice P and a morphism

h = (h∗, h∗) : N → P such that h f factors through the 0 object. Then h∗(f∗(0)) = 0 ∀x ∈

P , so that h∗(x) = f∗(f∗(h∗(x))) ≤ f∗(1). Thus h∗ factors through [f∗(1)] by a morphism

k∗ : [f∗(1)] → P . Define a lattice morphism k∗ : P → [f∗(1)] where k∗(x) = h∗(x) ≤ f∗(1). k∗and k∗ form an adjoint pair: take x ≥ f∗(1) and y ∈ P . Then

x ≤ f∗(y) ⇔ x ≤ h∗(y) ⇔ h∗(x) ≤ y ⇔ k∗(x) ≤ y

Thus g = cok(f) in Lat.

P

M N [f∗(1)]f

h

g

k


Lemma 39. The normal monomorphisms f : M → N in Lat are all equivalent (as subob-jects) to (x)→ N for some x ∈ N .

Lemma 40. The conormal morphisms f : M → N in Lat are all equivalent (as quotientobjects) to M → [x] for some x ∈M .

Definition 41. Let Mii∈I be complete lattices. We form the direct sum lattice ⊕i∈IMi

from the cartesian product P = (xi)i∈I : xi ∈Mi where (xi)i∈I ≤ (yi)i∈I when xi ≤ yi ∀i ∈I. Take X ⊆ P . It follows that

∩x∈Xx =(∩x=(xj)j∈I∈Xxi

)i∈I∑

x∈X

x =

∑x=(xj)j∈I∈X

xi

i∈I

So that ⊕i∈IMi is complete. There are canonical morphisms ιi : Mi → ⊕i∈IMi. Define ιi asfollows: set ιi∗(x) = (xj)j∈I where xi = x and xj = 0 for j 6= i. Set ιi

∗((xi)i∈I) = xi clearlyιi∗ a ιi∗.

Lemma 42. Let Mii∈I be complete lattices. ⊕i∈IMi forms a coproduct of the lattices Mi.

Proof. Let ιi : Mi → qi∈IMi be the canonical maps. Take a complete lattice P and mor-phisms fi : Mi : P . To make the following diagram commute we must define f : qi∈IMi → Pby setting f∗((xi)i∈I) =

∑i∈I fi(xi).

Mi

qi∈IMi P

fiιi

f

f∗ is then left adjoint by the adjoint functor theorem for posets, as it distributes over colimits.

Lemma 43. Let Mii∈I be complete lattices. ⊕i∈IMi forms a product of the lattices Mi.

Proof. Similar to the case of coproducts.

Definition 44. Let M and N be complete lattices. We form the lattice HomLat(M,N)from the set of adjoint pairs

A = f : f = (f∗, f∗) : M → N, f∗ a f∗, f∗ : M → N, f∗ : N →M

where f ≤ g when f∗(x) ≤ g∗(x) ∀x ∈ M and f∗(x) ≥ g∗(x) ∀x ∈ N . To show thatHomLat(M,N) is complete it suffices to show that HomLat(M,N) has arbitrary joins, bythe adjoint functor theorem for posets.

Let φii∈I be elements of HomLat(M,N). Define α∗(x) =∑


∗(x). α∗ and α∗ are poset functors M → N ; since colimit is a functor,

x ≤ y ⇒∑i∈I

(φi)∗(x) ≤∑i∈I

(φi)∗(y)



x ≤ y ⇒⋂i∈I

(φi)∗(x) ≤

⋂i∈I

(φi)∗(y)

To show α∗ a α∗ we must show∑

i∈I(φi)∗(x) ≤ y ⇔ x ≤⋂i∈I(φi)


(φi)∗(x) ≤ y

⇔ (φi)∗(x) ≤ y ∀i ∈ I⇔ x ≤ (φi)

∗(y) ∀i ∈ I

⇔ x ≤⋂i∈I

(φi)∗(y)

A 0 element for HomLat(M,N) is (φ∗, φ∗) where φ∗(x) = 0 for each x ∈M and φ∗(x) = 1

for each x ∈ N .

Definition 45. Let Mini=1 be lattices. Let f∗ :

qni=1Mi → P be a set map from the

cartesian product

qni=1Mi to a lattice P . We say f∗ is multilinear if, for each 1 ≤ i ≤ n,

and for each choice of elements X = (x1, ..., xi−1, xi, xi+1, ..., xn) with xi omitted, the mapfX∗ : Mi → P sending xi to f∗(x1, ..., xn) is left adjoint.

Definition 46. Take lattices M and N , which we can view as I-corollas. We form the tensorproduct of lattices M ⊗I N as follows: Take M ⊗I N = (⊕e∈M×NI) / ∼, where M × N isthe cartesian product and ∼ is the intersection of all equivalence relations such that(∑

i∈I

xi, y

)∼∑i∈I

(xi, y) and (0, y) ∼ 0(x,∑i∈I

yi

)∼∑i∈I

(x, yi) and (x, 0) ∼ 0

xi ∼ yi∀i ∈ I ⇒∑i∈I

xi ∼∑i∈I

yi,⋂i∈I

xi ∼⋂i∈I

yi

Notice there is a canonical map M ×N → M ⊗I N where (m,n) is sent to the equivalenceclass generated by (m,n).

Lemma 47 (Universal Property of Tensor Product). Let M and N be elements of Lat. Letφ : M × N → M ⊗I N be the canonical map. For each multilinear map f∗ : M × N → Pthere is a unique lattice map g : M ⊗I N → P such that g φ = f .

Theorem. Lat forms a monoidal category under tensor product.

Definition 48. Define for each set S the lattice Slat whose underlying set is the power setP(S) and such that U ≤ V in P(S) when U ⊆ V . This makes (−)lat into a functor fromSet to Lat which has as a right adjoint the forgetful functor Lat→ Set.

3.2. The Category Cal.

Definition 49. The unique calyx with a single element is called the 0 calyx. It is a terminalobject in Cal.


Definition 50. We have the calyx I = 0, 1 = Fcal for any field F. I is an initial object inthe category Cal.

Lemma 51. For an E-corolla M and an element x ∈ M there is a unique E-corolla mor-phism fx : E →M such that fx∗(1) = x.

Proof. We must have fx∗(a) = afx∗(1) = ax. Defining fx∗ in this way, we have

fx∗

(∑i∈I

ai

)=

(∑i∈I

ai

)x =

∑i∈I

(aix) =∑i∈I

fx∗

(∑i∈I

ai

)and fx∗(0) = 0, so that fx∗ is indeed left adjoint. Clearly fx∗(ab) = afx(b).

Remark. For an E-corolla M and an element x ∈M , there is not always a perennial corollamorphism f : E → M such that f(1) = x. We call such elements principal elements ofM . For a ring A and an A module M , the regular morphisms from Acal to M cor resembleelements of M up to units. Take a vector space V over a field F. The nonzero perennialcorolla morphisms from I = Fcal to V cor correspond to elements of projective space over V .

Definition 52. For a morphism f : E → F of calyxes we define ker(f) = (f∗(0)). It is asubcorolla, a notion to be defined later. Define im(f) = f∗(a) : a ∈ E and coim(f) =f∗(a) : a ∈ F. im(f) is a subcalyx of F and coim(f) is a subcalyx of E, a notion to bedefined later.

Example 53. Let E = Zcal and F = Qcal = I. The embedding Z→ Q induces a morphismf : E → F of calyxes where f∗(a) = 1 for a 6= 0, f∗(0) = 0, f∗(0) = 0, f∗(1) = 1. ker(f) = 0.im(f) = I, coim(f) = 0, 1. Even though ker(f) = 0, f∗ is not injective, and the obstructionto this is the difference between E and coim(f).

Lemma 54. Let f : E → F be a calyx morphism. f is a monomorphism in Cal if and onlyif it is a monomorphism in Lat.

Proof. If f is a monomorphism in Lat then a priori it is a monomorphism in Cal. Con-versely, by lemma 3.1 it suffices to show that if f∗ : E → F is not injective then f is nota monomorphism in Cal. So take a, b in E such that f∗(a) = f∗(b). Then by 51 there aremorphisms g : E → E and h : E → E such that g(1) = a and h(1) = b. Then f g = f hbut g 6= h.

Lemma 55. An epimorphism in Lat is an epimorphism in Cal

Proof. Follows from 3.1

Definition 56. Let E be a calyx. A subcalyx of E is a set F ⊆ E closed under sums andproducts and contianing 0. For each x ∈ E, (x) = y ∈ E : y ≤ x forms a subcalyx, calledthe principal subcalyx with respect to x. It follows that (x), or any subcalyx of E, forms acalyx in its own right, as an adjoint pair is determined by a left adjoint. The quotient andintersection in a subcalyx of E is distinct from the quotient and intersection in E.

Lemma 57. Take a calyx E. The category Sub(E) of subobjects of E is equivalent to thecategory C of subcalyxes X of E whose morphisms f : X → Y are calyx epimorphismsf : X → Y making the following diagram commute:


X Y

E

f

Proof. Define a functor Φ : Sub(E)→ C where a subobject f : X → E is sent to im(f). Foran inverse functor define Ψ : C → Sub(E) where a subcalyx X ⊆ E is sent to the morphismf : X → E whose left adjoint is the inclusion poset functor X → E.

Definition 58. Let E be a calyx. A quotient calyx of E is a set F ⊆ E closed underintersections and ideal quotients, and containing 1. For each x ∈ E, [x] = y ∈ E : y ≥ xforms a quotient calyx, called the principal quotient calyx with respect to x. It follows that[x], or any subcalyx of E, forms a calyx in its own right, as an adjoint pair is determined bya right adjoint. The product and sum in a quotient calyx of E is distinct from the productand sum in E.

Lemma 59. Take a calyx E. The category Quot(E) of quotient objects of E is equivalent tothe category D of quotient calyxes of E whose morphisms f : X → Y are calyx epimorphismsf : X → Y making the following diagram commute:

E

X Yf

Proof. Define a functor Φ : Quot(E) → D where a quotient object f : E → X is sent tocoim(f). For an inverse functor define Ψ : D → Quot(E) where a quotient object X ofE is sent to the morphism f : E → X whose right adjoint is the inclusion poset functorX → E.

Theorem. Cal forms a complete category.

Definition 60 (Tensor product of calyxes).

Definition 61. There is a functor Φ : Lat→ Cal which sends a complete lattice E to thecalyx ⊕∞i=0E

⊗n where

(a1 ⊗ · · · ⊗ an)(an+1 ⊗ · · · ⊗ an+m) = a1 ⊗ · · · ⊗ an+m

3.3. The Category E-Cor.

3.3.1. Morphisms in E-Cor.

Definition 62. Let f : M → N be a (possibly annual) morphism of E-corollas. We definethe following sets, called the kernel, cokernel, image, and coimage respectively:

ker(f) = (f∗(0))

cok(f) = [f∗(1)]

im(f) = (f∗(1))

coim(f) = [f∗(0)]

We will later see that ker(f) and im(f) are subcorollas, while cok(f) and coim(f) are quotientcorollas.


Definition 63. Let f : M → N be a morphism of E-corollas. The following are equivalent:

(1) im(f) = (f∗(1))(2) f∗ is injective on (f∗(1))

If these hold we say f∗ is perennial.

Definition 64. Let f : M → N be a morphism of E-corollas. The following are equivalent:

(1) coim(f) = [f∗(0)](2) f∗ is injective on [f∗(0)]

If these hold we say f∗ is perennial.

Definition 65. If f∗ and f∗ are perennial we say f is perennial.

Lemma 66. An Acal-corolla morphism induced by an A-module morphism is perennial.

Proof.

Lemma 67. Let f : M → N be a perennial E-corolla morphism. Then

ker(f) = x ∈M : f∗(x) = 0cok(f) = y ∈ N : f∗(y) = 1im(f) = f∗(x) : x ∈Mcoim(f) = f∗(y) : y ∈ N

Proof.

3.4. The Category E-Cor.

Lemma 68. Let f : M → N be a perennial morphism of E-corollas. ker(f) is the obstructionto the injectivity of f∗ (equivalently, the surjectivity of f∗) and cok(f) is the obstruction tothe surjectivity of f∗ (equivalently, the injectivity of f∗). More precisely, the following areequivalent:

(1) f∗ f∗ = idE(2) f is a monomorphism.(3) f∗ is injective.(4) f∗ is surjective.(5) ker(f) = 0(6) coim(f) = 1

Proof. Obvious.

Lemma 69. Analogously, the following are equivalent for a perennial morphism f : M → Nof E-corollas: and the following are equivalent:

(1) f∗ f∗ = idE(2) f is an epimorphism.(3) f∗ is surjective.(4) f∗ is injective.(5) im(f) = 1(6) cok(f) = 0

Proof. Obvious.


Definition 70. Let M be an E-corolla. For an element x ∈ M let (x) = y ∈ M : y ≤ xand let [x] = y ∈M : y ≥ x. We show that (x) and [x] are E-corollas in their own right.

Lemma 71. Let M be an E-corolla with element x ∈ M . (x) can be made into an E-corolla in its own right. It is already a complete lattice. Define φ : E → EndLat((x)) whereφ(a)∗(y) = ay, where ay is multiplication in the corolla M . This determines an adjoint mapφ(a)∗ : (x)→ (x) which is possibly distinct from the quotient operation in M . We have

φ(a)∗(y) = supz∈(x),z≤φ(a)∗(y)

z = supz∈(x),az≤y

z

Lemma 72. Let M be an E-corolla with element x ∈ M . The quotient (y : a) in (x) isequal to (y : a) ∩ x where · is multiplication in M .

Proof. By the uniqueness of adjoints it suffices to show that and y 7→ ya is left adjoint toy 7→ (y : a) ∩ x in [x].

Lemma 73. Let M be an E-corolla with element x ∈ M . [x] can be made into an E-corolla in its own right. It is already a complete lattice. Define φ : E → EndLat([x]) whereφ(a)∗(y) = (y : a), where (y : a) is the quotient operation in M . This determines an adjointmap φ(a)∗ : (x)→ (x) which is possibly distinct from the quotient operation in M . We have

φ(a)∗(y) = infz∈[x],φ(a)∗(y)≤z

z = infz∈[x],y≤(z:a)

z

Lemma 74. Let M be an E-corolla with element x ∈ M . The multiplication ay in [x] isequal to a · y + x where · is multiplication in M

Proof. By the uniqueness of adjoints it suffices to show that y 7→ a · y + x is left adjoint toy 7→ (y : a) in [x].

Lemma 75. Let M be an E-corolla and take x ∈ M . There is a canonical E-corollamorphism (x)→M and a canonical E-corolla morphism M → [x].

Proof. Take the canonical map i : (x) → M in Lat and π : M → [x] in Lat and note thati(ay) = ay = ai(y) and

π(ay) = ay + x = a(y + x) + x = ay + ax+ x = a(π(y))

Definition 76. We have the zero corolla 0, the unique E-corolla with one element.

Proof. Take an E-corolla M . It suffices to note that the unique lattice morphisms 0 → Mand M → 0 are in fact corolla moprhisms.

Lemma 77. Let f : M → N be an E-corolla morphism. Then the canonical map g :ker(f)→M is a kernel for f in the categorical sense.

Lemma 78. Let f : M → N be an E-corolla morphism. Then the canonical map g : N →cok(f) is a cokernel for f in the categorical sense.

Lemma 79. Let f : M → N be an E-corolla morphism. Then the canonical map g : M →im(f) is an image for f in the categorical sense.

Lemma 80. Let f : M → N be an E-corolla morphism. Then the canonical map g :coim(f)→ N is a coimage for f in the categorical sense.


Lemma 81. Let f : M → N be a perennial morphism of E-corollas. f is normal if and onlyif f is a monomorphism.

Lemma 82. Let f : M → N be a perennial morphism of E-corollas. f is conormal if andonly if f is an epimorphism.

Definition 83. Let Mii∈I be E-corollas. We form the corolla ⊕i∈IMi from the cartesianproduct P = (xi)i∈I : xi ∈ Mi where (xi)i∈I ≤ (yi)i∈I when xi ≤ yi ∀i ∈ I. Take X ⊆ P .It follows that

∩x∈Xx =(∩x=(xj)j∈I∈Xxi

)i∈I∑

x∈X

x =

∑x=(xj)j∈I∈X

xi

i∈I

Take x = (xi)i∈I ∈ P . We defineax = (axi)i∈I

(x : a) = ((xi : a))i∈IIt is routine to check that this satisfies the requirements of an E-corolla. There are canonicalmonomorphisms ιi : Mi → ⊕i∈IMi where xi maps to the element (xi)i∈I where xj = 0 forj 6= i. Likewise, there are canonical morphisms πi : ⊕i∈IMi →Mi where (xi)i∈I maps to xi.

Lemma 84. Let Mii∈I be E-corollas. ⊕i∈IMi forms a coproduct for Mii∈IProof. Let P be an E-corolla and take E-corolla morphisms fi : Mi → P . We must definef : ⊕i∈IMi → P such that f ιi = fi. Take an element (xi)i∈I in ⊕i∈IMi and let Xi be theelement whose ith entry is xi and which is 0 elsewhere. Then we must have

f∗((xi)i∈I) = f∗

(∑i∈I

Xi

)=∑i∈I

f∗(Xi) =∑i∈I

f∗(ιi∗(xi)) =∑i∈I

fi∗(xi)

Defining f∗ in this way produces a poset functor which distributes over coproducts andpreserves the initial object 0, so that it is left adjoint. Moreover f∗(a(xi)i∈I) = af∗((xi)i∈I),so that the lattice morphism f induced by f∗ is an E-corolla morphism, as claimed.

Lemma 85. Let Mii∈I be E-corollas. ⊕i∈IMi forms a product for Mii∈IProof. Let P be an E-corolla and take E-corolla morphisms fi : P → Mi. We must definef : P → ⊕i∈IMi such that πi f = fi. Take an element (xi)i∈I in ⊕i∈IMi and let Xi be theelement whose ith entry is xi and which is 1 elsewhere. Then we must have

f∗((xi)i∈I) = f∗

(⋂i∈I

Xi

)=⋂i∈I

f∗(Xi) =∑i∈I

f∗(πi∗(xi)) =

⋂i∈I

fi∗(xi)

Defining f∗ in this way produces a poset functor which distributes over products and preservesthe terminal object 1, so that it is right adjoint. Moreover

f∗(ax) =∑i∈I

fi∗(ax) =∑i∈I

afii(x) = a

(∑i∈I

fi∗(x)

)= af∗(x)

so that the lattice morphism f induced by f∗ is an E-corolla morphism, as claimed.

Theorem. E-Cor forms a complete category.


Definition 86. Let M and N be E-corollas. We form the corolla HomE(M,N) as theset of E-corolla morphisms f : M → N where f ≤ g when f∗(x) ≤ g∗(x) ∀x ∈ M andf∗(x) ≥ g∗(x) ∀x ∈ N . To show that HomE(M,N) is complete it suffices to show thatHomE(M,N) has arbitrary joins, by the adjoint functor theorem for posets.

Let φii∈I be elements of HomE(M,N). Define α∗(x) =∑


∗(x). α∗ and α∗ are poset functors M → N ; since colimit is a functor,

x ≤ y ⇒∑i∈I

(φi)∗(x) ≤∑i∈I

(φi)∗(y)


x ≤ y ⇒⋂i∈I

(φi)∗(x) ≤

⋂i∈I

(φi)∗(y)

Moreover, ∑i∈I

(φi)∗(ax) =∑i∈I

a(φi)∗(x) = a∑i∈I

(φi)∗(x)

and ⋂i∈I

(φi)∗((x : a)) =

⋂i∈I

((φi)∗(x) : a) =

(⋂i∈I

(φi)∗(x) : a

)To show α∗ a α∗ we must show

∑i∈I(φi)∗(x) ≤ y ⇔ x ≤

⋂i∈I(φi)


(φi)∗(x) ≤ y

⇔ (φi)∗(x) ≤ y ∀i ∈ I⇔ x ≤ (φi)

∗(y) ∀i ∈ I

⇔ x ≤⋂i∈I

(φi)∗(y)

A 0 element for HomE(M,N) is (φ∗, φ∗) where φ∗(x) = 0 for each x ∈M and φ∗(x) = 1

for each x ∈ N .

Lemma 87. For a commutative calyx E, HomE(E,M) ∼= M as E-corollas.

Proof. Define a lattice morphism Φ∗ : M → HomE(E,M) where x ∈ M is sent to the mor-phism fx ∈ HomE(E,M) where a 7→ ax. Then Φ∗(

∑i∈I xi) =

∑i∈I Φ∗(xi) and Φ∗(ab) =

aΦ∗(b).

Φ is surjective and injective, so Φ = (Φ∗,Φ∗) is an isomorphism, where Φ∗ is the right

adjoint of Φ∗.

Definition 88. Let Mini=1 be E-corollas. Let f∗ :

qni=1Mi → P be a set map from the

cartesian product

qni=1Mi to a lattice P . We say f∗ is multilinear if, for each 1 ≤ i ≤ n,

and for each choice of elements X = (x1, ..., xi−1, xi, xi+1, ..., xn) with xi omitted, the mapfX∗ : Mi → P sending xi to f∗(x1, ..., xn) is a left adjoint morphism of E-corollas.


Definition 89. Take E-corollas M and N . We form the tensor product of corollas M ⊗ENas follows: Take M ⊗E N = (⊕e∈M×NE) / ∼, where M ×N is the cartesian product and ∼is the intersection of all equivalence relations such that(∑

i∈I

xi, y

)∼∑i∈I

(xi, y), (ax, y) ∼ a(x, y), and (0, y) ∼ 0(x,∑i∈I

yi

)∼∑i∈I

(x, yi), (x, ay) ∼ a(x, y), and (x, 0) ∼ 0

xi ∼ yi∀i ∈ I ⇒∑i∈I

xi ∼∑i∈I

yi,⋂i∈I

xi ∼⋂i∈I

yi

x ∼ y ⇒ ax ∼ ay

Notice there is a canonical multilinear map M ×N → M ⊗I N where (m,n) is sent to theequivalence class generated by (m,n).

Lemma 90 (Universal Property of Tensor Product). Let M and N be elements of E-Cor.Let φ : M × N → M ⊗I N be the canonical multilinear map. For each multilinear mapf∗ : M × N → P there is a unique morphism of E-corollas g : M ⊗I N → P such thatg φ = f∗.

Lemma 91. The functor M →M ⊗E N is left adjoint to the functor M → HomE(M,N).

Theorem. E-Cor forms a monoidal category under tensor product.

Remark. Lat is categorically equivalent to I-Cor. As such, Lat can be viewed as a monoidalcategory.

4. Commutative Calyxes and their Corollas

Definition 92. A calyx E is called commutative if ab = ba ∀a, b ∈ E. In this section, allcalyxes are assumed to be commutative.

4.1. Maximal, Minimal, Irreducible, Co-Irreducible, and Prime Elements [-1].

Definition 93. Let E be a calyx with element p ∈ E. we make the following two symmetricaldefinitions:

(1) p is called irreducible if⋂i∈I ai ⊆ p⇒ ai ⊆ p for some i ∈ I.

(2) p is called co-irreducible if p ⊆∑

i∈I ai ⇒ p ⊆ ai for some i ∈ I.

Notice that 0 is irreducible when⋂i∈I ai = 0 ⇒ ai = 0 for some i ∈ I and that 1 is co-

irreducible when∑

i∈I = 1⇒ ai = 1 for some i ∈ I. Notice that p is irreducible in E if andonly if 0 (the smallest element in [x], i.e. x) is irreducible in [x] (the quotient calyx) and p isco-irreducible if and only if 1 (the largest element in (x), i.e. x) is co-irreducible in (x) (thesubcalyx).

Example 94. In the calyx I = 0, 1, 0 is irreducible and co-irreducible and 1 is irreducibleand co-irreducible.

Definition 95. Let E be a calyx with element x ∈ E. We make the following two symmet-rical definitions:

(1) x is called maximal if x < 1 and x < y ⇒ y = 1. Equivalently, if [x] ∼= I.


(2) x is called minimal if x > 0 and x > y ⇒ y = 0. Equivalently, if (x) ∼= I.

Lemma 96. Notice that maximal elements are irreducible and minimal elements are co-irreducible, since [x] ∼= I for a maximal element and (x) ∼= I for a minimal element. Thus,if an element m is maximal, then

⋂i∈I m ≤ x⇒ y ≤ x or z ≤ x

Lemma 97. Suppose E is a calyx. Then for every element a ∈ E there is a maximal elementm ∈ E such that a ≤ m and a minimal element ζ ∈ E such that ζ ≤ a. This follows fromZorn’s lemma. We can form the intersection of all maximal elements containing an elementand the sum of all minimal elements contained in an element.

Lemma 98. Suppose a ∈ E has a + m = 1 for each maximal element m ∈ E. Then a = 1.

Proof. By contrapositive: if a 6= 1 then a is contained in some maximal element m, so thata + m = m ( 1.

Lemma 99. Suppose a ∈ E has a ∩ ζ = 0 for each minimal element ζ ∈ E. Then a = 0

Proof. By contrapositive: if a 6= 0 then a contains some minimal element ζ, so that a ∩ ζ =ζ ) 0.

We next establish the notions of prime elements.

Definition 100. Let E be a calyx. We say an element p ∈ E is prime if ab ⊆ p⇒ a ⊆ p orb ⊆ p. Equivalently, if we have ab = 0⇒ a = 0 or b = 0 in [p].

Lemma 101. Let E be a calyx. If p ∈ E is prime then p is irreducible.

Proof. If a ∩ b ⊆ p then ab ⊆ a ∩ b ⊆ p, so that a ⊆ p or b ⊆ p.

Lemma 102. Let S ⊆ E−0 be closed under multiplication. Suppose that ∀x, y ∈ E, x ≤y, x ∈ S ⇒ y ∈ S. Then there is a prime ideal p ⊆ E not in S.

Proof. The set E − S is nonempty since it contains 0, and closed under suprema of tosets.By Zorn’s lemma, there is a maximal ideal p among those not contained in S. Take a, b 6≤ psuch that ab ⊆ p. Since a, b 6⊂ p, a, b ∈ S, so ab ∈ S. Thus p ≥ ab, so that p ∈ S.

Lemma 103. Let E be a calyx. A maximal element m ∈ E is prime.

Proof.

m is maximal ⇔ E/m ∼= I ⇒ (ab = 0⇒ a = 0 or b = 0) in E/m ⇔ m is prime

A priori a maximal element m ∈ E is irreducible.Thus we have the following diagram of implications in any calyx:

Minimal Co-Irreducible

Maximal Prime Irreducible

Definition 104. We say that x ∈ M is irreducible if y ∩ z ⊆ x ⇒ y ⊆ x or z ⊆ x. We sayx ∈M is coirreducible if x ⊆ y + z ⇒ x ⊆ y or x ⊆ z.


4.2. The Isomorphism Theorems [0]. Remark: perenniality of a morphism f : M → Nis exactly the condition necessary for the first isomorphism theorem to hold. That is, for aperennial map f : M → N of E-corollas M and N , [f∗(0)] ∼= (f∗(1)).

Theorem (The Second Isomorphism Theorem). Let M be a modular E-corolla. Then [x ∩y, y] ∼= [x, x+ y].

Theorem (The Third Isomorphism Theorem). Let M be an E-corolla with elements x ≤ y.[y] ∼= [z] where z ∈ [x] is the image of y.

4.3. Duality [1].

Definition 105. For a lattice E define Eop as the opposite category of E viewed as a posetcategory. Eop is complete when E is. The category Lat is isomorphic to Latop. There is acontravariant functor op from Lat to Latop.

Lemma 106. EndLat(M,N) and EndLat(Nop,M op) are isomorphic as lattices.

Proof. Define Φ∗ : EndLat(M,N) → EndLat(Nop,M op) as follows. Take an element f ∈

EndLat(M,N). Define g∗ : N op →M op by g∗(xop) = f∗(x)op. Then, for x, y ∈ N ,

xop ≤ yop ⇒ y ≤ x⇒ f∗(y) ≤ f∗(x)⇒ f∗(x)op ≤ f∗(y)op ⇒ g∗(xop) ≤ g∗(y

op)

Define g∗ : M op → N op by g∗(xop) = f∗(x)op. Then, for x, y ∈M ,

xop ≤ yop ⇒ y ≤ x⇒ f∗(y) ≤ f∗(x)⇒ f∗(x)op ≤ f∗(y)op ⇒ g∗(xop) ≤ g∗(yop)

To check that g∗ a g∗, take x ∈M and y ∈ N .

g∗(yop) ≤ xop

⇔ x ≤ g∗(yop)op = f∗(y)

⇔ f∗(x) ≤ y

⇔ x ≤ f∗(y)

⇔ f∗(x) ≤ y

⇔ g∗(xop)op ≤ y

⇔ yop ≤ g∗(xop)

Φ∗ is bijective. Thus it is left adjoint to a morphism Φ∗ : EndLat(Nop,M op)→ EndLat(M,N),

and the pair Φ = (Φ∗,Φ∗) is an isomorphism of lattices.

Lemma 107. EndLat(M,M) and EndLat(Mop,M op) are isomorphic as calyxes.

Proof. Define the lattice morphism Φ∗ : EndLat(M,M) → EndLat(Mop,M op) as before,

where for an element f ∈ EndLat(M,M), Φ(f)∗ = g∗ : N op → M op is given by g∗(xop) =

f∗(x)op. Define Φ(f)∗ = g∗ : M op → M op by g∗(xop) = f∗(x)op. To see that Φ is an isomor-phism of calyxes it suffices to show that Φ preserves composition. Take lattice morphisms


f, g : M →M

Φ(f g)∗(xop)

=((f g)∗(x))op

=(f∗ g∗(x))op

=Φ(f)∗(g∗(x)op)

=Φ(f)∗ Φ(g)∗(xop)

Definition 108. An E-corolla M induces an E-corolla M op with sturcture morphism E →EndLat(M) → EndLat(M

op). There is a contravariant functor op : E − -Cor → E − -Corwhere M mapsto M op with the mentioned structure map (it should be clear what it does tomorphisms). It is an involution. If M is an E-corolla with structure map µ : E → M , thenwe write µop : E →M op for the opposite corolla. Notice that µop(a)∗(x)op = (x : a) and thatµop(a)∗(x)op = ax.

replacement...?

4.4. Principal Elements [2]. Reminder: A morphism f : M → N of E-corollas is calledperennial if f∗ is surjective onto (f∗(1)) and f∗ is surjective onto [f∗(0)].

Definition 109. Let M be an E-corolla. An element x ∈ M is called principal if thecanonical map E →M sending 1 to x is perennial. We often write ‘ζ ∈ a’ for ‘ζ ≤ a, a ∈ E,ζ principal’. Note the distinction between principal ideals of a realizable calyx and principalelements.

Lemma 110. If ζ ∈ E is principal and x ∈M is principal then ζx is principal.

Proof. To see this, let µx be the E-corolla map sending 1 to x. Since composition of perennial

corolla maps is perennial, the map Eµ(ζ)∗→ E

µx

M is perennial, so that ζx is principal.

Lemma 111. If ζ ∈ E is principal and x ∈M is principal then (x : ζ) is principal.

Definition 112. Let f : M → N be a morphism of E-corollas. We say f is perennial if anyof the equivalent statements are true:

(1) The canonically induced morphism g : [f∗(0)] ∼= (f∗(1)) is an isomorphism of E-corollas.

(2) f∗ is injective on [f∗(0)] and f∗ is injective on (f∗(1)).(3) f∗ is surjective onto (f∗(1)) and f∗ is surjective onto [f∗(0)].(4) (f∗ f∗)(f∗(1)) = id|(f∗(1)) and (f∗ f∗)[f∗(0)] = id|[f∗(0)]

Proof. Clearly (1)⇒ (2), and (4)⇒ (1).To show (2) ⇒ (3), take x ∈ [f∗(0)]. f∗(x) = f∗(f

∗(f∗(x))) so x = f∗(f∗(x)). Thus f∗ issurjective onto [f∗(0)]. Conversely, take x ∈ (f∗(1)). f∗(x) = f∗(f∗(f

∗(x))) so x = f∗(f∗(x)).

Thus f∗ is surjective onto (f∗(1)).

To show (3)⇒ (4), take x ∈ (f∗(1)). Then x = f∗(y) for y ∈ N . Then

f∗ f∗(x) = f∗ f∗ f∗(y) = f∗(y) = x


Thus (f∗ f∗)(f∗(1)) = id|(f∗(1)). Similarly, taking x ∈ [f∗(0)], write x = f∗(y). Then

f∗ f∗(x) = f∗ f∗ f∗(y) = f∗(y) = x

Thus (f∗ f∗)[f∗(0)] = id|[f∗(0)].

Definition 113. Let E be a calyx with structure map µ : E → EndLat(E). We say ζ ∈ Eis principal if µ(ζ) is a perennial E-corolla morphism. In other words, we say an elementζ of a calyx E is principal if the (unique) E-corolla map E → E whose left adjoint sends1 to a is perennial. The set of perennial maps in EndE-Cor(E) forms a submonoid undermultiplication, and multiplication of principal elements ζ, η ∈ E corresponds to compositionof maps µ(ζ)µ(η). We say an element a of a calyx E is coprincipal if the (unique) E-corollamap E → Eop whose left adjoint is multiplication by aop is perennial.

Lemma 114. Let E be a calyx. For a minimal element a ∈ E, Ia(∑

i∈I ai)

=∑

i∈I Ia(ai)and Ia(0) = 0, so that Ia is (both right and) left adjoint.

Proof. Define Ia∗ : E → E where x 7→ x ∩ a. To show that a is co-regular, it suffices toshow that Ia∗ preserves colimits. Take aii∈I .

∑i∈I a ∩ ai ≤ a ∩

∑i∈I ai ≤ a. Either

a ∩∑

i∈I ai ≤ a = 0 or a ∩∑

i∈I ai ≤ a = a. If a ∩∑

i∈I ai ≤ a = 0 then∑

i∈I a ∩ ai = 0.Otherwise, a ⊆

∑i∈I ai, so that a ⊆ ai for some i ∈ I since a minimal element is co-irreducible

by lemma 96. So a ≤ a ∩ ai ≤∑

i∈I a ∩ ai ≤ a, so that∑i∈I

a ∩ ai ≤ a ∩∑i∈I

ai ≤ a

Moreover Ia(0) = 0, so that Ia preserves colimits.

Lemma 115. Let E be a calyx. For a maximal element m ∈ E, Pm is (both left and) rightadjoint.

Definition 116. Let E be a calyx. For any a ∈ E let Ia be the right adjoint lattice morphismsending b ∈ E to b ∩ a. For each a ∈ E, let Pa be the left adjoint lattice morphism sendingb ∈ E to b + a.

Definition 117. Let E be a calyx with structure map µ : E → EndLat(E). ζ ∈ E isprincipal in Dilworth’s sense if for each a ∈ E, the following two diagrams commute:

E E

E E

µ(ζ)∗

I(a:ζ) Ia

µ(ζ)∗

E E

E E

µ(ζ)∗

Pζa Pa

µ(ζ)∗

In other words, the following diagram of adjoint pairs commutes for each a ∈ E:

E E

E E

µ(ζ)

(Pζa,I(a:ζ)) (Pa,Ia)

µ(ζ)

Lemma 118. Let E be a calyx. ζ ∈ E is principal if and only if it is principal in Dilworth’soriginal sense.


Proof. Suppose ζ ∈ E is principal. Let Ia : E → E be the right adjoint map sending x tox∩a. Ia can be viewed as a map [x]→ [x] for each x ≤ a. Let Pa : E → E be the left adjointmap sending x to x + a. Pa can be viewed as a map (x) → (x) for each x ≥ a. The map

µ(ζ)∗ : E → (ζ) factors through [(0 : ζ)] as EP(0:ζ)→ [(0 : ζ)]

θ→ (ζ) by the first isomorphismtheorem. Since ζ is principal, µ(ζ) is perennial, so that θ is an isomorphism. Take a ∈ E.Since (0 : ζ) ≤ (a : ζ), the following diagram commutes:

E [(0 : ζ)]

E [(0 : ζ)]

P(0:ζ)

I(a:ζ) I(a:ζ)

P(0:ζ)

The following diagram also commutes since θ is an isomorphism:

[(0 : ζ)] (ζ)

[(0 : ζ)] (ζ)

Iθ−1(a)

θ

Ia

θ

Since θ−1(a) = (a : ζ), the following diagram commutes for each a ∈ E:

E [(0 : ζ)] (ζ)

E [(0 : ζ)] (ζ)

P(0:ζ)

I(a:ζ) I(a:ζ)

θ

Ia

P(0:ζ) θ

Thus the following diagram commutes for each a ∈ E:

E E

E E

µ(ζ)∗

I(a:ζ) Ia

µ(ζ)∗

Commutativity of the dual diagram is shown similarly.

Suppose next that ζ is Dilworth principal. We show that ζ is principal. To show thatµ(ζ)∗ is surjective onto (µ(ζ)∗(1)), take a ≤ ζ. Since ζ is dilworth principal, the followingdiagram commutes:

E (ζ)

E (ζ)

µ(ζ)∗

I(a:ζ) Ia

µ(ζ)∗

Therefore

µ(ζ)∗((a : ζ)) = µ(ζ)∗(I(a:ζ))(1) = Ia(µ(ζ)∗(1)) = Ia(ζ) = a

So that µ(ζ)∗ is surjective onto (µ(ζ)∗(1)). The dual aspect of principality is shown similarly.


Lemma 119. f is principal if and only if f∗ f∗ = If∗(1) and f∗ f∗ = If∗(0). This is establishesthat principal is equivalent to weakly principal.

Proof. If f∗ f∗ = If∗(1) and f∗ f∗ = If∗(0) then f∗ f∗|(f∗(1)) = id(f∗(1)) and f∗ f∗|[f∗(0)] = id[f∗(0)],which is one of the equivalent statements in definition 112. Conversely, suppose f is principal.By lemma 118, the following diagrams commute

E E

E E

µ(ζ)∗

I(a:ζ) Ia

µ(ζ)∗

E E

E E

µ(ζ)∗

Pa Pζa

µ(ζ)∗

Evalutating at 1 in the first diagram, we see that

1 ζ

(a : ζ) ζ(a : ζ) = ζ ∩ a

Evaluating at 0 in the second diagram, we see that

0 ann(ζ)

aζ (ζa : ζ) = a + ann(ζ)

These imply that ζ is weakly principal.

Lemma 120. Let M be an E-corolla. Suppose ζ ∈ M is principal and f : M → N is aperennial map of E-corollas. Then f∗(ζ) is principal.

Proof. Let µ : E → M be the perennial map sending 1 to x. f µ is the composition ofperennial maps and therefore perennial.

4.5. Exact Sequences [3].

Lemma 121. Let M,N,P be E-corollas and let f : M → N and g : N → P be perennial

maps. We say Mf→ N

g→ P is exact at N if the following equivalent conditions hold:

(1) im(f) = ker(g)(2) coker(f) = coim(g)

Proof.

im(f) = ker(g) ⇔ f∗(1) = g∗(0) ⇔ cok(f) = coim(g)

Definition 122. We say a sequence

· · · →Mi−1 →Mi →Mi+1 → · · ·

of perennial E-corolla morphisms is exact if it is exact at every object.

Lemma 123 (Snake Lemma). Take a commutative diagram of E-corollas as below


N M L 0

0 N ′ M ′ L′

f

α

g

β γ

f′ g′

There is an exact sequence

(α∗(0))δ→ (β∗(0))

ε→ (γ∗(0))∂→ [α∗(1)]

ζ→ [β∗(1)]η→ [γ∗(1)]

so that the following diagram commutes and has exact rows and collumns:

(α∗(0)) (β∗(0)) (γ∗(0))

N M L 0

0 N ′ M ′ L′

[α∗(1)] [β∗(1)] [γ∗(1)]

δ ε

∂f

α

g

β γ

f′ g′

ζ η

Proof. We construct the left adjoint in ∂ as follows: for x ∈ L such that γ(x) = 0, writex = g(y) for y ∈ M . g′(β(y)) = 0. Write β(y) = f′(z). Viewing z as an element of N ′, take∂∗(x) = z. ∂∗ distributes over colimits and is therefore left adjoint.

Exactness at M : clearly = 0. Take x ∈ (β∗(0)) such that ε∗(x) = 0. Then g′∗ β∗(x) = 0.Exactness at L:Exactness at N ′:Exactness at M ′:Exactness at L′:

4.6. Hom and Tensor [10, 11].

Definition 124 (The Hom Functor). Let f : M ′ → M and g : N → N ′ be E-corolla mor-phisms. There are morphisms of E-corollas HomE(f, N) : HomE(M,N) → HomE(M ′, N)sending h to h f and HomE(M, g) : HomE(M,N)→ HomE(M,N ′) sending h to g h.

Lemma 125 (not done).

4.7. Deciduous, Coniferous, and Cupulate Calyxes [4].

Definition 126. A corolla is called deciduous if every element is the sum of principal ele-ments. Every realizable corolla is deciduous. A corolla is called coniferous if every elementis the intersection of coprincipal elements. Note the existence of deciduous coniferous treesin biology. A corolla is deciduous if and only if its opposite corolla is coniferous.

Example 127. Not every corolla is deciduous or coniferous. For example take the followingI-corolla:


·

·

·

Does every module induce a coniferous corolla?

Lemma 128. Let M be a deciduous noetherian corolla. Then every element is the sum offinitely many principal elements.

Proof. Suppose x ∈ M is not the sum of finitely many principal elements. Then we canconstruct a strictly increasing chain x0 < x1 < x2 < · · · where xi+1 = xi + ζi for someprincipal element ζi ≤ x, ζi 6≤ xi.

Lemma 129. Let M be a coniferous artinian corolla. Then every element is the intersetionof finitely many coprincipal elements.

Proof. Suppose x ∈ M is not the intersection of finitely many coprincipal elements. Thenwe can construct a strictly decreasing chain x0 > x1 > x2 > · · · where xi+1 = xi ∩ ζi forsome coprincipal element ζi ≥ x, ζi 6≥ xi.

Definition 130. Take a finite collection of elements x1, ..., xn ⊆ M for an E-corolla M .Let En be the coproduct of n copies of E, with ei the element which is 1 in the ith slot and0 elsewhere. We say x1, ..., xn is a generating set if the map En → E where ei 7→ xi issurjective.

Lemma 131. Suppose M is a Noetherian deciduous E-corolla over a calyx E. For everyelement x ∈M , (x) has a finite generating set.

Proof.

4.8. Modularity.

Definition 132. Let E be a lattice. The following conditions are equivalent:

(1) a ≤ a′ implies a + (b ∩ a′) = (a + b) ∩ a′ ∀a, a′, b ∈ E.(2) [a ∩ b, b] ∼= [a, a + b] as latices for each a, b ∈ E.(3) a ≤ a′, a + b = a′ + b, a ∩ b = a′ ∩ b⇒ a = a′ ∀a, a′, b ∈ E.

If any of the three above conditions hold we say E is modular. Note that every ring inducesa modular calyx and every A-module induces a modular Acal-corolla.

Example 133. The calyx induced by a semiring is not necessarily modular

4.9. Perianths.

Definition 134. Let E be a calyx. An E-perianth is a calyx F with a calyx morphismE → F . We often write simply F when the calyx morphism E → F is understood. Amorphism of perianths E → F and E → G is a morphism of calyxes F → G such thatE → F → G = E → G. There is a functor from A-algebras to Acal-perianths for any ringA, a forgetful functor from E-perianths to E-corollas for any calyx E.

Lemma 135. There is a left adjoint functor Φ : Cal→ E −Per which sends a calyx


Lemma 136. By composing the left adjoints Set → Lat → Cal → E − Per we get theE-algebra E[S]. We write E[x1, ..., xn] for free E-perianth in n variables. In the case ofn = 1 we have the free perianth in a singe variable:

E[x] ∼= E ⊗I ⊕n∈N≥0⊗ni=1 I

Definition 137. We say an E-perianth F is finitely generated if there is a perennial mapE[x1, ..., xn]→ F such that E → E[x1, ..., xn]→ F is the canonical structure map.

4.10. Finitely Generated Perianths, Finitely Generated Elements.

4.11. Finite Corollas, Finite Elements, and Finitely Presented Corollas [5]. fromstacks project

Definition 138. An E-corolla M is said to be finite over M if it is the finite sum of principalelements ζ1, ..., ζn. An element x ∈ M is said to be finite if it is the finite sum of principalelements.

4.12. Calyx Maps of Finite Type and Finite Presentation.

4.13. Finite Calyx Maps.

4.14. Compact Elements and Cupulate Calyxes.

Definition 139. An element x of an E-corolla M is said to be compact if x ≤∑

i∈I xiimplies that x ≤

∑i∈F xi for some finite subset F ⊆ I. In particular, we say a ∈ E is

compact if a ≤∑

i∈I ai implies that a ≤∑

i∈F ai for some finite subset F ⊆ I.

Lemma 140. The sum of finitely many compact elements is compact.

Proof. Let ζ1, ..., ζn be compact elements and take aii∈I in E such that∑n

i=1 ζi ≤∑

i∈I ai ≤E. Then there are finite subsets Fini=1 of I such that ai ≤

∑i∈Fi ai. Take F = ∪ni=1Fi.∑n

i=1 ζi ≤∑

i∈I ai.

Lemma 141. Let M be an E-corolla in which 1 is compact. Then if an element ζ ∈ E isprincipal and ζ =

∑i∈I xi, then ζ =

∑i∈F xi for a finite subset F ⊆ I.

Proof. Take a principal element ζ ∈ M with perennial map µ : E → M such that µ(1) = ζ.Suppose ζ =

∑i∈I xi for xii∈I in M .

xi ≤ ζ, so we can write xi = µ∗(ai) for ai ∈ E. µ∗(∑

i∈I ai)

=∑

i∈I µ∗(ai) = ζ, so∑i∈I ai = 1. Thus

∑i∈F ai = 1 for some finite set F ⊆ I. Therefore ζ = µ∗

(∑i∈F ai

)=∑

i∈F µ∗ (ai) =∑

i∈F xi.

Definition 142. We say an element x of an E-corolla M is finitely generated if x =∑n

i=1 ζifor finitely many principal elements ζ1, ..., ζn in M . We say M is finitely generated if 1 isfinitely generated in M .

Definition 143. A calyx is called cupulate if it satisfies the following:

(1) Every element is the sum of principal elements.(2) Every element is the sum of compact elements.(3) 1 is compact.

A realizable calyx is always cupulate.


Lemma 144. If E is cupulate, then compact is equivalent to finitely generated.;

Proof. Let a ∈ E be compact. a =∑

i∈I ζi for principal elements ζi. By compactnessA =

∑i∈F ζi for a finite set F ⊆ I.

Conversely, suppose a ∈ E is principal. Express a =∑

i∈I ci for compact elements cii∈I .By lemma 141, a =

∑i∈F ci for a finite subset F ⊆ I, which must be compact by lemma

140. Thus a is compact, so that any principal element is compact. Thus the sum of finitelymany principal elements is compact.

Corollary 145. The product of compact elements is compact in a cupulate calyx.

Theorem. Let E be a calyx in which every element is the sum of principal elements andevery element is compact. Further suppose that a ≤ b⇒ a = cb for some c ∈ E. Then E isrealizable as a semiring.

Proof.

homework 2 problem 1.direct sum preserves joins direct sum is a monad direct sum is a direct sum in the opposite

category

4.15. Localization [9]. For the purposes of this section all calyxes are assumed to be com-mutative.

Definition 146. For each a ∈ E, and each localization set S ⊆ E, we define the saturationaS =

∑σ∈S(x : σ).

Definition 147. Let E be a calyx. A localization set S is a set S ⊆ E of principal elementswhich is multiplicatively closed and which contains 1.

Lemma 148. Let E be a calyx with localization set S. If a, b, p ∈ E with p prime and notcontained in S, then the following hold:

(1) a ≤ b⇒ aS ≤ bS

(2) a ≤ aS

(3) p = pS

(4) (aS : σ) = aS) for each σ ∈ S.(5) a(b : σ) = (ab : σ) for σ ∈ S and a ∈ E such that (0 : σ) ⊆ a.(6) (aS)S = aS

Proof. (1) Suppose a ≤ b and take c such that cσ ≤ a for some σ ∈ S. Then cσ ≤ b trivially.

(2) aσ ≤ a for any σ ∈ S, so that a ≤ (a : σ) for any σ ∈ S, so that a ≤∑

σ∈S(a : σ).

(3) Suppose b ≤ pS. Then bσ ≤ p for some σ ∈ S. Thus b ≤ p since p is prime and σ 6≤ p.

(4) Suppose xσ ≤ aS. Then xστ ≤ a for some τ ∈ S. So x ∈ aS.

(5) This can be seen as a consequence of (4). Alternatively, suppose b ≤ (aS)S. Thenbστ ≤ a for some σ, τ ∈ S. So b ∈ aS since στ ∈ S.


(6) Follows since σ is principal.(7)

µ(aS)∗(lim−→s∈S

(b : s)) = lim−→s∈S

µ(aS)∗(b : s) = lim−→s∈S

(aSb : s) = (aSb)S

Lemma 149. Let E be a calyx with localization set S and structure map µ : E → EndLat(E).The set S−1E = aS : a ∈ E forms a calyx.

Proof. To show that S−1E forms a complete lattice, it suffices to show that S−1E has arbi-trary joins (note that the join of elements ai

S will possibly be distinct from the join∑

i∈I aiS

in E). We show that(∑

i∈I ai)S

is a colimit for aiS in S−1E. Clearly aSi ≤

(∑i∈I ai

)S.

Suppose that for some b ∈ E, aSi ≤ bS for each i ∈ I. Take x such that xσ ≤∑

i∈I ai. Then

xσ ≤∑

i∈I ai ≤ bS, so that xστ ∈ b for some τ ∈ S, so that x ∈ bS. Thus(∑

i∈I ai)S ≤ bS.

Note that 0S acts as an initial object. Note also that intersection in S−1E is possibly distinctfrom intersection in E.

Define a map ν : S−1E → EndLat(S−1E) where aS is sent to the map whose left adjoint

ν(aS)∗ has ν(aS)∗(xS) = (aSxS)S. Clearly ν(aS)∗ preserves initial objects. To show that it

preserves all colimits, take aii∈I in E. Then

ν(aS)∗

(lim−→i∈I

xSi

)

=

aS

(lim−→i∈I

xSi

)SS

=

(aS lim−→i∈I

xSi

)SS

=

(lim−→i∈I

aSxSi

)SS

= lim−→i∈I

aSxSi

= lim−→i∈I

ν(aS)(xSi )

So ν(aS)∗ is indeed left adjoint. To show that ν itself is left adjoint, take aii∈I in E. Foreach b ∈ E, we can use commutativity to note that

ν

(lim−→i∈I

aSi

)∗

(b) = ν (b)∗ (lim−→i∈I

aSi ) = lim−→i∈I

ν (b)∗ (aSi ) = lim−→i∈I

ν(aSi)∗ (b) =

(lim−→i∈I

ν(aSi))∗

(b)

so that ν(

lim−→i∈I aSi

)∗

=(

lim−→i∈I ν(aSi))∗, so that ν

(lim−→i∈I a

Si

)=(

lim−→i∈I ν(aSi))

.


Note that this method of proof automatically gives a calyx morphism φS : E → S−1E,which we call the structure map of the localization.

Lemma 150. Let M be an E-corolla (recall that this means that M is the finite sum ofprincipal elements). Let S ⊆ E be a multiplicative set. Then for any finitely generatedelement x ∈M , ann(x)p = ann(xp).

Proof. Write x =∑n

i=1 ζi for principal elements ζini=1. For each 1 ≤ i ≤ n, we have anexact sequence of perennial morphisms 0 → ann(ζi) → E → (ζi) → 0. Since localizationpreserves exactness, we have that 0 → ann(ζi)

S → S−1E → (ζSi )S−1M → 0 is exact. Thusann(ζSi ) = ann(ζi)

S. Now

S−1

(ann

(n∑i=1

ζi

))

=S−1

(n⋂i=1

ann(ζi)

)

=n⋂i=1

S−1(ann(ζi))

=n⋂i=1

ann(S−1ζi)

=ann

(n∑i=1

S−1ζi

)

***

Corollary 151. Suppose a, b ∈M are finitely generated elements of an E-corolla M and bis finitely generated. Then (ap : bp) = (a : b)p for each p ∈ Spec(E).

Proof. (a : b) = ann([b, a + b]) so we can apply lemma 150.

Lemma 152. Let f : E → F be a calyx morphism and S ⊆ E a localization set. Thenf∗(a

S) = f∗(a)f∗(S).

Proof. Using 151, we have

f∗(aS) = f∗

(∑σ∈S

(a : σ)

)=∑σ∈S

f∗ ((a : σ)) =∑σ∈S

(f∗(a) : f∗(σ)) =∑

σ∈f∗(S)

(f∗(a) : σ)

Lemma 153 (UMP of Localization). Let S be a localization set of a calyx E. The calyxS−1E with the structure map φS : E → S−1E is universal among calyxes F with morphismsg : E → F such that f∗(a) = 1 for each a ∈ S.

Proof. Let F be a calyx with a morphism g : E → F such that f∗(a) = 1 for each a ∈ S.Define a map h : S−1E → F where aS 7→ g(a).


Lemma 154. Let S and T be multiplicative subsets of a calyx E with S ⊆ T . Let φS : E →S−1E and φT : E → T−1E be the localization maps. Then φS(T )−1S−1E ∼= T−1E as calyxes.

Proof. Follows from the universal property of localization.

Lemma 155. Let E be a calyx with localization set S. If a, b, p ∈ E with p prime and notcontained in S, then The following hold:

(1) x ≤ y ⇒ xS ≤ xS

(2) x ≤ xS

(3) (xS : σ) = xS for each σ ∈ S.(4) x(y : σ) = (xt : σ) for σ ∈ S and x ∈ E such that (0 : σ) ⊆ x.(5) (xS)S = xS

(6) xSyS = (xSy)S

Lemma 156. Let E be a calyx with localization set S and structure map µ : E → EndLat(E).Suppose that every element in E is the sum of principal elements. The set S−1E = aS :a ∈ E forms a calyx.

Definition 157. Let M be an E-corolla and let S ⊆ E be a localization set. For an elementx ∈ M we form xS =

∑σ∈S(x : σ) and S−1M = xS : x ∈ M We say xS ≤ yS in S−1M

when xS ≤ yS in M .

Theorem. S−1M forms an S−1E corolla.

Proof. First we show that S−1M forms a complete lattice, for which it suffices to show thatS−1M has arbitrary joins (note that the join of elements xiSi∈I will possibly be distinct

from the join∑

i∈I xiS in M). We show that

(∑i∈I xi

)Sis a join for xi

S in S−1M . Clearly

xSi ≤(∑

i∈I xi)S

. Suppose that for some y ∈ E, xSi ≤ yS for each i ∈ I. Take x such that

xσ ≤∑

i∈I xi. Then xσ ≤∑

i∈I xi ≤ yS, so that x ≤ (yS)S = yS. Thus(∑

i∈I xi)S ≤ yS.

Note that 0S acts as an initial object. Note also that intersection in S−1M is possibly distinctfrom intersection in M .

Define a map ν : S−1E → EndLat(S−1M) where aS is sent to the map whose left adjoint

ν(aS)∗ has ν(aS)∗(xS) = (aSxS)S. Clearly ν(aS)∗ preserves initial objects. To show that it

preserves all colimits, take xii∈I in M . Then

ν(aS)∗

(∑i∈I

xSi

)=

aS

(∑i∈I

xSi

)SS

=

(aS∑i∈I

xSi

)SS

=

(∑i∈I

aSxSi

)SS

=

(∑i∈I

(aSxi)S

)S

Thus ν(aS)∗ is indeed left adjoint. To show that ν itself is left adjoint, take aii∈I in E.For each b ∈ E, we can use commutativity to note that

ν

(lim−→i∈I

aSi

)∗

(b) = ν (b)∗ (lim−→i∈I

aSi ) = lim−→i∈I

ν (b)∗ (aSi ) = lim−→i∈I

ν(aSi)∗ (b) =

(lim−→i∈I

ν(aSi))∗

(b)

so that ν(

lim−→i∈I aSi

)∗

=(

lim−→i∈I ν(aSi))∗, so that ν

(lim−→i∈I a

Si

)=(

lim−→i∈I ν(aSi))

.

Note that this method of proof automatically gives a calyx morphism φS : E → S−1E,which we call the structure map of the localization.


Lemma 158. S−1M is the universal S−1E-corolla with a E-corolla morphism M → S−1M .Note that S−1E-corolla is in particular an E-corolla by restriction of scalars.

Proof. Define the map φ∗ : M → S−1M where x 7→ xS. φ∗ distributes over coproducts andis therefore left adjoint. φ∗ induces a corolla map φ : M → S−1M whose left adjoint is φ∗.Take an S−1E corolla N with an E-corolla morphism ψ : M → N . To show that ψ factorsthrough S−1M it suffices to show that ψ(x) = ψ(xS) for each x ∈M . Clearly ψ(x) ≤ ψ(xS).Suppose σy ≤ x for some σ ∈ s. Then ψ(y) = ψ(σy) ≤ ψ(x). Thus ψ(xS) = ψ(x).

Lemma 159. Suppose 0 → Mf→ N

g→ L → 0 is an exact sequence of E-corollas. Then0→ S−1M → S−1N → S−1L→ 0 is exact.

Definition 160 (The Localization Functor). Take a calyx E with localization set S. Themap S−1(−) on E-corollas sending M to S−1M can be made into a functor E-Cor →S−1E-Cor: a map φ : M → N induces a map M → N → S−1N , which, by the universalproperty in lemma 158, induces a morphism S−1φ : S−1M → S−1N . Thus S−1φ∗(x

S) =(φ(x)∗)

S and S−1φ∗(xS) = (φ(x)∗)S for a morphism φ : M → N . It should be clear that thismakes S−1(−) into a functor.

Lemma 161. Localization is an exact functor. Also, localization commutes with takingsubrings and quotient ring.

Proof. Let E be a calyx. Take an exact sequence 0 → (x) → M → [x] → 0, where Mis an E-corolla and x ∈ M is an element. We show that the induced map 0 → S−1(x) →S−1M → S−1[x]→ 0 is perennial and exact. It suffices to note that this sequence is naturallyisomorphic to the exact sequence 0 → (xS) → S−1M → [xs] → 0. In particular, there arenatural isomorphisms S−1(x) → (xS) and S−1[y] → [yS] making the following diagramcommute:

0 S−1(x) S−1M S−1 [y] 0

0 (xS)M S−1M[yS]M

0

We show that the map S−1(x)→ (xS)M , which sends∑

σ∈S(y : σ)(x) to∑

σ∈S(y : σ)M , is anisomorphism.

Lemma 162. Being 0 is a local property. That is, if Mp = 0 for each prime ideal p ∈ E,then M = 0, and vice versa.

Proof. Clearly if M = 0 then Mp = 0 for each prime p ∈ E. Suppose Mp = 0 for eachprime p ∈ E, and suppose for a contradiction that there is x 6= 0 in M . Then ann(x) 6= E,so that it is contained in a maximal element m. Now Mm = 0, so that xS = 0S. So∑

σ∈S(x : σ) ≤∑

σ∈S(0 : σ). So, for each τ ∈ S,

τx ≤ τ(x : τ) ≤∑σ∈S

σ(x : σ) = σ∑σ∈S

(x : σ)

=σ∑σ∈S

(0 : σ) =∑σ∈S

σ(0 : σ) = 0

Thus σx = 0 for each σ ∈ S. But σ 6≤ ann(x) for each σ ∈ S, a contradiction.


Lemma 163. Having 0 kernel is a local property of corollas. That is, if φp : Mp → Np haszero kernel for each prime element p ∈ E, then φ has zero kernel, and vice versa.

Proof. One direction is clear. Suppose φ : M → N is locally injective, so that φm is injectivefor each maximal ideal m (we only need this for the maximal ideals). Let ψ : L → M bethe kernel of φ. Then Lm = 0 for each maximal ideal m since localization is exact by lemma161, so that L = 0. Thus φ has zero kernel.

Lemma 164. Having 0 cokernel is a local property of corollas. That is, if φp : Mp → Np

has zero cokernel for each prime element p ∈ E, then φ has zero cokernel, and vice versa.

Proof. One direction is clear. Suppose φ : M → N is locally surjective, so that φm issurjective for each maximal ideal m (we only need this for the maximal ideals). Let ψ : N →P be the cokernel of φ. Then Pm = 0 for each maximal ideal m since localization is exact bylemma 161, so that P = 0. Thus φ has zero cokernel.

Lemma 165. Let E be a calyx. For each prime p, write ap for aσ:σ 6⊂p. We show that ifap = 1 for each prime p then a = 1.

Proof. Take an element a ∈ E such that ap = 1 for each prime p. Suppose for a contradictionthat a 6= 1. Then a ⊆ m for some maximal element m. By assumption am = 1. Recall thatam =

∑σ 6⊂m(a : σ). But, for each b ∈ E,

b ≤ (a : σ) ⇔ bσ ≤ a⇒ bσ ≤ m⇒ b ≤ m

So∑

σ 6⊂m(a : σ) ≤ m, a contradiction.

Lemma 166 (Not Done). Let E be a calyx and M an E-corolla. For each prime p, writexp for xσ:σ 6⊂p. We show that ‘Being 1 is a local property,’ now for modules i.e. if xp = 1for each prime p then x = 1.

Proof. Take an element x ∈M such that xp = 1 for each prime p. Suppose for a contradictionthat x 6= 1. Then annop(x) ⊆ m for some maximal element m. Recall that annop(x) = a ∈E : (x : a) = 1. By assumption xm = 1. Recall that xm =

∑σ 6⊂m(x : σ).

annop(∑

σ 6⊂m(x : σ))

=∑

σ 6⊂m annop ((x : σ)) for each σ 6⊂ m, annop((x : σ)) ≤ m. So

annop(∑

σ 6⊂m(x : σ))≤ m,

so that∑

σ 6⊂m(x : σ) 6= 1, a contradiction.

Lemma 167. Let M be an E-corolla with finitely generated elements x and y. Supposexp = yp for each p ∈ E. Then x = y.

Proof. Without loss of generality, we show that x ≤ y, so that it suffices to show (y : x) = 1.It follows from corollary 151 that (y : x)p = 1 for each p ∈ Spec(E). By lemma 165,(y : x) = 1.

Corollary 168. Let M and N be E-corollas, with M deciduous. Suppose f, g : M → N areE-corolla morphisms such that fp = gp for each prime p ∈ E. f = g.

Proof. Certainly by lemma 167, f∗(ζ) = g∗(ζ) for principal elements ζ ∈ M . Since allelements in M are the sum of principal elements, f∗ = g∗ en

Corollary 169 (Not Done). If fp is an isomorphism for each p ∈ Spec(E) then f is anisomorphism.


Proof. Take a map f : M → N such that f : M

Lemma 170. Let E be a calyx and ζ, η ∈ E principal elements such that ζ + η = 1. Letα : E → Eζ and β : E → Eη be the canonical localization maps. The canonical localization

map γ : E → Eζη factors through Eζ and Eη as Eα→ Eζ

δ→ Eζη and Eβ→ Eη

ε→ Eζη. Weshow that the following diagram of canonical maps is a pullback:

E Eζ

Eη Eζη

α

β δ

ε

Proof. Note that δ(a) = (a : η) for a ∈ Eζ and ε(a) = (a : ζ) for a ∈ Eη. Let P be thepullback of the following diagram:

Eζ

Eη Eζη

δ

ε

The pullback is constructed from the set Eη ×Eζ as all the elements (a, b) such that δ(a) =ε(b). There is a map φ : E → P such that the following diagram commutes:

E

P Eζ

Eη Eζη

α

β

φ

δ

ε

We show that φ has an inverse map. Define the inverse map ψ : P → E where (a, b) 7→ a∩b.

To show φ ψ = id, take a ∈ Eζ and b ∈ Eη such that (a : η) = (b : ζ). Then

(a ∩ b : ζ) = (a : ζ) ∩ (b : ζ) = (a : ζ) ∩ (a : η) = (a : ζ + η) = a

and(a ∩ b : η) = (a : η) ∩ (b : η) = (b : ζ) ∩ (b : η) = (b : ζ + η) = b

so that φ ψ([(a, b)]) = [(a, b)].

To show ψ φ = id, take a ∈ E. (a : ζ) ∩ (a : η) = (a : ζ + η) = a.

Corollary 171. Let E be a calyx with principal elements ζ, η ∈ E. The following diagramis a pullback:

Eζ+η Eζ

Eη Eζη

Proof. Passing to Eζ+η we see that ζ + η = 1, so that we can apply lemma 170.


Lemma 172. Let E be a calyx and ζii∈I ∈ E principal elements. E∑i∈I ζi

= lim←−F⊆I E∑i∈F ζi

.

Proof. For each finite subset S ⊆ I write ES for E∑i∈S ζi

. When S ⊆ T there is a canonicalmap ET → ES which localizes at the element

∑i∈S ζi (the canonical map E → ES factors

through ET ). Let C be the poset category of finite subsets of I. There is thus a contravariantfunctor Φ : C → Cal. We show that EI ∼= lim←−Φ. Passing to EI , we see that it suffices toassume

∑i∈I ζi = 1 and prove that E ∼= lim←−Φ. The canonical localization maps E → ES

make the following diagram commute for each S and T :

E ES

ET ES∩T

This induces a map α : E → lim←−Φ. Define a set map β : lim←−Φ → E as follows: for anelement aFF∈Obj(C) in lim←−Φ take β(aFF∈Obj(C)) =

⋂∀∈Obj(C) aF . We check that this con-

stitutes an inverse function for α.

Take an equivalence class [aFF∈Obj(C)] ∈ Obj(C). α(⋂

F⊆I finite aF)

= [(⋂G⊆I finite aG :∑

i∈F ζi)F∈Obj(C)]. It suffices to show that(⋂

F⊆I finite aF : ζi)

= aζi . Thus it suffices to showthat (aG : ζi) ≥ aζi ∀G ∈ Obj(C). But this is obvious.

Conversely, to show βα = id, take a ∈ E. a =(a :∑

F∈Obj(C)∑

i∈F ζi

)=⋂F∈Obj(C)

(a :∑

i∈F ζi).

Lemma 173. Let E be a calyx and let S be the set of principal elements not contained in agiven prime p. S is a multiplicative set. View S as a category whose morphisms are pairs(λ, s) written λ : s→ λs of elements λ, s ∈ S with source s and target λs. Let Φ : S → Calbe the functor which on objects has Φ(ζ) = Eζ and on morphisms λ : s → λs gives thecanonical localization map Es → (Es)λ ∼= Esλ. lim−→Φ ∼= Ep.

Proof. It is clear that lim−→Φ has the an equivalent universal property to Ep.

Lemma 174 (Not finished). Let p and q be prime elements in a deciduous calyx E neitherof which is less than or equal to the other. Let S = ζ ∈ E : ζ 6≤ p, ζ 6≤ q. pS and qS aremaximal in S−1E.

Proof. Take a nonzero element a ∈ S−1E. a + p + q

Theorem (Not finished). Let pini=1 be prime elements in a deciduous calyx E. Let S =ζ ∈ E : ζ 6≤ pi ∀1 ≤ i ≤ n. Maximal elements of the set p1, ..., pn are maximal in S−1E.

Proof. Clearly the claim holds for n = 1. Suppose n = 2. Let φS : E → S−1E be thecanonical map. It suffices to show that I ∼= [φS∗(p1)].≤ φS∗(a), p

Theorem (Prime Avoidance). [Not finished] Let E be a deciduous calyx and take a ∈ E.Let pii∈I be a finite set of prime ideals in E. If for each principal element ζ ∈ a there isi ∈ I such that ζ ≤ pi, then a ≤ pi for some i ∈ I.


Proof. We argue by contrapositive. Suppose a 6≤ pi ∀i ∈ I. Let S = ζ ∈ E : ζ 6≤ pi ∀i ∈ I.We induct on |I| ∈ N≥1 to show that a is a unit in S−1E. It will follow that there is aprincipal element ζ ∈ a such that ζ 6≤ pi ∀i ∈ I. The case for |I| = 1 is clear.

The case for |I| = 2. Let p and q be prime elements. Take a such that a 6≤ p and a 6≤ q.Let S = ζ ∈ E : ζ 6≤ pi ∀i ∈ I.

For the induction step, take n ∈ N and say |I| = n+ 1. Take i ∈ I arbitrary.For each j ∈ I − i take a principal element ζ ≤ a ζ 6≤ pi, ζ ≤ pj.Since a 6≤ pia + pi ≤ pj + pi a ≤ pj. (a + pi) ∩ pi ≤ (pj + pi) ∩ pi

Theorem. Let A be a ring. The functor F : A-mod → Acal-Cor commutes with localiza-tion.

Proof. Follows from the characterization of ideals in a localization.

Theorem (Not Done). Being perennial is a local property.

Proof. Take a map f : M → N of E-corollas and suppose that fp is perennial for each primep ∈ E. Then each map fp induces an isomorphism gp : [f∗p(0)] → (f∗p(1)) on calyxes. Themap f induces a map g : [f∗(0)] → (f∗(1)) whose localization at p is gp since localizationcommutes with taking subcalyxes and quotient calyxes. Since isomorphism of corollas is alocal property and gp is an isomorphism for each p ∈ Spec(E), g is an isomorphism.

Theorem (Not Done). If an ideal a of a ring A is locally principal then it is Dilworthprincipal. Note that this is not true for typical principal elements. In any Dedekind domain,any ideal is locally principal but not principal.

Proof. Suppose an ideal a ∈ E is locally principal. Then ap is principal in Acalp for eachp ∈ Spec(E). Thus it is locally Dilworth principal. That is, for each p, there is a perennialmap fp : Acal → Acal such that fp induces an isomorphism [f∗p(0)] ∼= (ap∗(1)).

Lemma 175 (Not Done). Let A be a ring. There is a faithful exact functor F : Acal-Cor→q

p∈Spec(A)Rcalp -Cor.

4.16. Base Change [13]. c.f. stacks project.

Let f : E → F be a map of calyxes. Let M be an F -corolla. Let M be an F -corolla. Letφ : E → E ′ be a map of calyxes. The base change of f by φ is the map E ′ → F ⊗E E ′. Thebase change of the F -corolla M is the F ′-corolla M ⊗E E ′.

4.17. The Chinese Remainder Theorem [14].

Lemma 176. Let E be a calyx with elements a, b ∈ E such that a+b = 1. [a∩b] ∼= [a]

q

[b].

Proof. Passing to [a ∩ b], it suffices to show that E ∼= [a]

q

[b]. We show that the canonicalmap φ : E → [a]

q

[b] is an isomorphism. Define a set map ψ : [a]

q

[b] → E where


(x, y) 7→ ya + xb. To show ψ φ = 1, observe that

ψ φ(c, d)

= (ya + xb + a, ya + xb + b)

= (xb + a, ya + b)

= (xb + xa + a, ya + yb + b)

= (x(a + b) + a, y(a + b) + b)

= (x+ a, y + b)

= (x, y)

And for the other direction,

φ(ψ(x)) = φ(x+ a, x+ b) = (x+ a)b + (x+ b)a = xb + xa = x

Lemma 177. Let E be a calyx with elements aini=1 in E such that ai + aj = 1 for 1 ≤ i 6=j ≤ n. [∩ni=1ai]

∼=

qni=1[ai].

Proof. We show this by induction on n. For n = 1 the claim is trivial. Suppose the claimholds for some n ∈ N and take aini=1 such that ai + aj = 1 for i 6= j. Then

[∩ni=1ai] = [an ∩n−1⋂i=1

ai] = [an]

q

[n−1⋂i=1

ai] = [an]

q( qn−1i=1 [ai]

) ∼= qni=1[ai]

4.18. The Spectrum of a Calyx [16].

Definition 178. For a calyx E with element a ∈ E, let V (a) be the set of prime ideals ofE.

Lemma 179. Take elements a, b of a clayx E. rad(a) ≤ rad(b) if and only if V (b) ≤ V (a).

Proof. Note that⋂V (a) = rad(a). If V (a) ⊆ V (b) then rad(b) =

⋂V (b) ≤

⋂V (a) =

rad(a). Conversely, if rad(b) ≤ rad(a) and p ∈ V (a), then p ≥ rad(b) ≥ b, so thatV (a) ⊆ V (b).

Lemma 180. Let a, b, aii∈I be elements of a calyx E. Then

(1) V (ab) = V (a ∩ b) = V (a) ∪ V (b)(2) V

(∑i∈I ai

)=⋂i∈I V (ai)

Proof. Clearly V (a) ∪ V (b) ⊆ V (a ∩ b) ⊆ V (ab). And if a prime p has p 6≥ a, p 6≥ b, thenp 6≥ ab. So V (ab) ⊆ V (a) ∪ V (b).


For the second claim, note that

p ∈ V

(∑i∈I

ai

)⇔ p ≥

∑i∈I

ai

⇔ p ≥ ai ∀i ∈ I⇔ p ∈ V (ai) ∀i ∈ I

⇔ p ∈⋂i∈I

V (ai)

Corollary 181. The sets of the form V (a) ⊆ Spec(E) form a topology for a calyx E.

Definition 182. For a calyx E with principal element ζ ∈ E define D(ζ) = Spec(E)−V (ζ).The open sets D(ζ) : ζ is principal form a basis for Spec(E). We refer to them as theprincipal open sets of Spec(E).

For an element a ∈ E we write D(a) = Spec(E)− V (a). Every open set U ⊆ Spec(E) isof this form.

4.19. Completions. c.f. atiyah and macdonald.

4.20. Graded Calyxes and Corollas [55, 57]!!!

Definition 183. Let Eii∈N≥0be lattices. A graded calyx E over E0-corollas Eii∈N≥0

isa calyx E which has a direct sum decomposition as a corolla over E0 as ⊕i∈N≥0

Ei, such thataiaj ∈ Ei+j for ai ∈ Ei and aj ∈ Ej, for each i, j ∈ N≥0. We can form the irrelevant ideal as∑

i∈N≥1Ei, denoted E+. We write ai for (aj)j∈N≥0

when aj = 0 for j 6= i. Such elements are

called homogeneous.

A graded E-corolla is a E0-corolla endowed with a structure map φM : E⊗E0M →M . Wewrite (an)∞n=0(xn)∞n=0 for φM((an)∞n=0 ⊗ (xn)∞n=0). An element (xj)

∞j=0 is called homogeneous

if there is an i ∈ N≥1 such that xj = 0 for i 6= j.Clearly E forms a graded corolla over itself.

Definition 184. For a graded E-corolla M and i ∈ N≥0 let M [i] be the shifted gradedE-corolla, i.e. where M [i]n = Mi+n.

Definition 185. For a graded calyx E, set E(d) = ⊕i≥0Mni, which forms a graded calyx inits own right.

Definition 186. Let E be a graded calyx. For graded E-corollasM andN letHomgr(M,N)0

be the E0-corolla of graded E-corolla morphisms from M to N . We define a graded E-corollaHomgr(M,N) by endowing the E0-corolla ⊕i∈N≥0

Homgr(M,N [i]) with the structure of agraded E-corolla.


Lemma 187. Let E be a graded calyx with homogeneous elements ai in E+. We showthat

(aini=1) = E+ ⇔ E0[aii∈I ] = E

Proof. One direction is obvious. So suppose that∑

i∈I ai = E+. We claim that E[aii∈I ] =E. Replacing each ai with homogeneous elements ai1, ..., aim such that

∑mj=1 aij = ai, it

is clear that no generality is lost in assuming ai to be homogeneous. Take a homogeneouselement a ∈ Em. We show by induction on m that a ∈ E0[aii∈I ]. Obviously if a ∈ E0 thenwe are done. Otherwise, we can write a =

∑ni=1 biai with bi homogeneous of lower degree

and apply the induction hypothesis.

Definition 188. A graded E-corolla M is said to be finite over E (or just finite) if there areprincipal elements ζ1, ..., ζn in M (where we view M as an E0 corolla) and elements a1, ..., anof E such that

∑ni=1 aiζi = 1 in M .

Lemma 189. Let F be a modular graded calyx and set E = F0. Suppose that E is aNoetherian calyx. Suppose there are elements ζ1, ..., ζn of F which are principal (where weview F as an E corolla) and aini=1 in F such that

∑ni=1 aiζi = F .

Proof. Write b0 = 0 and

Lemma 190. Let E be a graded calyx and take a collection ζii∈I of homogeneous principalelements in E+. The following are equivalent:

(1) The E0-corolla map E0[xii∈I ]→ E where xi 7→ ζi sends 1 to 1 (has trivial cokernel).(2) There are elements aii∈I in E+ such that

∑i∈I aiζi = E+.

4.21. Noetherian Calyxes [30]!!!

Definition 191. A calyx R is Noetherian if every element is the sum of principal elements.

Lemma 192. Let E be a deciduous calyx. The following are equivalent:

(1) Every ascending chain a1 ≤ a2 ≤ · · · ≤ an ≤ · · · stabilizes (ACC).(2) Every element is the sum of principal elements.(3) Every collection of elements has a maximal element.

Lemma 193. If E is a Noetherian calyx then [x] is noetherian for each x ∈ E.

Lemma 194. A finitely generated perianth over a noetherian calyx is noetherian as a calyx.

Proof. It suffices to show that E[x] is noetherian for a calyx E. And for this it suffices toshow that x is principal. The E-corolla map µx : E → E[x] where µx(a) = ax. This map isclearly perennial, so that x is principal.

In light of the above lemma, we can think of E[x1, ..., xn] as adjoining principal elementsto E in the most general way.

Lemma 195. Any finitely generated calyx over a Noetherian calyx is Noetherian.

Proof. Follows from lemma 193 and lemma 194.

Lemma 196. Any localization of a Noetherian calyx is Noetherian.

Corollary 197. A finitely generated algebra over I is Noetherian.


The following is a key lemma in the use of Noetherian calyxes.

Lemma 198. Let E be a Noetherian calyx and M a modular E-corolla, finitely generatedby elements ζ1, ..., ζn. Take x0 = 0, xi =

∑ij=1 ζj. Take x, y ∈ M with x ≤ y. If x = y in

[xi, xi+1] for each 0 ≤ i < n then x = y.

Proof. We show this by induction on n ∈ N≥0. For n = 0 the claim is trivial. For theinduction step, take n ∈ N≥0 and x ≤ y in M with x = y in [xi, xi+1]. By the inductionhypothesis x = y in (xn−1). Thus x ∩ xn−1 = y ∩ xn−1 and x + xn−1 = y + xn−1, so thatx = y by definition 132.

Lemma 199. Let E be a Noetherian calyx. Let M be a finitely generated deciduous modularE-corolla. Then M is Noetherian.

Proof. Take principal elements ζ1, ..., ζn and let x0 = 0, xi =∑i

j=1 ζj. By lemma 192, itsuffices to show that every ascending chain y0 ≤ y1 ≤ y2 ≤ · · · terminates. By lemma 198,it suffices to show that every ascending chain y0 ≤ y1 ≤ y2 ≤ · · · terminates in [xi, xi+1] foreach 0 ≤ i < n. But this is true since [xi, xi+1] is noetherian as there is a perennial mapE → [xi+1].

4.22. Local Calyxes [17].

Definition 200. Define a local calyx is a pair (E,m) where m is the unique maximal elementof E. We often write E for (E,m). The quotient [m] is always I. A local morphism of localcalyxes (E,m) and (F, n) is a morphism φ : E → F such that φ∗(m) ≤ n. Equivalently,a local morphism of local calyxes (E,m) and (F, n) is a morphism φ : E → F such thatφ∗(n) = m Note that I is a local ring and that Ep is local. The unique maximal element ofEp is the extension of p by the canonical map E → Ep.

Lemma 201. For a calyx E, the following are equivalent:

(1) E is local.(2) E has a maximal ideal m such that a 6≤ m⇒ a = 1(3) Spec(E) has exactly one closed point.

Proof. Omitted.

Lemma 202. Let φ : E → F be a morphism of calyxes. Take an element p ∈ F . Thecanonically induced map Eφ∗(p) → Fp is a local morphism.

Theorem. Suppose A is a local ring. If a ∈ Acal is Dilworth principal then it is principal.

Proof. Take a perennial corolla map φ : Acal → Acal. φ∗(1) is an ideal a in A. There is acalyx isomorphism [φ∗(0)] → (φ∗(1)). (φ∗((a)) : a ∈ a generates the local calyx [φ∗(0)].By Nakayama’s lemma, it has a single generator as a minimal generating set, call it φ∗((a)).(a) is then a generator of (φ∗(1)) by the correspondence [φ∗(0)]→ (φ∗(1)).

Definition 203. We say a set map of abelian monoids f : G→ H is a homomorphism up tounits if for each g, h ∈ G, f(g)f(h) = uf(gh) for some invertible element u ∈ H. We say f issurjective up to units if for each h ∈ H there is g ∈ G and a unit u in H such that f(g)u = h.We say a set map of commutative rings f : A → B is a homomorphims up to units if foreach a, b ∈ A, f(ab) = uf(a)f(b) for some unit u ∈ S and f(a+ b) = v(f(a) + f(b)) for someunit v ∈ S. We say f is surjective up to units if for each b ∈ B, there is a unit u and anelement a ∈ A such that f(a)u = b. We can still take kernels of such maps.


Construction: Let P be the monoid of nonzero principal ideals of some domain A. LetM be the multiplicative monoid corresponding to A, without the zero element. There is aset map α : P → M which is a homomorphism up to units. α induces a setmap of ringsβ : ZP → ZM which is a homomorphism up to units. There is a surjective homomorphismγ : ZM → A with kernel a ⊆ ZM , so that ZM/a is isomorphic to A by the map inducedby γ. β induces a map δ : ZP/b → ZM/a which is an injective homomorphism up tounits. Every element in ZM/a is of the form a+ a for a ∈M , so δ is surjective up to units.Thus δ induces an isomorphism of calyxes f : (ZP/b)cal → (ZM/a)cal. Since (ZM/a) ∼= A,(ZP/b)cal ∼= Acal. Thus Acal ∼= [b] for some b ∈ (ZP )cal.

dil(Acal) ∼= dil((ZP/b)cal)?dil((ZP/b)cal) ∼= P ⇔ ZP is local with maximal ideal

Theorem. For which a is P the set of Dilworth principal ideals of (ZP/a)cal?

Rcal ∼= Scal and the group of units in R is isomorphic to the group of units in S impliesthe corresponding group rings are isomorphic up to units. (ZP/a) ∼= (ZM/b) up

R and S are the same quotient of the same group ring.Let R be a commutative domain whose units form an abelian group U . Let M be the

multiplicative monoid of R with the same underlying set, but with 0 taken out. Let P be theabelian monoid of nonzero principal ideals of R. Let F : Cmon→ Grp be the functor fromgroup monoids with cancelation to groups. M is an abelian monoid with cancellation, andso is P . There is a canonical exact sequence of Z-modules 0 → U ι→ F(M)

π→ F(P) → 0.Thus, with φ : P → F(P) the canonical morphism, M = (φπ)−1(P). Thus, given the unitsU and principal ideals P of a domain, we see the monoid of the ring is an element of

M ∩ (φ π)−1(P) : M ∈ ext1(U ,F(P))

Take a local noetherian ring R whose calyx is E. Eprinc is the set of principal elementsof R. Let M be the abelian monoid of R, let U be the units of R, and let P = Eprinc. M isin

N ∩ (φ π)−1(P) : N ∈ ext1(U ,F(P))

The map M → P induces a map ZM → ZPAcal ∼= Bcal, Acal/mAcal

Theorem (Krull’s Intersection Theorem). Let a be an element of a deciduous Noetheriancalyx E. Let b =

⋂n∈N a

n. Then ab = b, so that b = 0 when b ≤ Jac(E) by lemma 213.

Proof. If ab = 1, then b = 1 = ab. Otherwise, we can take a primary decompositionab =

⋂ni=1 aib where each ai is pi primary.

Take 1 ≤ i ≤ n. If a ≤ pi. Since E is noetherian, there is n ∈ N≥0 such that pni ≤ ai, sothat

b =⋂

m∈N≥0

am ≤ an ≤ pni ≤ ai

Suppose a 6≤ pi. Then there ζ ∈ a, ζ 6≤ pi. If b 6≤ ai then take η ≤ b, η 6≤ ai. Sinceηζ ≤ ab ≤ ai, η 6≤ b, we must have ζn ≤ ai for some n ∈ N≥0. Thus ζ ∈ rad(ai) = pi. Thusb ≤ ai ∀1 ≤ i ≤ n, so that b ≤ ab.


4.23. The Nilradical and the Jacobson Radical [18].

Definition 204. The nilradical nil(E) ∈ E of a calyx E is defined as the intersection of allthe prime elements in E. If nil(E) = 0 we say E is reduced.

Lemma 205. (nil(E)) = a ∈ E : an = 0

Proof. Suppose an = 0. Then a ≤ p for every prime p ∈ E, so that a ≤ nil(P ).

On the other hand, suppose an 6= 0 ∀n ∈ N. Localize at S = ai : i ∈ N and find amaximal element in PS. Take its image p under the monomorphism PS → P . p is prime asit is the image of a prime under a ring-lattice morphism. So a 6≤ nil(P ).

Definition 206. Define the radical of an ideal a ∈ A as ∩p∈[p]p. So nil([a]) = rad(A).

Lemma 207. The following are properties of radical ideals:

(1) rad(rad(a)) = rad(a)(2) rad(ab) = rad(a ∩ b) = rad(a) ∩ rad(b)(3) rad(a) = (1) ⇔ a = (1).(4) rad(a + b) = rad(rad(a) + rad(b))(5) If p is prime then rad(pn) = p.

Lemma 208. Let E be a calyx and a ∈ E an element. Let φ : E → [a] be the canonicalmap. rad(φ(b)) = φ(rad(b))

Lemma 209. Let a be an element of a calyx E. rad(a) = b ∈ E : bn ≤ a

Proof. Follows from the previous claim.

Definition 210. For a ring-lattice P define Jac(E) = ∩m maximalm.

Lemma 211. Let E be a calyx and a ∈ E an element. Let φ : E → [a] be the canonicalmap. Jac([a]) = φ(Jac(E))

4.24. Nakayama’s Lemma [19].

Lemma 212. Let (E,m) be a local calyx with finitely generated E-corolla M . If mM = Mthen M = 0.

Proof. Suppose for a contradiction that M 6= 0, while mM = M . Let ζ1, ..., ζn ∈ M beprincipal elements such that M =

∑ni=1 ζi in M , with n ∈ N≥1 minimal such that this is

possible. Since M = mM ,∑n

i=1 ζi = ζ1m +∑n

i=2 ζi. Let y =∑n

i=2 ζi. We pass to [y], in

which 1ζ1 = mζ1. ζ1 is principal in [y]. So ζ1 = 0. Thus ζ1 + y = y, so ζ1 ≤∑n

i=2 ζi,contradicting the minimality of n. Thus a = 0.

Lemma 213 (Nakayama’s Lemma). [not done] Let E be a calyx and let M be the set ofmaximal elements in E. Let M be a finitely generated E-corolla. If Jac(E)M = M thenM = 0.

Proof. Suppose ma = a for each m ∈M. Thus, ammm = am for each m ∈M, so that am = 0for each m ∈M since am is finitely generated. By lemma 162 a = 0.

Corollary 214. Let E be a calyx and let M be a finitely generated E-corolla with elementx ∈M . If M = x+ Jac(E)M , then x = M .


Proof. If M = x + Jac(E)M then x + M = x + Jac(E)M , so that [x] = Jac(E)[x]. Thus[x] = 0. We conclude x = M .

Corollary 215. If M is a finitely-generated corolla over a calyx E with principal elementsxii∈I . For each i ∈ I, let yi = xi + Jac(E)M be the image of xi in [Jac(E)M ]. If∑

i∈I yi = 1 in [Jac(E)M ], then∑

i∈I xi = 1 in M .

Proof. Suppose∑

i∈I yi = 1 in [Jac(E)M ]. Then∑

i∈I xi+Jac(E)M =∑

i∈I(xi+Jac(E)M) =∑i∈I(xi + Jac(E)M) = M so

∑i∈I xi = M by corollary 215.

4.25. Zerodivisors, the Corolla Quotient [24].

Definition 216. Let E be a calyx. ζ ∈ E is called a zerodivisor if ζa = 0 for some nonzeroa ∈ E. Denote the set of zerodivisors and 0 in E by Z(E).

Definition 217. Let E be a calyx. a ∈ E is called nilpotent if an = 0 for some n ∈ N. Anonzero nilpotent is a zerodivisor. We write nil(E) = a ∈ E : an = 0 and nilprinc.(E) =ζ ∈ E principal : ζn = 0.

Definition 218. For elements x, y ∈ M we define (x : y) = a ∈ E : ay ≤ x. The map(x : −) : M → Eop is left adjoint to the map (x : −)Eop →M .

Lemma 219. E ∼= Eop.

Proof. Consider the E-corolla map θ : Eop → E where a 7→ (0op : a). Take σ : E → Eop,x 7→ xop and τ : Eop → E, xop 7→ x.

τ θ σ(x) = (0op : aop)op = a1 = a

Thus τ θ σ is a lattice isomorphism, so that θ is a lattice isomorphism, so that it issurjective and injective. A surjective and injective corolla morphism is an isomorphism ofcorollas.

Definition 220. Let M be an E-corolla with structure map µ : E →M . An element a ∈ Eis said to be a zerodivisor of M if a 6= 0 and µ(a)∗ is injective. Let the set of zerodivisors ofM and zero itself be denoted by Z(M).

Definition 221. An element a ∈ E is said to be a co-zerodivisor of a corolla M withstructure map µ : E → M if it is a zerodivisor in the opposite module M op. To see this interms of requirements on M , observe:

⇔ µ(a)∗(xop) = µ(a)∗(y

op)⇒ xop = yop

⇔ µ(a)∗(xop)op = µ(a)∗(y

op)op ⇒ x = y

⇔ (x : a) = (y : a)⇒ x = y

Thus a nonzero element a ∈ E is a co-zerodivisor of a corolla M if and only if µ(a)∗ isinjective. Let the set of co-zerodivisors of M and 0 itself be denoted by Zop(M).

Definition 222. Let E be a calyx and M an E-corolla. We say a ∈ E is M -nilpotentif an1 = 0 ∈ M . The set of M -nilpotent elements of E is rad(ann(M)). In other wordsµ(a)n = 0 for some n ∈ N. Clearly if a is nilpotent then it is M -nilpotent.

Lemma 223. Nonzero M-nilpotent elements of E are zerodivisors.


Proof. Suppose a ∈ E is M -nilpotent. Then µ(a)∗ µ(a)∗ · · · µ(a)∗ = 0 so µ(a)∗ mustsend some nonzero element to 0. Thus a is a zerodivisor.

Lemma 224. Nonzero nilpotent elements of E are co-zerodivisors.

Proof. Suppose a ∈ E is nilpotent. Then µ(a)∗ µ(a)∗ · · · µ(a)∗ = 1 so µ(a)∗ must sendsome element x 6= 1 to 1.

Definition 225. For a corolla M , define

Z(M) = a ∈ E : ax = 0 for some nonzero x ∈M

4.26. Supports and Annihilators [39].

4.27. Valuation Rings [49]. discrete valuation rings and dedekind domains (example andclassification)

Lemma 226. Let A be a valuation ring with surjective valuation v : A → Ω. Take idealsa, b ∈ A.

v(ab) = v(x) + v(y) : x ∈ v(a), y ∈ v(b) = v(a) + v(b)

Proof. Suppose t ∈ v(ab). Then t = v (∑n

i=1 xiyi) ≥ min v(xiyi) : 1 ≤ i ≤ n = min v(xi) + v(yi) : 1 ≤ i ≤ nfor xi ∈ a, yi ∈ b, so that t ∈ v(a) + v(b). On the other hand, suppose t ∈ v(a) + v(b). Thent = v(x) + v(y) = v(xy) ∈ v(ab) for x ∈ a, y ∈ b.

Lemma 227. Let A be a valuation ring with surjective valuation v : A → Ω. Take idealsa, b ∈ A. Let H be the upper closure of max(v(x) − v(y), 0) : x ∈ v(a), y ∈ v(b). Thenv((a : b)) = H.

Proof. Suppose t ∈ v((a : b)). Then t = v (x) for x ∈ A such that x ⊆ (a : b). For eachy ∈ b, there is z ∈ a such that v(xy) ≥ v(z). Then for each y ∈ b, there is z ∈ a such thatt+v(y) = v(xy) ≥ v(z), so that t ≥ v(z)−v(y), so that t ≥ maxv(z)−v(y), 0, so that t ∈ H.

Conversely, suppose t ∈ H. Then t+ v(y) ≥ v(x) for some y ∈ b, x ∈ a.So t ≥. for xi ∈ a, yi ∈ b, so that t ∈ v(a) + v(b). On the other hand, suppose

t ∈ v(a) + v(b). Then t = v(x) + v(y) = v(xy) ∈ v(ab) for x ∈ a, y ∈ b.

Theorem. Suppose A is a valuation ring with valuation v : A→ Ω, with Ω a totally orderedabelian group. Then, for any a ∈ A, aA ∈ Acal is Dilworth principal.

Proof. Let ζ = aA and put v(a) = t. We show that µ(ζ)∗ is injective (note that A is adomain so that (0 : ζ) = 0). Take ideals a and b in A such that ζa = ζb. Then

t+ v(x) : x ∈ a = v(ζa) = v(ζb) = t+ v(y) : y ∈ bso that v(a) = v(b), so that a = b.

Next we show that µ(ζ)∗ is injective on the set of ideals contained in ζ = ζA. Take idealsa and b in A such that (a : ζ) = (b : ζ) and such that a, b ⊆ ζ. Then

max(v(x)− t, 0) : x ∈ a = v((a : ζ)) = v((b : ζ)) = max(v(y)− t, 0) : y ∈ bSince a, b ⊆ ζ, max(v(x)− t, 0) = v(x)− t for each x ∈ a and max(v(y)− t, 0) = v(y)− t foreach y ∈ b. This gives

v(x)− t : x ∈ a = v(y)− t : y ∈ b


so that

v(x) : x ∈ a = v(y) : y ∈ bi.e. v(a) = v(b). Thus a = b.

4.28. Primary Decomposition. Credit to http://www.math.uiuc.edu/ r-ash/ComAlg/ComAlg1.pdffor the proofs I analogized.

Set a calyx E with corolla M throughout.

Definition 228. Let M be an E-corolla. M is primary if Z(M) ⊆ rad(ann(M)). In otherwords, M is primary if for every a ∈ E greater than 0 such that ax = ay for some x 6= y ∈M ,an1 = 0 in M . We say x ∈ M is p-primary if x < 1 and [x] is a primary E-module andrad(annM((x))) = p, where ann[x] is the annihilator of x in the modu. Note that the oper-ations in [x] are possibly distinct from the operations in M .

Specializing to the case where M = E, we say E is primary if Z(E) ⊆ rad(E). In otherwords, E is primary if for every a ∈ E greater than 0 such that ab = ac for some b 6= c ∈ E,anE = 0 in E. We say a ∈ M is p-primary if a < 1 and [a] is a primary E-module andrad(ann(a)) = p.

M is secondary if Zop(M) ⊆ radopM(M). In other words, M is secondary if for every a > 0in E such that (x : a) = (y : a) for x, y ∈ M with x 6= y, an1 = 0 in M . We say x ∈ M isp-secondary if x > 0 and (x) is a primary E-module and rad(annop(x)) = p. Note that theoperations in [x] are possibly distinct from the operations in M .

E is secondary if Zop(E) ⊆ rad(E). In other words, E is secondary if for every a > 0in E such that (b : a) = (c : a) for b, c ∈ M with b 6= c, an1 = 0 in M . We say a ∈ M isp-secondary if a > 0 and (a) is a primary E-module and rad(annop(a)) = p.

Lemma 229. Take a ∈ E and suppose m = rad(a) is a maximal element in E. Then a ism-primary.

Proof. Take x ∈ E with x /∈ m and suppose xy + a = xz + a for y, z ≥ a. Let h = xy + a =xz + a. x+ m = 1. Take k ∈ N such that mk ⊆ a. So

y = 1ky = (m + x)ky ≤ (mk + x)y ≤ (a + x)y ≤ a + xy = h

and similarly z ≤ h. But h = x · y ⊆ y and h = x · z ⊆ z where · is multiplication in [a]. Soy = z.

Corollary 230. If m is a maximal ideal then mn is m primary for each n ∈ N≥1.

Definition 231. A primary decomposition of an element x ∈ M is given by x = ∩ni=1xiwhere xi are pi-primary elements. The decomposition is reduced if pi are distinct and x isnot the intersection of any proper subcollection of xi. A secondary decomposition of anelement x ∈M is given by x =

∑ni=1 xi where xi are pi-secondary. Again, the decomposition

is reduced if pi are distinct and x is not the sum of any proper subcollection of xi.

Lemma 232. If x, y are p-primary, then x ∩ y is p-primary.


Proof. Let z = x ∩ y. p = rad(ann(x)) = rad(ann(y)).

To show that rad(ann(z)) = p, we show that p ⊆ rad(ann(z)) and rad(ann(z)) ⊆ p.The second is clear. Take a ∈ p. There are n,m ∈ N such that an1 ⊆ x and am1 ⊆ y. Thusan+m ⊆ x ∩ y = z, so that a ∈ radM(z).

To show that z is primary, note that

Z([z]) = Z([x ∩ y]) ⊆ Z([x]) ∩ Z([y]) ⊆ rad(ann(x)) ∩ rad(ann(y)) = rad(ann(z))

Lemma 233. If x, y are p-secondary, then x+ y is p-secondary.

Lemma 234. If x ∈M is irreducible and M is Noetherian then x is primary.

Proof. Suppose x is not primary. Then for some a ∈ E, the canonical map µ : [x] → [x]which sends y to ay + x is not injective. Consider the ascending chain

ker(id) ⊆ ker(µ) ⊆ ker(µ2) ⊆ · · · ⊆ ker(µn) ⊆ · · ·

Since M is noetherian the sequence terminates, say at ker(µn) for some n ∈ N. Let φ = µn.ker(φ) = ker(φ2). If x ∈ ker(φ)∩ im(φ) then x = φ(y) for some y ∈M φ2(y) = φ(x) = 0 soy ∈ ker(φ2) = ker(φ), so x = φ(y) = 0. ker(φ) is not injective, so ker(φ) 6= 0, and µ is notnilpotent, so im(φ) is nonzero. Thus [x] is irreducible.

Lemma 235. If x ∈M is coirreducible and M is artinian then x is secondary.

Theorem (Existence of Primary Decompositions). By lemma 234 and lemma 232, everyelement x ∈M of a noetherian corolla has a reduced primary decomposition.

Theorem (Existence of Secondary Decompositions). By lemma 235 and lemma 233, everyelement x ∈M of an artinian corolla has a reduced secondary decomposition.

Definition 236. A prime p of E is called an associated prime of M if p = ann(x) for someprincipal element x ∈ M . Equivalently, if there is a perennial corolla map E → M whosekernel is p. We denote the set of associated primes by Ass(M).

Definition 237. A prime element p of E is called a co-associated prime of M if p = annop(x)for some coprincipal element x ∈M . Reminder: annop(x) is defined as (x : 1). Equivalently,if there is a perennial corolla map M → Eop whose cokernel is p. Equivalently, if there isa perennial corolla map E → M op whose kernel is p. We denote the set of co-associatedprimes by Assop(M).

Lemma 238. A maximal element of ann(x) : x ∈ M,x 6= 0 is a prime ideal. If E isnoetherian and nonzero, then Ass(M) 6= .

Proof. Suppose ann(x) is maximal and take a, b ∈ E such that ab ≤ ann(x). Supposea, b 6≤ ann(x). Then ax 6= 0. But then ann(ax) 6= 1 and ann(ax) contains b and ann(x),contradicting the maximality of ann(x).

Lemma 239. A maximal element of annop(x) : x ∈ M,x 6= 1 is a prime ideal. If E isnoetherian and nonzero, then Assop is nonempty.


Proof. Suppose annop(x) is maximal and take a, b ∈ E such that ab ≤ annop(x). Supposea, b 6≤ annop(x). Then (x : a) 6= 1. But then annop((x : a)) 6= 1 and annop((x : a)) continasb and annop(x), contradicting the maximality of annop(x).

Lemma 240. For any element x ∈M , Ass((x)) ⊆ Ass(M) ⊆ Ass((x)) ∪ Ass([x]).

Proof. Clearly Ass((x)) ⊆ Ass(M). Suppose p = ann(y) for y ∈M but ann(y) /∈ Ass((x)).We show that ann(y) = (x : y) ∈ Ass([x]). Clearly ann(y) ⊆ (x : y) and if a ∈ A hasay ≤ x, ay 6= 0, then a /∈ ann(x). Since ann(x) is prime,

b ∈ ann(ax) ⇔ bb ∈ ann(x) ⇔ b ∈ ann(x)

Thus ann(x) = ann(ax) ∈ Ass((x)), a contradiction. So ass(M) ⊆ Ass((x))∪Ass([x]).

Lemma 241. For any element x ∈M , Assop((x)) ⊆ Assop(M) ⊆ Assop((x)) ∪ Assop([x]).

Lemma 242. Suppose 0 = x0 ≤ x1 ≤ · · · ≤ xn = M is a chain of elements in an E-corolla M . Then Ass(M) ⊆ ∪ni=1Ass([xi, xi+1]) and Assop(M) ⊆ ∪ni=1Ass

op([xi, xi+1]) .Suppose M1, ...,Mn are E-corollas with M = ⊕ni=1Mi. Then Ass(M) = ∪ni=1Ass(Mi) andAssop(M) = ∪ni=1Ass

op(Mi).

Proof. Follows from induction on the previous claim.

Lemma 243. If p ∈ E is prime then AssE([p]) = p.

Proof. Clearly p ∈ AssE([p]). For the other direction it suffices to show that ann(x) = p foreach nonzero x ∈ [p]. Take nonzero x ∈ [p] and suppose yx = 0. Then y ≤ p since x 6≤ p.

Lemma 244. If p ∈ E is prime then AssopE ((p)) = p.

Remark. For any prime p, p ∈ Z(M) and p ∈ Zop(M).

Theorem. Let M be a nonzero deciduous Noetherian corolla. Express the zero module 0 asthe reduced intersection of primary elements xini=1 in M with rad(ann(xi)) = pi for primespi. Ass(M) = pini=1.

Proof. Let p ∈ AssE(M) and take a principal element x ∈ M such that x 6= 0, p = ann(x).Reorder the xi such that x ≤ xi for 1 ≤ i ≤ j and x ≤ xi for j + 1 ≤ i ≤ n. Take inte-

gers nini=1 such that pni ≤ ann(xi). Then pni 1 ≤ xi so that(⋂j

i=1 pnii

)x ≤

⋂ni=1(pnii x) ≤⋂n

i=1 xi = 0. Thus⋂ji=1 p

nii ≤ ann(x) = p, so that pi ≤ p for some 1 ≤ i ≤ j. We show

that p ≤ pi. Take a ≤ p. Then ax = 0 and x 6≤ xi, so that the map φ : [xi] → [xi] whichmultiplies by a is not injective. Since xi is primary, φ is nilpotent. So a ≤ rad(ann(xi)) = pi.

Conversely, each pi is an associated prime. Without loss of generality we take i = 1.Since the chosen primary decomposition of 0 is minimal,

⋂ni=2 xi 6≤ x1. Choose x ≤

⋂ni=2 xi

with x 6≤ x1. Take n ∈ N such that pn1x ⊆ x1 but pn−11 x 6⊆ x1 (take p0

1 = E). Take a principalelement y ≤ pn−1

1 x such that y 6≤ x1. We show that p1 = ann(y). p1y ⊆ p1pn−11 x = pn1x.

Since x ≤⋂ni=2 xi, so p1y ⊆

⋂ni=2 xi. Thus p1y ⊆

⋂ni=1 xi = 0, so p1 ⊆ ann(y). On the other

hand, if a ∈ E and ay = 0, then ay ∈ x1 but y /∈ x1, so that the map φ : [x1] → [x1] whichmultiplies by a is not injective and therefore is nilpotent. Thus a ∈ rad(ann(x1)) = p1.

Corollary 245. Let M be a Noetherian deciduous corolla. If x =⋂i∈I xi is a reduced

primary decomposition of x, and xi is pi-primary, then pi are uniquely determined by x.


Lemma 246. Suppose every element of E is the sum of finitely many principal elementsand S is a localization set in E. Then

AssS−1E(S−1M) = S−1(p) : p ∈ Ass(M) : ζ /∈ p ∀ζ ∈ SProof. Suppose p ∈ Ass(M) and ζ 6≤ p ∀ζ ∈ S. Then there is a perennial exact sequence0→ p→ E → M of E-corollas. Thus 0→ S−1p→ S−1E → S−1M is an exact sequence ofS−1E corollas. Since ζ 6≤ p ∀ζ ∈ S, S−1p is a prime of S−1E. Thus S−1p is an associatedprime of S−1M .

Convsersely, suppose q ∈ AssS−1E(S−1M). Then q = S−1p for some p ∈ Spec(E) withζ 6≤ p∀ζ ∈ S and q = ann(xS) for some s ∈M . Write q =

∑ni=1 ζi where ζi are principal.

Write µi : E → M for the canonical corolla morphism sending 1 to ζi. Let µx : E → Msend 1 to x.

Eµi→ E →M → S−1M

4.29. Dimension [59].

4.30. Associated Primes [62]. see stacks project

Definition 247. Let E be a calyx and M an E-corolla. A prime p ∈ E is called an associatedprime of M if there is a principal element ζ ∈ M such that ann(ζ) = p. We denote the setof all such primes as AssE(M) or just Ass(M).

Lemma 248. Let E be a calyx and M an E-corolla. Ass(M) ⊆ Supp(M).

Proof. Take p ∈ AssE(M). If xp = 0p in Mp then (x : ζ) = 0 for each principal elementζ ∈Mp. Thus ζx ≤ ζ(x : ζ) = 0, so that ζ ∈ ann(x). Thus xp 6= 0.

Lemma 249 (not done). Let E be a calyx and take an exact sequence 0→ Nf→M

g→ L→ 0of E-corollas. Ass(M) ⊆ Ass(N) ∪ Ass(L).

Proof. Take p ∈ Ass(M) and an element x ∈M such that ann(x) = p. If x ≤ f∗(1) then p ∈Ass(N). Otherwise we show p ∈ Ass(L). It suffices to show that ann(x) = ann(g∗(x)).

4.31. Completion [95]. c.f. the stacks project.

Definition 250. Let E be a calyx with element a. Form the poset category Z≤0, oredered inthe usual way. Define a functor Φ : Z≤0 → Cal where Φ(n) = [a−n] ∀n ∈ Z≤0. We define fora morphism α : n→ m set Φ(α) to be the canonical quotient morphism [a−n]→ [a−m]. From

this functor Φ we form the completion of E with respect to a: E = lim←−Φ. An element b of Ecan be represented as a sequence bnn≥0 where bn ∈ [an] for each n ∈ N≥0 and bn = bn+1 in

[an]. There is a canonical morphism E → E. IfM is an E-corolla, we define the completion asfollows: define a functor Φ : Z≤0 → E− -Cor where Φ(n) = [a−nM ] ∀n ∈ Z≤0. We define fora morphism α : n→ m set Φ(α) to be the canonical quotient morphism [a−nM ]→ [a−mM ].

From this functor Φ we form the completion of E with respect to a: E = lim←−Φ. An element

x of M can be represented as a sequence xnn≥0 where xn ∈ [anM ] for each n ∈ N≥0 and

xn = xn+1 in [anM ]. We can view M as an E corolla. Thus there are canonical maps M → M

and therefore M ⊗E R→ M . Moreover, completion forms a functor E-Cor→ E-Cor.


why can we view M as an E corolla?

Lemma 251.

4.32. Regular Local Rings [105].

Definition 252. We say a local Noetherian calyx E is a regular local calyx if dimI(m/m2) =

dim(E).

Definition 253 (The Discrete Valuation Calyx). The discrete valuation calyx is the calyxwhose underlying lattice N≥0 is ordered such that n ≥ m when n ≥ m as integers (fornonzero n,m) and 0 ≤ n ∀n ∈ N. Multiplication nm of nonzero elements is defined as sum,and we set 0n = 0 for all n ∈ N≥0. The prime elements of a discrete valuation calyx are 2and 0.

Example 254. The

Lemma 255 (not done). Any regular local ring is a domain.

Proof. By Krull’s intersection theorem, lemma 4.22,⋂n∈N≥0

= 0. Let a, b ∈ E be such that

ab = 0. Take n,m ∈ N≥0 maximal such that a ∈ mn, b ∈ mm.

Theorem. A calyx E is a regular local calyx of dimension 1 if and only if it is isomorphicto the discrete valuation calyx N

Proof. One direction is obvious. So let (E,m) be a local calyx of Krull dimension one, withI ∼= [m2,m]. m 6= m2, so 0 ≤ m2 < m. In particular 0 6= m. Take a principal element ζ ≤ mwith ζ 6≤ m2. Then ζ+m2 = m. By corollary 215 ζ = m. So m is principal. By theorem 4.22⋂n∈N m

n = 0. Take a ∈ E not equal to 1. If a ≤ mn for each n ∈ N≥0 then a = 0. Otherwise,a 6≤ mn for some n ∈ N≥0. Take n minimal such that a 6≤ mn+1. (a : ζn) 6≤ (ζn+1 : ζn) = ζsince ζ is principal. Thus (a : ζn) 6≤ m, so that (a : ζn) = 1. Then a = ζn.

Guess: a regular local calyx of dimension d is isomorphic to I[[x1, ..., xn]] for some n.is a regular local calyx automatically complete?

5. Schemes

5.1. Intrinsic Definitions. credit to hartshorne.

Definition 256. Let E be a cupulate calyx. Recall that for a prime ideal p in E, Ep isthe localization of E at p, and that, for an element s ∈ E, Es is the localization at themultiplicative set sn : n ∈ N≥0. For an open set U ⊆ Spec(E), define EU to be the set offunctions φ : U → qp∈UEp such that φ(p) ∈ Ep and such that φ is locally an element of E.More precisely, we require that for each p ∈ U there is a neighborhood V ⊆ U of p and anelement a ∈ E such that φ(q) = φq(a) for each q ∈ V , where φq : A → Ap is the canonicalmap. This construction determines a calyx EU .

Definition 257. Let E be a calyx. The spectrum of E is a pair consisting of of the topo-logical space Spec(E) together with the sheaf of rings Ω where Ω(U) = EU . It should beclear what the restriction maps are.


Lemma 258. Let E be a calyx with elements ζ1, ..., ζn ∈ E such that∑n

i=1 ζi = E. Formthe poset category Λ = 0, 1n − (0, 0, ..., 0) ordered where (a1, ..., an) ≤ (b1, ..., bn) whenai ≤ bi ∀1 ≤ i ≤ n for aini=1, bini=1 in 0, 1. Form the functor Φ : Λ → Cal where(a1, ..., an) 7→ E q

1≤i≤n,ai=1ζiand 0 7→ Eζ1···ζn. For each morphism λ : (a1, ..., an) ≤ (b1, ..., bn)

in Λ, let Φ(λ) : E q

1≤i≤n,ai=1ζi→ E q

1≤i≤n,bi=1ζibe the canonical localization map. We show

that lim←−Φ ∼= E.

Proof. The canonical localization maps E → Φ(a) for a ∈ Obj(Λ) form a cone for Φ, andthus a map φ : E → lim←−Φ such that E → lim←−Φ → Φ(a) = E → Φ(a) for each a ∈ Obj(Λ).We construct an inverse map ψ : lim←−Φ→ E for φ. Write Φa≤b for Φ(λ) where λ : a→ b is amorphism in Λ. Take a collection of objects aa ∈ Φ(a) such that Φa≤b(aa) = ab for each paira ≤ b in Λ. Let Φ([aaa∈Λ]) =

⋂a∈A aa.

Clearly ψ φ = id.

To show φ ψ = id, take [aaa∈Λ] in Λ and let x = Φ([aaa∈Λ]). Write ai for theelement of Λ which is 1 in the ith slot and 0 elsewhere, and write ai for aai . Observe that(aj : ζi) = (ai : ζj), so that

(x : ζj)

=

(⋂a∈Λ

aa : ζj

)

=

(n⋂i=1

ai : ζj

)

=n⋂i=1

(ai : ζj)

=n⋂i=1

(aj : ζi)

=

(aj :

n∑i=1

ζi

)=(aj : 1) = aj

Theorem. Let E be a cupulate calyx (recall definition 143) and let (Spec(E),Ω) its spectrum.For any principal element ζ ∈ E, the calyx Ω(D(ζ)) is isomorphic to Eζ. In particular, takingζ = 1, Ω(Spec(E)) = Eζ.

Proof. Passing to (D(ζ),Ω|D(ζ)), which is also cupulate, it suffices to show that Ω(Spec(E)) ∼=E. Define φ : E → Ω(Spec(E)) by sending a to the section α on E assigning to each primep the element ap ∈ Ep.


To show φ is injective, take a, b ∈ E such that ap = bp for each p in Spec(E). a = b bylemma 167.

Take α ∈ Ω(Spec(E)). Cover Spec(E) with open sets Uii∈I such that there are aii∈Iin E with α(p) = (ai)p for p ∈ Ui. Without loss of generality, Uii∈I can be taken to beprincipal open sets D(ζi), since such sets form a basis for Spec(E). Since 1 ∈ E is compact,Spec(E) is compact as a topological space, so that I can be taken to be a finite set.

We aim to show there is a ∈ E with ap = α(p) for each prime p ∈ Spec(E).

First we show by induction on |F | that for each finite F ⊆ I there is a ∈ E∑i∈F ζi

suchthat aζi = α(p). The case for |F | = 1 is automatic. Suppose the claim is true for some finiteset F ⊆ I and take i ∈ I. Write ζ = ζi and η =

∑j∈F ζj. Passing to Eζ+η we have ζ + η = 1

and ζη = ζ ∩ η. Let a ∈ Eζ be such that ap = α(p) for p ∈ D(ζ). Let b ∈ Eη be such thatbp = α(p) for each p ∈ D(η). Then bp = ap for p ∈ D(ζ) ∩D(η). The following diagram ofcanonical maps commutes:

E Eζ

Ω(Spec(E)) Ω(D(ζ))

Eη Eζη

Ω(D(η)) Ω(D(ζ) ∩D(η))

ω

Take x = b ∩ a. Since (a : η) = (b : ζ), we have

(x : ζ) = (a ∩ b : ζ) = (a : ζ) ∩ (b : ζ) = (a : ζ) ∩ (a : η) = (a : ζ + η) = a

and

(x : η) = (a ∩ b : η) = (a : η) ∩ (b : η) = (b : ζ) ∩ (b : η) = (b : ζ + η) = b

With ω as in the diagram above, ω(x) = α|Spec(E), as claimed.

Now the claim follows from lemma 172.

Lemma 259. Take a calyx E and let U be an open set of Spec(E). Let Ω be the structuresheaf of Spec(E). Ω(U) ∼= E∑

ζ∈Eprinc.,ζ 6≤p ∀p∈U ζ.

Proof. By lemma 172

Ω(U) ∼= lim←−ζ 6≤p ∀p∈U

Ω(D(ζ)) ∼= lim←−ζ 6≤p ∀p∈U

Eζ ∼= E∑ζ 6≤p ∀p∈U

Lemma 260. Let E be a calyx and (Spec(E),Ω) its spectrum. Then the stalk Ωp is isomor-phic to Ep.


Proof. For each p ∈ Spec(E) let Pp be the set of principal elements not contained in p.Ωp∼= lim−→ζ∈Pp

Ω(D(ζ)) ∼= lim−→ζ∈PpEζ ∼= Ep by lemma 173.

Definition 261. A sheaf of calyxes is called a calical sheaf. A morphism (X,ΩX)→ (Y,ΩY )is a pair (f, f#) where f : X → Y is a continuous map of topological spaces and f# : ΩY →ΩX f−1 is a morphism of sheaves, where f−1 is viewed as a poset functor from the topologyof Y to the topology of X. Thus the class of calical sheaves with calical sheaf morphismsforms a category. Write f for a morphism (f, f#) of calical sheaves.

Definition 262. Let (X,Ω) be a calical sheaf. If furthermore Ωp is a local calyx for eachp ∈ X then we say (X,Ω) is a locally calical sheaf or a locally calical space. The morphisms oflocally calical spaces are not arbitrary morphisms of calical sheaves, but instead morphisms(f : (X,ΩX) → (Y,ΩY ) such that for each p ∈ X, the canonical map k : ΩXp → ΩY f(p) hask∗(mY ) = mX where mX is the maximal element in ΩXp and mY is the maximal element inΩY p. This construction forms a category.

Lemma 263. The forgetful functor Φ from locally calical spaces to topological spaces dis-tributes over limits and colimits.

Lemma 264. The forgetful functor Φ from locally ringed spaces to topological spaces dis-tributes over limits and colimits.

Lemma 265. There is a covariant functor Φ from calyxes to locally calical spaces.

Proof. It should be clear how Φ acts on objects. We construct Φ(f) for a morphism of calyxesf : E → F as follows: define α : Spec(F )→ Spec(E) where α(p) = f∗(p). Then

α−1(V (a))

=p ∈ F : α(p) ∈ V (a)=p ∈ F : a ≤ f∗(p)=p ∈ F : f∗(a) ≤ p=V (f∗(a))

so that α is continuous. For each p ∈ Spec(F ), f induces a canonical map fp : Af∗(p) →Bp. Now for any open set U ⊆ Spec(E), there is a calyx morphism α# : Spec(E)(U) →α∗Spec(F )(U) where α#(s)∗(p) = fp∗(s(α(p))). This makes (α, α#) a map of locally calicalspaces.

Lemma 266 (not done yet). The functor Φ as in ?? is full and faithful.

Proof. It should be clear that Φ is faithful. Take a morphism of locally calical spaces(α, α#). α# induces a map α : ΩSpec(E)(Spec(E)) → ΩSpec(F )(Spec(F )). By lemma 5.1ΩSpec(E)(Spec(E)) ∼= E and ΩSpec(F )(Spec(F )) ∼= F . Thus α : E → F . Similarly, by lemma260, the map of stalks αp : (ΩSpec(E))α(p) → ΩSpec(F )p

is really a map αp : Ep → Fp.

Remark. Let (X,Ω) be a scheme. Since the underlying topological space of Φ(X) is the sameas the underlying topological space of X, Φ(X) is determined by Φ(X)p for primes p.

Definition 267. An affine carpel is a locally calical space (X,ΩX) which is isomorphic tothe spectrum of some calyx. A carpel is a locally calical space (X,ΩX) such that everypoint p has an open neighborhood Up such that (U,ΩX |U) is an affine calical space. X is the


underlying topological space of the carpel (X,ΩX) and ΩX is its structure sheaf. The classof carpels forms the class of objects of a category whose morphisms are morphisms of locallycalical spaces. Thus carpels form a full subcategory of the category of calical spaces.

Example 268. Take the calyx I = 0, 1, the unique domain calyx of dimension 0. Spec(I)has an underlying space consisting of a single point, ∗. The structure sheaf Ω of Spec(I)has Ω(∗) = I and Ω() = 0.

Lemma 269 (Coproducts in the category of locally calical spaces). . Let Si = (Ui,Ωi)i∈Ibe locally calical spaces. Form the sheaf S = qi∈ISi = (qi∈IUi, )i∈I ,Ω) where Ω(qi∈IVi) =

(σi)i∈I : σi ∈ Ωi(Vi). There are inclusion maps (αi, α#i ) : Si → S where αi : Ui → qi∈IUi

is the inclusion map and α#i : Ω→ α∗Ωi sends (sj)j∈I to si.

Lemma 270 (Pushouts in the category of locally calical spaces). .

Lemma 271.

5.2. Some Examples.

Example 272. Consider the calyx I. Spec(I) consists of a single point 0, and the local-ization of I at 0 is I, so that Ω0

∼= I where Ω is the structure sheaf of Spec(I).

Example 273. Let A be a discrete valuation ring. Acal ∼= F where F = N≥0 ∪ ∞ has theusual order of N, where sum in F is maximum of integers, intersection in F is minimum ofintegers, and product in F is sum of integers. F has two prime ideals, 0 and m where m ismaximal in A.

Example 274. Let E ∼= Zcal be the calyx corresponding to the integers. The prime elementsof E are the prime ideals (0), (2), (3), (5), (7), ... in Z. Of these, the nonzero prime idealsare all maximal and therefore closed. The ideal (0) is a generic point for Spec(E). Thelocalization of E at each element is a discrete valuation ring and therefore isomorphic to thecalyx N≥0 ∪ ∞.

Example 275. Similarly, k[x]cal (with k algebraically closed) is a dimension 1 calyx withprime ideals (x− a) for a ∈ k, and 0. Closed sets in k[x] correspond to finite sets of nonzero

prime ideals and the set of all elements. 0 is a generic point for k[x]cal.

Example 276. More generally, take any dedekind domain A. Acalp is a discrete valuationring for each prime ideal p ⊆ A. The topology on Spec(A) has as closed sets finite sets ofpoints and the whole set.

Example 277 (not done). Let S be a smooth algebraic curve over an algebraically closedfield.

5.3. Projective Calical Schemes.

5.4. First Properties of Schemes. (off of hartshorne) c.f. chapter 3.

Definition 278. A calical scheme is connected if its topological space is connected. A calicalscheme is irreducible if its underlying topological space is irreducible.

Lemma 279. Let X = Spec(E), the spectrum of a calyx. X is irreducible if and only ifnil(E) is prime.


Proof.

nil(E) is irreducible

V (a) ∪ V (b) = V (0)⇒ V (a) = 0 or D(b) = 0

V (ab) = V (0)⇒ V (a) = 0 or V (b) = 0

ab ≤ nil(E)⇒ a ≤ nil(E) or b ≤ nil(E)

Definition 280. A calical scheme X is reduced if for every open set U , the intersection ofprimes in the calyx ΩX(U) is 0.

Lemma 281 (not done). Let X = Spec(E), the spectrum of a calyx. X is reduced if andonly if nil(E) = 0.

Proof. One direction is obvious. Conversely, suppose nil(E) = 0 and take an open setD(a) ⊆ X.

Definition 282. A scheme X is integral if for every open set U ⊆ X, the calyx ΩX(U) is adomain.

Lemma 283. A calical scheme X is integral if it is reduced and irreducible.

Proof. An integral calical scheme is clearly reduced when it is integral.

Suppose X is reducible. Then X can be expressed as the disjoint union of open sets Uand V . ΩX(X) ∼= ΩX(U)

q

ΩX(V ).

Suppose X is reduced and irreducible. Let U ⊆ X be an open subset and supposea, b ∈ Ω(U) have ab = 0. Let Y = p ∈ U : ap and let Z = p ∈ U :.

5.5. Smooth Schemes.

5.6. Separated and Proper Morphisms.

5.7. Sheaves of Corollas. (taken from hartshorne)

Let (X,ΩX) be a calical space. A sheaf of ΩX-corollas.

5.8. Divisors.

5.9. Projective Morphisms.

5.10. Differentials.

5.11. Formal Calical Schemes.

6. Cohomology of Calical Sheaves

6.1.


References

[1] R. P. Dilworth: Abstract Commutative Ideal Theory, Pacific Journal of Mathematics, 12, 481-498 (1962).[2] D. D. Anderson: Abstract commutative Ideal Theory Without Chain Condition, Algebra Universalis, 6,

131-145 (1976).[3] D. D. Anderson, E. W. Johnson, Dilworth’s Principal Elements, Algebra Universalis, 36, 392-404 (1996).

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CALYXES AND COROLLAS - College of LSA | U-M LSA U-M ... · monoids by Mlat. Note that, while the monoid Mmay not be commutative in general, joins and meets in Mlat are still commutative