3.20pt. Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

Transcription

1 3.20pt Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

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

3 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) associated to an equivariant cohomology ring Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

4 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-module homomorphisms In general, equivariant cohomology rings have non-standard gradings. i.e. not generated by degree 1 elements over the degree 0 part. Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

5 Introduction Example Let F be a graded ring. The following are equivalent: Every nonzero homogeneous element in F is invertible. F 0 is a field and either F = F 0, or there exists a d > 0 and an x F d such that F = F 0 [x, x 1 ] as a graded ring. In fact, in this last case, d > 0 is the smallest positive degree with F d 0. The only graded ideals in F are F and 0. A ring satisfying any of these three equivalent conditions is called a graded field. Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

6 Basic Commutative Algebra Results Let A be a Z-graded Noetherian ring with graded ideals I and Ĩ, and let M grmod(a). A 0 is Noetherian and A is a finitely generated A 0 -algebra by a set of homogeneous elements V (I) = V (Ĩ) if and only if I = Ĩ The associated primes of M are graded Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

7 Basic Commutative Algebra Results Lemma For M grmod(a), there exists a graded filtration F of M, 0 = M 0 M 1 M N 1 M N = M, with the property that for each 1 n N, there exist graded primes p i such that, (A/p i )(d i ) = M i /M i 1. Let S F be the set of primes(graded) which appear in the filtration, then {minimal primes of M} Ass A (M) S F Supp A (M) Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

8 Graded Krull Dimension and Length Lemma If A is a Z-graded Noetherian ring, then therefore if M grmod(a), dim(a) dim(a) dim(a) + 1; dim A (M) dim A (M) dim A (M) + 1. Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

9 Graded Krull Dimension and Length Lemma Let S be a positively graded ring, and M grmod(s), m a graded ideal in S. Then, m *maximal if and only if m maximal dim S (M) = dim S (M) l S (M) = l S (M) Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

10 Localization Different Methods Usual Localization Graded Localization The degree 0 part Construction Graded localization: take a multiplicatively closed subset T A containing entirely homogeneous elements. Construct localization T 1 M as usual. Graded localization at a prime p denoted M [p] Degree zero localization (T 1 M) 0. Degree zero localization at a prime denoted M (p) Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

11 Results Comparing Localization Methods Positively Graded Case Graded primes p Proj(S), q 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 Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

12 Results Comparing Localization Methods Z-graded Case p Spec(A) graded, minimal l A[p] (M [p] ) = l Ap (M p ) l A(p) (M (p) ) l A[p] (M [p] ) = l Ap (M p ) If there is a homogeneous element of degree 1 (or, equivalently, -1) in A p l A(p) (M (p) ) = l A[p] (M [p] ) = l Ap (M p ) Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

13 Poincare Series Definition Suppose S is a positively graded ring, with S 0 Artinian, and M is a finitely generated graded S-module. Then the Poincaré series of M is the formal Laurent series with integer coefficients P M (t) = i Z l S0 (M i )t i, where l S0 (M i ) is the length of the finitely generated module M i over the Artinian ring S 0. The order of the pole of the Poincaré series at t = 1 is equal to the graded (and ungraded) Krull dimension of M Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

14 Poincare Series Example Let R = k[x 1,..., x u ], where the grading is non-standard, say that the degree of x i is equal to d i, and f is a homogeneous element of degree d. We compute, P R/(f ) (t) = P R (t) P R( d) (t) = 1 t d (1 t d 1 ) (1 t d u) (1 t d 1 ) (1 t d u) (1 t)(1 + t + + t d 1 ) = (1 t) u (1 + t + + t d1 1 ) (1 + t + + t du 1 ) [ t + + t d 1 ] = (1 t) u 1 (1 + t + + t d1 1 ) (1 + t + + t du 1. ) Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

15 *Local Algebra For finitely generated modules over Noetherian rings, standard measures from local algebra include: System of Parameters Hilbert-Samuel functions Regular systems and Koszul Complex Multiplicities Theorems from local algebra to connect all of these measures Question: Working in grmod( ), do the same results hold if we use graded analogues? Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

16 *Samuel Polynomial Construction Graded ideal of definition I, l A (M/IM) < Note: a module can be *Artinian without being Artinian *Samuel function n l A (M/F n (M)) is polynomial-like for n >> 0 *Samuel multiplicity p(m, I, n) = e(m,i,d) d! n d + lower order terms. Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

17 Samuel Polynomial Example Consider R = k[x 0,..., x u ]/(f ) as a module over itself where deg(x i ) = 1 for each i, and f is homogenous of degree d. Define S = k[x 0,..., x u ]. We have a graded short exact sequence 0 S/m n f S/m n+d S/(m n+d + (f )) 0. One may show that S/(m n+d + (f )) = R/m n+d, and then use additivity of vector space dimension over short exact sequences to compute: p(r, m, n + d) = d u! nu + L.O.T. Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

18 Fundamental Theorem of Graded Dimension Theory Theorem If (A, N ) is a *local Noetherian ring and M grmod(a), then dim A (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 elements x 1,..., x j N with l A (M/(x 1,..., x j )M) < Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

19 Fundamental Theorem of Graded Dimension Theory S positively graded, S 0 Artinian, M grmod(s) dim S (M) Krull dimension of M d 1 (M) The least j s.t. there exists positive integers n 1,..., n j with j i=1 (1 tn i )P M (t) Z[t, t 1 ] s 1 (M) The least j s.t. there exist homogeneous elements x 1,..., x j S + with M finitely generated over S 0 x 1,..., x j dim S (M) Graded Krull dimension of M s(m) The least j such that there exist homogeneous elements x 1,..., x j S + with l S (M/(x 1,..., x j )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 this common dimension, which we will call D(M). Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

20 *Samuel Multiplicity Corollary M grmod(a), I a GIOD i) I is a GIOD for A/p i and p(a/p i, I, n) exists, for 1 i N. ii) If D. = max{deg( p(a/p i, I, n)). = d i 1 i N} and D(M ). = {p j d j = D}, e(m, I, D) = p D(M ) n p (M )( e(a/p, I, D)). In other words, the *Samuel multiplicity of M can be decomposed by the isolated primes in Spec(A) which contain Ann A (M) and have top dimension. Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

21 *Koszul Mulitplicity Construction (M, ) a graded complex in grmod(a) χ(m) =. i ( 1)i l A (M i ) (*Euler Characteristic) Define graded Koszul complex. Sequence x of homogeneous elements (*Koszul Multiplicity) χ A ( x, M) = i ( 1) i l A (H i ( x, M)) Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

22 Connecting the *Multiplicities *Local Case (A, N ) *local, M grmod(a), x a homogeneous sequence in N generating a GIOD, ) j 0, l A (Hj A ( x, M) < χ A ( x, M) = e(m, I, dim A (M)) R positively graded, R 0 = k Let x be a GSOP for M grmod(r), χ k x (M, k)(1) = χ k x ( x, M) = e R (I, M, D(M)) = χ R ( x, M) Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

23 *Samuel Multiplicity Results Lemma For S positively graded, S 0 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) Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

24 *Samuel Multiplicity Results Lemma For (A, N ) *local, such that A N has a homogeneous element of degree 1, I a GIOD: l A0 (M 0 /I n 0 M 0) = l A (M/I n M) < for each n p(m, I, n) = p(m 0, I 0, n) For d deg( p(m, I, n)), e(m, I, d) = e(m 0, I 0, d) Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

25 The Degree Definition If S is a positively graded Noetherian ring, S 0 is Artin, M grmod(r), M 0 and D(M) = dim R (M), then deg S (M). = lim t 1 (1 t) D(M) P M (t) is a well-defined, strictly positive, rational number. For convenience, define deg S (0) = 0. Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

26 The Degree Example Let R = k[x, y, z]/(zy 2 x 3 ) have the standard grading. We have that P R (t) = 1 + t + t2 (1 t) 2. If we expand as a Laurent series about t = 1 we get P R (t) = 3 (1 t) t + 1. Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

27 Example Let R = k[x, y]/(y 2 x 3 ) be a module over itself where deg(x) = 2 and deg(y) = 3. deg(r) = 1 [ ] t + + t 5 P R (t) = (1 t) 1 (1 + t)(1 + t + t 2 ) 1 = (1 t) 1 t. Compute that e(r, x) = 2 and e(r, y) = 3 Notice e(r,x) 2 = e(r,y) 3 = deg(r) Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

28 The Degree Theorem Suppose that M grmod(r), x 1,... x D(M) form a GSOP for M, where deg(x i ). = d i, and I is the graded ideal generated by the x i s. Then, deg(m) = e R(M, I, D(M)) d 1 d D(M). Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

29 The Degree The Algebraic Degree Sum Formula Let M grmod(r). Then, deg(m) = l R[p] (M [p] ) deg(r/p). p D(M) Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

30 Applications to Topology G a topological group with a continuous action on topological space X Construct (singular) equivariant cohomology: HG (X, R) where R is a commutative ring Contravariant functor: Pairs (G, X ) to graded R-algebras Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

31 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 is isomorphic to the group cohomology. If G = Z, and X point, then BG = S 1. Therefore, H Z = R[x]/(x 2 ), where deg(x) = 1. Let p 2 be a prime, G = Z/pZ, k = Z/pZ. H Z/p = Z/p[t] Z/p Λ[s], where t H 2, s H 1, and Λ is the exterior algebra over Z/p. For p = 2, H Z/2 = Z/2[s], where s H 1. Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

32 Equivariant Cohomology Examples Example For G = S 1, BS 1 = CP, and H (CP, Z) = Z[c] where c has degree two. More generally, if G = U(n) then BG = G n (C ) (the Grassmananian on C ) and H (G n (C ), Z) = Z[c 1,..., c n ] where deg(c i ) = 2i. Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

33 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 HG (X ) is a finitely generated k-algebra (where k is the field of coefficients for HG (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 and H (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. Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

34 Finite Generation For p odd, HG (X ) is not in general a commutative ring Denote the even degree part of HG (X ) as H G (X ) H G (X ) is commutative, and it follows from the theorems above that H G (X ) grmod(h G (X )) Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

35 Example For any compact Lie group G, there exists G U(n) for some n. H (BU) = k[c 1,..., c n ], deg(c i ) = 2i H G (X ) grmod(h U ) Apply Hilbert-Serre Theorem: P H G (X )(t) = q(t) n i=1 (1 t2i ) Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

36 Main Theorems of Quillen Theorem ([15] Theorem 7.7) Assume (G, X ) satisfies property, and that H (X ) is finite dimensional. Then, the Krull dimension of HG (X ) equals the maximal rank of an elementary abelian p-subgroup such that X A. Recall that all cohomology is taken with coefficients in k = Z/pZ. Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

37 Main Theorems of Quillen Definition Q(G, X ) is the set of pairs (A, c) where A is an elementary abelian p-subgroup, X A, and c is a component of X A. Definition Assume property for the pair (G, X ). Let (A, c) Q(G, X ), and pick a point x 0 c. We define p A,c Spec(H G (X )) as the kernel of the following composition H G (X ) resg A H A (x 0 ) H A (x 0 )/ 0. Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

38 Main Theorems of Quillen 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 order on the set of equivalence classes by [A, c] [A, c ] if and only if h G such that hah 1 A and hc c. Theorem ([16] Proposition 11.2) With property on (G, X ) there is a one-to-one correspondence [A, c] p (A,c) between the set of maximal classes [A, c] and the set of minimal primes for H G (X ). Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

39 Example Let G be the extra-special 5 group of exponent 5, which may be realized as the group of all matrices 1 a b G = 0 1 c : a, b, c Z/ Compute the maximal elementary abelian 5-subgroups: 1 a b 1 0 b A α = 0 1 α a, B = 0 1 c Each A α and B are normal, and isomorphic to Z/5 Z/5. Quillen s theorem implies that dim(hg ) = 2 and there are 6 minimal primes. Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

40 Localization Historically, localization of equivariant cohomology rings has yielded interesting results about the topology of G-actions Example Let X = CP n, G = S 1 S 1, k = Q, R = H G = k[t 1,..., t m ] H G (X ) = R[ξ]/ ((ξ α 1 ) m1 (ξ α s ) ms ) By localizing at a certain set of homogeneous elements, Hsiang [11] shows that the fixed point set X G has the cohomology type of CP m CP ms 1 Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

41 Lynn Results [12] Theorem For X a compact, smooth manifold with G a compact Lie group acting smoothly on X. There exists a G-manifold F such that: deg(hg (X )) = [A,c] B max (G,X ) deg(h G (G (c F A )) Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

42 Lynn Results [12] Using the results of the previous theorem, Lynn deduces: Theorem Let G be a compact Lie group, and let B max (G) be the set of conjugacy classes of maximal rank elementary abelian p-groups of G. Then, deg(h G ) = [A] B max (G) 1 W G (A) deg(h C G (A) ). Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

43 Example G is extra-special 5 group of exponent 5 Compute each centralizer and normalizer of each A. It turns out that the centralizer of each maximal elementary abelian subgroup is isomorphic to A = Z/5 Z/5 and the normalizer of each is G. Furthermore, W G (A) = G/A = Z/5 Lynn s theorem implies deg(h G ) = [A] B(G) 1 W G (A) deg(h C G (A) ) = = 6 5. Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

44 Centerpiece Results The following theorem generalizes Lynn s result, and it s proof does not require any smoothness hypothesis Theorem The proof of the theorem uses localization results of Duflot [6] Fix a prime p, and let all cohomology coefficients be in Z/pZ. Let G be a compact Lie group, and let X be a Hausdorff topological space on which G acts continuously, assume also that X is either compact or has finite mod p cohomological dimension. Then, deg(hg (X )) = [A,c] B max (G,X ) 1 W G (A, c) deg(h C G (A,c) (c)). Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

45 Centerpiece Results Proposed Theorem Assume the same hypotheses on G and X as above. By the previous theorem, deg(hg (X )) = [A,c] B max (G,X ) 1 W G (A, c) deg(h C G (A,c) (c)). From our result on the additivity of degree in the graded category, we have: deg(hg (X )) = deg H G (X )(HG (X )) = l HG (X ) [p] (HG (X ) [p]) deg(h G (X )/p). p D(HG (X )) We propose that the two summations are equal term-by-term. Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

46 Centerpiece Results Methodology of Proof of Proposed Theorem The index sets of the two summations are the same using Quillen s identification [16] W G (A, c) acts freely on the components of H C G (A,c) (c F A ) [6] Using this, Duflot s localization results [6], and our commutative algebra results it seems like we can compare the graded lengths from the algebraic sum formula to the corresponing summand in the geometric sum formula Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

47 Avramov, L. and Buchweitz, R., Lower Bounds for Betti Numbers, Compositio Mathematica, 86 (1993). Atiyah, M. and Macdonald, I.G., Introduction to Commutative Algebra. Addison-Wesley Publishing Company, Inc., Reading, Massachusetts,(1969). Benson, D. Representations and Cohomology II: Cohomology of Groups 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 Algebraic Geometry. Springer-Verlag, New York, (1995). Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

48 Green, D., Grobner Bases and the Computation of Group Cohomology. Springer-Verlag, Berlin Heidelberg, (2003). Grothendieck, A., Elements de geometrie algebrique (rediges avec la collaboration de Jean Dieudonne) : II. Etude globale elementaire, de quelques classes de morphismes, Publications mathematiques de li.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 Transformation Groups. Springer-Verlag, Berlin, Heidelberg, (1975). Lynn, B., A Degree Formula for Equivariant Cohomology. Transactions of the American Mathematical Society, 366 (2014). Maiorana, J. A., Smith Theory for p-groups. Transactions of the American Mathematical Society, 223 (1976), Milnor, J., Construction of Universal Bundles I and II. Annals of Mathematics, 63 (1956), , Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45

49 Quillen, D., The Spectrum of an Equivariant Cohomology Ring I. Annals of Mathematics, 94, No. 3 (1971), Quillen, D., The Spectrum of an Equivariant Cohomology Ring II. Annals of Mathematics, 94, No. 3 (1971), Smoke, W., Dimension and Multiplicity for Graded Algebras. Journal of Algebra, 21 (1972). Serre, J-P., Local Algebra. Springer-Verlag, Berlin (2000). Symonds, P., On the Castelnuovo-Mumford Regularity of the Cohomology Ring of a Group. Journal of the American Mathematical Society, Vol. 23, No. 4 (2010), Mark Blumstein Commutative Algebra of Equivariant Cohomology Rings Spring / 45