Commutative Algebra of Equivariant Cohomology Rings€¦ · Introduction Example Let F be a graded ring. The following are equivalent: Every nonzero homogeneous element in F is invertible

3.20pt

Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring 2017 1 / 45

Commutative Algebra of Equivariant Cohomology Rings

Mark Blumstein

Spring 2017


Introduction

Overview

Objects of interest: Topological spaces with group actions

Study these spaces via equivariant cohomology rings

Consider commutative algebra in graded category of modules

Results on prime spectrum, Krull dimension, localization

Main theorem: Formula to compute a number (the degree) associatedto an equivariant cohomology ring


Introduction

The Category grmod(A)

A is a ring with properties: Z-graded, commutative, unital,Noetherian

Objects of grmod(A): Finitely generated Z-graded A-modules

Morphisms of grmod(A): Degree preserving A-modulehomomorphisms

• In general, equivariant cohomology rings have non-standard gradings.i.e. not generated by degree 1 elements over the degree 0 part.


Introduction

Example

Let F be a graded ring. The following are equivalent:

Every nonzero homogeneous element in F is invertible.

F0 is a field and either F = F0, or there exists a d > 0 and an x ∈ Fdsuch that F ∼= F0[x , x−1] as a graded ring. In fact, in this last case,d > 0 is the smallest positive degree with Fd 6= 0.

The only graded ideals in F are F and 0.

A ring satisfying any of these three equivalent conditions is called agraded field.


Basic Commutative Algebra Results

Let A be a Z-graded Noetherian ring with graded ideals I and I, and letM ∈ grmod(A).

A0 is Noetherian and A is a finitely generated A0-algebra by a set ofhomogeneous elements

∗V (I) = ∗V (I) if and only if√I =

√I

The associated primes of M are graded


Basic Commutative Algebra Results

Lemma

For M ∈ grmod(A), there exists a graded filtration F of M,

0 = M0 ⊂ M1 ⊂ · · ·MN−1 ⊂ MN = M,

with the property that for each 1 ≤ n ≤ N, there exist graded primes pisuch that,

(A/pi )(di ) ∼= M i/M i−1.

Let SF be the set of primes(graded) which appear in the filtration, then

{minimal primes of M} ⊆ AssA(M) ⊆ SF ⊆ ∗SuppA(M)


Graded Krull Dimension and Length

Lemma

If A is a Z-graded Noetherian ring, then

∗ dim(A) ≤ dim(A) ≤ ∗ dim(A) + 1;

therefore if M ∈ grmod(A),

∗ dimA(M) ≤ dimA(M) ≤ ∗ dimA(M) + 1.


Graded Krull Dimension and Length

Lemma

Let S be a positively graded ring, and M ∈ grmod(S), m a graded ideal inS . Then,

m *maximal if and only if m maximal

∗ dimS(M) = dimS(M)

∗`S(M) = `S(M)


Localization

Different Methods

Usual Localization

Graded Localization

The degree 0 part

Construction

Graded localization: take a multiplicatively closed subset T ⊆ Acontaining entirely homogeneous elements. Construct localizationT−1M as usual.

Graded localization at a prime p denoted M[p]

Degree zero localization (T−1M)0.

Degree zero localization at a prime denoted M(p)


Results Comparing Localization Methods

Positively Graded Case

Graded primes p ∈ Proj(S), q 6∈ Proj(S)

dim(S[p]) = ∗ dim(S[p]) + 1

dim(S[q]) = ∗ dim(S[q])

dim(S(p)) = ∗ dim(S[p])

= dim(S[p])− 1


Results Comparing Localization Methods

Z-graded Case

p ∈ Spec(A) graded, minimal

∗À[p](M[p]) = Àp(Mp)

À(p)(M(p)) ≤ ∗À[p]

(M[p])

= Àp(Mp)

If there is a homogeneous element of degree 1 (or , equivalently, -1)in A− p

À(p)(M(p)) = ∗À[p]

(M[p]) = Àp(Mp)


Poincare Series

Definition

Suppose S is a positively graded ring, with S0 Artinian, and M is a finitelygenerated graded S-module. Then the Poincare series of M is the formalLaurent series with integer coefficients

PM(t) =∑i∈Z

`S0(Mi )ti ,

where `S0(Mi ) is the length of the finitely generated module Mi over theArtinian ring S0.

• The order of the pole of the Poincare series at t = 1 is equal to thegraded (and ungraded) Krull dimension of M


Poincare Series

Example

Let R = k[x1, . . . , xu], where the grading is non-standard, say that thedegree of xi is equal to di , and f is a homogeneous element of degree d .We compute,

PR/(f )(t) = PR(t)− PR(−d)(t)

=1

(1− td1) · · · (1− tdu)− td

(1− td1) · · · (1− tdu)

=(1− t)(1 + t + · · ·+ td−1)

(1− t)u(1 + t + · · ·+ td1−1) · · · (1 + t + · · ·+ tdu−1)

=1

(1− t)u−1

[1 + t + · · ·+ td−1

(1 + t + · · ·+ td1−1) · · · (1 + t + · · ·+ tdu−1)

].


*Local Algebra

For finitely generated modules over Noetherian rings, standardmeasures from local algebra include:

I System of ParametersI Hilbert-Samuel functionsI Regular systems and Koszul ComplexI Multiplicities

Theorems from local algebra to connect all of these measures

Question: Working in grmod(−), do the same results hold if we usegraded analogues?


*Samuel Polynomial

Construction

Graded ideal of definition I, ∗À(M/IM) <∞Note: a module can be *Artinian without being Artinian

*Samuel function n 7→ ∗À(M/Fn(M)) is polynomial-like for n >> 0

*Samuel multiplicity ∗p(M, I, n) = ∗e(M,I,d)d! nd + lower order terms.


Samuel Polynomial Example

Consider R = k[x0, . . . , xu]/(f ) as a module over itself wheredeg(xi ) = 1 for each i , and f is homogenous of degree d . DefineS = k[x0, . . . , xu]. We have a graded short exact sequence

0→ S/mn ·f−→ S/mn+d → S/(mn+d + (f ))→ 0.

One may show that S/(mn+d + (f )) ∼= R/mn+d , and then useadditivity of vector space dimension over short exact sequences tocompute:

∗p(R,m, n + d) =d

u!nu + L.O.T .


Fundamental Theorem of Graded Dimension Theory

Theorem

If (A,N ) is a *local Noetherian ring and M ∈ grmod(A), then

∗ dimA(M) = ∗d(M) = ∗s(M) .

∗d(M) is the degree of ∗p(M,N , n)

∗s(M) is the least j such that there exist homogeneous elementsx1, . . . , xj ∈ N with ∗À (M/(x1, . . . , xj)M) <∞


Fundamental Theorem of Graded Dimension TheoryS positively graded, S0 Artinian, M ∈ grmod(S)

dimS(M) Krull dimension of M

d1(M) The least j s.t. there exists positive inte-gers n1, . . . , nj with

∏ji=1(1 − tni )PM(t) ∈

Z[t, t−1]

s1(M) The least j s.t. there exist homogeneous el-ements x1, . . . , xj ∈ S+ with M finitely gen-erated over S0〈x1, . . . , xj〉

∗ dimS(M) Graded Krull dimension of M

∗s(M) The least j such that there exist homo-geneous elements x1, . . . , xj ∈ S+ with∗`S (M/(x1, . . . , xj)M) <∞

∗d(M) The degree of the Samuel polynomial of M

• Also, the order of the pole of the Poincare series at t = 1 is equal to thiscommon dimension, which we will call D(M).


*Samuel Multiplicity

Corollary

M ∈ grmod(A), I a GIOD

i) I is a GIOD for A/pi and ∗p(A/pi , I, n) exists, for 1 ≤ i ≤ N.

ii) If D.

= max{deg(∗p(A/pi , I, n)).

= di | 1 ≤ i ≤ N} andD(M•)

.= {pj | dj = D},

∗e(M, I,D) =∑

p∈D(M•)

np(M•)(∗e(A/p, I,D)).

• In other words, the *Samuel multiplicity of M can be decomposed by theisolated primes in Spec(A) which contain AnnA(M) and have topdimension.


*Koszul Mulitplicity

Construction

(M, ∂) a graded complex in grmod(A)

∗χ(M).

=∑

i (−1)i ∗ À(Mi ) (*Euler Characteristic)

Define graded Koszul complex. Sequence x of homogeneous elements

(*Koszul Multiplicity)

∗χA(x ,M) =∑i

(−1)i ∗ À(Hi (x ,M))


Connecting the *Multiplicities

*Local Case

(A,N ) *local, M ∈ grmod(A), x a homogeneous sequence in Ngenerating a GIOD,

∀j ≥ 0, ∗À(HAj (x ,M)

)<∞

∗χA(x ,M) = ∗e(M, I, ∗ dimA(M))

R positively graded, R0 = k

Let x be a GSOP for M ∈ grmod(R),

χk〈x〉(M, k)(1) = ∗χk〈x〉(x ,M)

= ∗eR(I,M,D(M))

= ∗χR(x ,M)


*Samuel Multiplicity Results

Lemma

For S positively graded, S0 Artin, M ∈ grmod(S), I positively graded:

∗p(M, I, n) = p(M, I, n)

For d ≥ deg(∗p(M, I, n)),

∗e(M, I, d) = e(M, I, d)


*Samuel Multiplicity Results

Lemma

For (A,N ) *local, such that A−N has a homogeneous element of degree1, I a GIOD:

À0(M0/In0M0) = ∗À(M/InM) <∞ for each n

∗p(M, I, n) = p(M0, I0, n)

For d ≥ deg(∗p(M, I, n)),

∗e(M, I, d) = e(M0, I0, d)


The Degree

Definition

If S is a positively graded Noetherian ring, S0 is Artin,M ∈ grmod(R),M 6= 0 and D(M) = ∗ dimR(M), then

degS(M).

= limt→1

(1− t)D(M)PM(t)

is a well-defined, strictly positive, rational number. For convenience, definedegS(0) = 0.


The Degree

Example

Let R = k[x , y , z ]/(zy2 − x3) have the standard grading. We have that

PR(t) =1 + t + t2

(1− t)2.

If we expand as a Laurent series about t = 1 we get

PR(t) =3

(1− t)2− 3

1− t+ 1.


Example

Let R = k[x , y ]/(y2 − x3) be a module over itself where deg(x) = 2 anddeg(y) = 3.

PR(t) =1

(1− t)1

[1 + t + · · ·+ t5

(1 + t)(1 + t + t2)

]=

1

(1− t)1− t.

deg(R) = 1

Compute that ∗e(R, x) = 2 and ∗e(R, y) = 3

Notice ∗e(R,x)2 = ∗e(R,y)

3 = deg(R)


The Degree

Theorem

Suppose that M ∈ grmod(R), x1, . . . xD(M) form a GSOP for M, wheredeg(xi )

.= di , and I is the graded ideal generated by the xi ’s. Then,

deg(M) =∗eR(M, I,D(M))

d1 · · · dD(M).


The Degree

The Algebraic Degree Sum Formula

Let M ∈ grmod(R). Then,

deg(M) =∑

p∈D(M)

∗`R[p](M[p]) · deg(R/p).


Applications to Topology

G a topological group with a continuous action on topological spaceX

Construct (singular) equivariant cohomology: H∗G (X ,R) where R is acommutative ring

Contravariant functor: Pairs (G ,X ) to graded R-algebras


Equivariant Cohomology Examples

Example

G acts freely on X , then H∗(X/G ) ∼= H∗G (X )

Example

The groups used in this example are discrete, so the cohomology of BG isisomorphic to the group cohomology.

If G = Z, and X point, then BG = S1. Therefore, H∗Z∼= R[x ]/(x2),

where deg(x) = 1.

Let p 6= 2 be a prime, G = Z/pZ, k = Z/pZ.H∗Z/p

∼= Z/p[t]⊗Z/p Λ[s], where t ∈ H2, s ∈ H1, and Λ is the exterior

algebra over Z/p. For p = 2, H∗Z/2∼= Z/2[s], where s ∈ H1.


Equivariant Cohomology Examples

Example

For G = S1, BS1 = CP∞, and H∗(CP∞,Z) = Z[c] where c has degreetwo. More generally, if G = U(n) then BG = Gn(C∞) (the Grassmananianon C∞) and H∗(Gn(C∞),Z) ∼= Z[c1, . . . , cn] where deg(ci ) = 2i .


Finite Generation

Theorem

([15]) Let G be a compact Lie group, and let X be a G -space. If H∗(X )is a finitely generated k-vector space, then H∗G (X ) is a finitely generatedk-algebra (where k is the field of coefficients for H∗G (X ).)

Theorem

([15]) With the same hypotheses as the previous theorem, if(u, f ) : (G ,X )→ (G ′,X ′) is a morphism such that u is injective andH∗(X ) is a finitely generated k-module, then (u, f )∗ : H∗G ′(X

′)→ H∗G (X )is finite. i.e. H∗G (X ) is a finitely generated H∗G ′(X

′)-module.


Finite Generation

For p odd, H∗G (X ) is not in general a commutative ring

Denote the even degree part of H∗G (X ) as HG (X )

HG (X ) is commutative, and it follows from the theorems above thatH∗G (X ) ∈ grmod(HG (X ))


Example

For any compact Lie group G , there exists G ↪→ U(n) for some n.

H∗(BU) ∼= k[c1, . . . , cn], deg(ci ) = 2i

H∗G (X ) ∈ grmod(H∗U)

Apply Hilbert-Serre Theorem:

PH∗G (X )(t) =q(t)∏n

i=1(1− t2i )


Main Theorems of Quillen

Theorem

([15] Theorem 7.7) Assume (G ,X ) satisfies property †, and that H∗(X ) isfinite dimensional. Then, the Krull dimension of H∗G (X ) equals themaximal rank of an elementary abelian p-subgroup such that XA 6= ∅.Recall that all cohomology is taken with coefficients in k = Z/pZ.



Definition

Q(G ,X ) is the set of pairs (A, c) where A is an elementary abelianp-subgroup, XA 6= ∅, and c is a component of XA.

Definition

Assume property † for the pair (G ,X ). Let (A, c) ∈ Q(G ,X ), and pick apoint x0 ∈ c . We define pA,c ∈ Spec(HG (X )) as the kernel of thefollowing composition

HG (X )resGA−−→ HA(x0)→ HA(x0)/

√0.



Definition

Define an equivalence relation on Q(G ,X ): (A, c) ∼ (A′, c ′) if and only if∃g ∈ G such that gAg−1 = A′ and gc ′ = c . We can define a partial orderon the set of equivalence classes by [A, c] ≤ [A′, c ′] if and only if ∃h ∈ Gsuch that hAh−1 ⊆ A′ and hc ′ ⊆ c .

Theorem

([16] Proposition 11.2) With property † on (G ,X ) there is a one-to-onecorrespondence [A, c]↔ p(A,c) between the set of maximal classes [A, c]and the set of minimal primes for HG (X ).


Example

Let G be the extra-special 5 group of exponent 5, which may berealized as the group of all matrices

G =

1 a b

0 1 c0 0 1

: a, b, c ∈ Z/5

.

Compute the maximal elementary abelian 5-subgroups:

Aα =

1 a b

0 1 α · a0 0 1

, B =

1 0 b

0 1 c0 0 1

.

Each Aα and B are normal, and isomorphic to Z/5× Z/5.

Quillen’s theorem implies that dim(H∗G ) = 2 and there are 6 minimalprimes.


Localization

Historically, localization of equivariant cohomology rings has yieldedinteresting results about the topology of G -actions

Example

Let X = CPn, G = S1 × · · · × S1, k = Q, R = H∗G = k[t1, . . . , tm]

H∗G (X ) ∼= R[ξ]/ ((ξ − α1)m1 · · · (ξ − αs)ms )

By localizing at a certain set of homogeneous elements, Hsiang [11]shows that the fixed point set XG has the cohomology type ofCPm1−1 + · · ·+ CPms−1


Lynn Results [12]

Theorem

For X a compact, smooth manifold with G a compact Lie group actingsmoothly on X . There exists a G -manifold F such that:

deg(H∗G (X )) =∑

[A,c]∈Bmax (G ,X )

deg(H∗G (G · (c × FA))


Lynn Results [12]

• Using the results of the previous theorem, Lynn deduces:

Theorem

Let G be a compact Lie group, and let Bmax(G ) be the set of conjugacyclasses of maximal rank elementary abelian p-groups of G . Then,

deg(H∗G ) =∑

[A]∈Bmax (G)

1

|WG (A)|deg(H∗CG (A)).


Example

G is extra-special 5 group of exponent 5

Compute each centralizer and normalizer of each A. It turns out thatthe centralizer of each maximal elementary abelian subgroup isisomorphic to A ∼= Z/5× Z/5 and the normalizer of each is G .Furthermore, WG (A) = G/A ∼= Z/5

Lynn’s theorem implies

deg(H∗G ) =∑

[A]∈B(G)

1

|WG (A)|deg(H∗CG (A))

=1

5+

1

5+

1

5+

1

5+

1

5+

1

5

=6

5.


Centerpiece Results

The following theorem generalizes Lynn’s result, and it’s proof doesnot require any smoothness hypothesis

The proof of the theorem uses localization results of Duflot [6]

Theorem

Fix a prime p, and let all cohomology coefficients be in Z/pZ. Let G be acompact Lie group, and let X be a Hausdorff topological space on whichG acts continuously, assume also that X is either compact or has finitemod p cohomological dimension. Then,

deg(H∗G (X )) =∑


1

|WG (A, c)|deg(H∗CG (A,c)(c)).


Centerpiece Results

Proposed Theorem

Assume the same hypotheses on G and X as above. By the previoustheorem,

deg(H∗G (X )) =∑


1

|WG (A, c)|deg(H∗CG (A,c)(c)).

From our result on the additivity of degree in the graded category, we have:

deg(H∗G (X )) = degHG (X )(H∗G (X ))

=∑

p∈D(H∗G (X ))

∗`HG (X )[p](H∗G (X )[p]) · deg(HG (X )/p).

We propose that the two summations are equal term-by-term.


Centerpiece Results

Methodology of Proof of Proposed Theorem

The index sets of the two summations are the same using Quillen’sidentification [16]

WG (A, c) acts freely on the components of H∗CG (A,c)(c × FA) [6]

Using this, Duflot’s localization results [6], and our commutativealgebra results it seems like we can compare the graded lengths fromthe algebraic sum formula to the corresponing summand in thegeometric sum formula


Avramov, L. and Buchweitz, R., Lower Bounds for Betti Numbers,Compositio Mathematica, 86 (1993).

Atiyah, M. and Macdonald, I.G., Introduction to CommutativeAlgebra. Addison-Wesley Publishing Company, Inc., Reading,Massachusetts,(1969).

Benson, D. Representations and Cohomology II: Cohomology ofGroups and Modules. Cambridge University Press, Cambridge, U.K.,(1991).

Bruns, W. and Herzog,J., Cohen-Macaulay rings. Revised Edition.Cambridge University Press, Cambridge, U.K., (1998).

Dieck, T. Transformation Groups. Walter de Grutyer & Co., Berlin,(1987).

Duflot, J., Localization of Equivariant Cohomology Rings.Transactions of the American Mathematical Society, 288 (1984).

Eisenbud, D., Commutative Algebra with a View Toward AlgebraicGeometry. Springer-Verlag, New York, (1995).


Green, D., Grobner Bases and the Computation of Group Cohomology.Springer-Verlag, Berlin Heidelberg, (2003).

Grothendieck, A., Elements de geometrie algebrique (rediges avec lacollaboration de Jean Dieudonne) : II. Etude globale elementaire, dequelques classes de morphismes, Publications mathematiques delI.H.E.S. 8 (1961).

Hilton, P. and Stammbach, U., A Course in Homological Algebra.Springer Science+Business Media, New York, (1971).

Hsiang, W., Cohomology Theory of Topological TransformationGroups. Springer-Verlag, Berlin, Heidelberg, (1975).

Lynn, B., A Degree Formula for Equivariant Cohomology. Transactionsof the American Mathematical Society, 366 (2014).

Maiorana, J. A., Smith Theory for p-Groups. Transactions of theAmerican Mathematical Society, 223 (1976), 253−266.

Milnor, J., Construction of Universal Bundles I and II. Annals ofMathematics, 63 (1956), 272−284, 430−436.


Quillen, D., The Spectrum of an Equivariant Cohomology Ring I.Annals of Mathematics, 94, No. 3 (1971), 549−572

Quillen, D., The Spectrum of an Equivariant Cohomology Ring II.Annals of Mathematics, 94, No. 3 (1971), 573−602.

Smoke, W., Dimension and Multiplicity for Graded Algebras. Journalof Algebra, 21 (1972).

Serre, J-P., Local Algebra. Springer-Verlag, Berlin (2000).

Symonds, P., On the Castelnuovo-Mumford Regularity of theCohomology Ring of a Group. Journal of the American MathematicalSociety, Vol. 23, No. 4 (2010), 1159−1173.


Documents

Commutative Algebra of Equivariant Cohomology Rings€¦ · Introduction Example Let F be a graded ring. The following are equivalent: Every nonzero homogeneous element in F is invertible