Search

We prove a general structure theorem for the rational Borel equivariant cohomology ring of an equivariantly formal isotropy action, rederiving He’s recent computation of the equivariant cohomology of real Grassmannians as an illustration.

R. Sazeedeh showed that top local cohomology modules are Gorenstein injective in a Gorenstein local ring with at most two dimensions. In this paper, it is proved that the condition of dimension in his result cannot be relaxed and the conclusion in his result holds for complete local hypersurface rings with any dimension.

We define the deficiency of a cohomology class u u with respect to a vector field as the set of limit points in the ambient manifold of long almost closed orbits representing homology classes on which u u is nonpositive. We prove that, up to infinite cyclic coverings, the sole vector fields on closed manifolds exhibiting nonzero cohomology classes with finite deficiency are the gradient-like ones. We also prove that if the manifold is not a sphere, every singularity is hyperbolic and there is a closed transverse submanifold intersecting all regular orbits, then there is also a nonzero cohomology class with finite deficiency.

We prove a duality theorem for graded algebras over a field that implies several known duality results : graded local duality, versions of Serre duality for local cohomology and of Suzuki duality for generalized local cohomology, and Herzog-Rahimi bigraded duality., 6 pages

Let f be a degeneration of Kahler manifolds. The local invariant cycle theorem states that for a smooth fiber of the degeneration, any cohomology class, invariant under the monodromy action, rises from a global cohomology class. Instead of the classical cohomology, one may consider the non-abelian cohomology. This note demonstrates that the analogous non-abelian version of the local invariant cycle theorem does not hold if the first non-abelian cohomology is the moduli space (universal categorical quotient) of the representations of the fundamental group., Comment: 4 pages

We give a counterexample to the following conjecture of E. Bueler: “The de Rham cohomology of any complete Riemannian manifold is isomorphic to a weighted L 2 L^2 cohomology where the weight is the heat kernel."

For certain manifolds with an involution the mod 2 cohomology ring of the set of fixed points is isomorphic to the cohomology ring of the manifold, up to dividing the degrees by two. Examples include complex projective spaces and Grassmannians with the standard antiholomorphic involution (with real projective spaces and Grassmannians as fixed point sets). Hausmann, Holm and Puppe have put this observation in the framework of equivariant cohomology, and come up with the concept of conjugation spaces, where the ring homomorphisms arise naturally from the existence of what they call cohomology frames. Much earlier, Borel and Haefliger had studied the degree-halving isomorphism between the cohomology rings of complex and real projective spaces and Grassmannians using the theory of complex and real analytic cycles and cycle maps into cohomology. The main result in the present note gives a (purely topological) connection between these two results and provides a geometric intuition into the concept of a cohomology frame. In particular, we see that if every cohomology class on a manifold X X with involution is the Thom class of an equivariant topological cycle of codimension twice the codimension of its fixed points (inside the fixed point set of X X ), these topological cycles will give rise to a cohomology frame.

We construct a polarized Hodge structure on the primitive part of Chen and Ruan’s orbifold cohomology H o r b k ( X ) H_{orb}^k(X) for projective S L SL -orbifolds X X satisfying a “Hard Lefschetz Condition”. Furthermore, the total cohomology ⨁ p , q H o r b p , q ( X ) \bigoplus _{p,q}H_{orb}^{p,q}(X) forms a mixed Hodge structure that is polarized by every element of the Kähler cone of X X . Using results of Cattani-Kaplan-Schmid this implies the existence of an abstract polarized variation of Hodge structure on the complexified Kähler cone of X X . This construction should be considered as the analogue of the abstract polarized variation of Hodge structure that can be attached to the singular cohomology of a crepant resolution of X X , in light of the conjectural correspondence between the (quantum) orbifold cohomology and the (quantum) cohomology of a crepant resolution.

Let O λ \mathcal {O}_\lambda be a generic coadjoint orbit of a compact semi-simple Lie group K K . Weight varieties are the symplectic reductions of O λ \mathcal {O}_\lambda by the maximal torus T T in K K . We use a theorem of Tolman and Weitsman to compute the cohomology ring of these varieties. Our formula relies on a Schubert basis of the equivariant cohomology of O λ \mathcal {O}_\lambda , and it makes explicit the dependence on λ \lambda and a parameter in L i e ( T ) ∗ =: t ∗ Lie(T)^*=:\mathfrak {t}^* .

In this paper we study the Annihilator Theorem and the Local-global Principle for the annihilation of local cohomology modules over a (not necessarily finite-dimensional) Noetherian Gorenstein ring.

We present a proof that the equivariant intersection cohomology of any complete algebraic variety acted by a connected algebraic group G is a free module over H*(BG).

It is shown that the mod 3 cohomology of a homotopy associative mod 3 H-space which is rationally equivalent to the Lie group E 7 and which has integral 3-torsion is isomorphic to that of E7 as a Hopf algebra over the mod 3 Steenrod algebra.

We show that the real cohomology algebra of a compact toric variety of complex dimension 2 2 is determined, up to isomorphism, by the combinatorial data of its defining fan. Surprisingly enough, this is no longer the case when taking rational coefficients. Moreover, we show that neither the rational nor the real or complex cohomology algebras of compact quasi-smooth toric varieties are combinatorial invariants in general.

We generalize the classical theorem of Serre on the non-triviality of infinitely many homotopy groups of 1 1 -connected finite CW-complexes to CW-complexes where the cohomology groups either grow too fast or do not grow faster than a certain rate given by connectivity. For example, this result can be applied to iterated suspensions of finite Postnikov systems and certain spaces with finitely generated cohomology ring. In particular, we obtain an independent, short proof of a theorem of R. Levi on the non-triviality of k k -invariants associated to finite perfect groups. Another application concerns spaces where the cohomology grows like a polynomial algebra on generators in dimension n n , 2 n 2n , 3 n , … 3n, \ldots for a fixed number n n . We also consider spectra where we prove a non-triviality result in the case of fast growing cohomology groups.

We show that for a discrete group G, the equivariant cohomology of a G-space X with G-local coefficients M is isomorphic to the Bredon-Illman cohomology of X with equivariant local coefficients M.

We observe that work of Gugenheim and May on the cohomology of classical homogeneous spaces G/H of Lie groups applies verbatim to the calculation of the cohomology of generalized homogeneous spaces G/H, where G is a finite loop space or a p-compact group and H is a subgroup in the homotopical sense.

Let p be an odd prime number. The purpose of this paper is to provide a p-group G whose mod-p cohomology ring has a nilpotent element ξ E H*(G) satisfying ξ p ¬= 0.

We show that the first non-finitely generated local cohomology module Hi a (M ) of a finitely generated module M over a noetherian ring R with respect to an ideal a ⊆ R has only finitely many associated primes.

Suppose B B is a block of a group algebra k G kG with cyclic defect group. We calculate the Hochschild cohomology ring of B B , giving a complete set of generators and relations. We then show that if B B is the principal block, the canonical map from H ∗ ( G , k ) H^*(G,k) to the Hochschild cohomology ring of B B induces an isomorphism modulo radicals.

We will provide an example of a p-group G which has elements of order p 3 in some of its integral cohomology groups but which also has the property that p 2 annihilates ¯ H i (G; Z) for all sufficiently highi. This provides a counterexample to a conjecture of A. Adem which stated that if a finite group K has an element of order p n in one of its integral cohomology groups then it has such an element in infinitely many of its cohomology groups.

Let G G be a group, D D a division ring and V V a D G DG -module. V V is called finitary provided that V / C V ( g ) V/C_V(g) is finite dimensional for all g ∈ G g\in G . We investigate the first and second degree cohomology of finitary modules in terms of a local system for G G .

The generalized cyclic cohomology is introduced which is associated with q-deformed commutators xy − qyx. Some formulas related to the trace of the product of q-deformed commutators are established. The Chern character of odd dimension associated with q-deformed commutators is studied.

Let I be an ideal primary to the maximal ideal in a local ring. We utilize two well-known theorems due to J.-P. Serre to prove that the difference between the Hilbert function and the Hilbert polynomial of I is the alternating sum of the graded pieces of the graded local cohomology (with respect to its positively-graded ideal) of the Rees ring of I. This gives new insight into the higher Hilbert coefficients of I. The result is inspired by one due to J. D. Sally in dimension two and is implicit in a paper by D. Kirby and H. A. Mehran, where very different methods are used.

In this paper, we consider actions of G = Z p G = {Z_p} (with p an odd prime) on spaces X which are of cohomology type (0, 0) (i.e., have the mod - p \bmod \text {-}p cohomology of the one-point union of an n-sphere, a 2n-sphere and a a 3n-sphere, n odd). If X is not totally non-homologous to zero in X G {X_G} we determine the fixed set, give examples of all possibilities for the fixed set and compute the cohomology ring structure of the orbit space in the case where G acts freely. In [4], we considered fixed sets for related spaces, when X is totally non-homologous to zero in X G {X_G} .

We construct, for each value of n, a split extension of finite 2-groups, with complement isomorphic to Z/2 , for which the differential d n {d_n} is nonzero in the Lyndon-Hochschild-Serre spectral sequence.

The formalism of weights allows very simple analysis of the cohomology of hyperplane complements in a uniform fashion for different cohomology theories. An l-adic analogue of Arnold's conjecture on the torsion-freeness of these cohomology groups is one of the consequences. Let k be a field and let V = Spec(k[xl, x2, ... , Xd]) be an affine d-space over k. A hyperplane arrangement in V is a collection sv of affine subspaces of codimension 1 in V. We are concerned with the hyperplane complement M(v) := V (UHE5,f H) and its cohomology. Theorem 1. Let k = C. Then the mixed Hodge structure on Hi(M(V), Z) is pure of weight 2i and Hodge type (i, i) . Theorem 1'. Suppose the hyperplanes are defined over the finite field Fq and k is an algebraic closure, so that M(-V) = M(WV)o OFq k. Let I be a prime not dividing q. Then (a) the etale cohomology group Hi(M(SI), Z1) is pure of weight 2i, and (b) in fact, the geometric Frobenius Frq acts on HI(M(-V), Z1) as multiplication by qi. As the reader will see below, the proofs of Theorems 1 and 1' are rather trivial. In some respects, the first is also just a reformulation of known results. The cohomology of hyperplane complements has been studied extensively by several authors and the structure of the cohomology algebra has been explicitly determined [8, 9]. Numerous results and conjectures in topology, algebra, and combinatorics arising from this study can be found in the beautiful monograph of Orlik and Terao [9] which also contains a full bibliography. I believe, nonetheless, that Theorem 1 needs to be stated at least once in the formulation above for the following reasons: (1) The 'philosophy of weights' [1 ]renders both the statement and the proof of a 'soft' result regarding cohomology simpler and conceptually more transparent than a direct computation; (2) it reveals an essential underlying structure more basic than so involved an object as the full cohomology algebra; and Received by the editors June 24, 1992. 1991 Mathematics Subject Classification. Primary 14M99, 14F20. The authot was supported in part by NSF Grant DMS 9106444. i 1994 American Mathematical Society

Landweber provided two proofs of the existence of (level 2) elliptic cohomology (Lecture Notes in Math., vol. 1326, Springer-Verlag, New York, 1988, pp. 69-93). As Baker pointed out (J. Pure Appl. Algebra 63 (1990), 1-11), one of these proofs gives a level 1 elliptic cohomology theory as well. In this note we provide an alternative proof of the existence of level 1 elliptic cohomology. The idea here is to use Landweber’s direct proof of the existence of level 2 elliptic cohomology and an integrality argument to deduce the existence of level 1 1 elliptic cohomology from that.

For spaces with a group action, we introduce Bredon cohomology with local (or twisted) coefficients and show that it is invariant under weak equivariant homotopy equivalence. We use this new cohomology to construct a Serre spectral sequence for equivariant fibrations. Bredon (1) introduced what is now called Bredon cohomology, with the pur- pose of developing obstruction theory in the context of spaces equipped with an action of a fixed group G. The kind of coefficients needed in this theory are not abelian groups but rather contravariant functors from the orbit category tf (tr) into abelian groups. The purpose of this paper is to prove the existence of a spectral sequence for a Serre fibration of (7-spaces. As in the nonequivariant case, if no further restrictions are made, it is necessary to use cohomology with twisted coefficients. Such twisted coefficients are not functors on the orbit category cf(G), but on some augmentation of cf (G) depending on the base space of the fibration. Our approach is to give a new definition of Bredon cohomology in terms of cohomology of categories. Indeed, with a G-space X we shall associate a category AG(X) of "equivariant singular simplices in X." Any abelian group- valued functor Af from the orbit category cf(G) can be viewed as a system of coefficients on the category AG(X). A key result is that for any such Af, the cohomology groups H*(AGX, M) of this category are naturally isomorphic to the Bredon cohomology groups HG(X, M); see Theorem 2.2. The category AG(X) allows us to define cohomology groups of a G-space with more general coefficients. In particular, we shall construct a category HG(X) of "equivariant homotopy classes of paths in X," which sits in between AG(X) and cf (G) by functors AG(X) -► UG(X) -► tf(G). A twisted or local system of coefficients Af on X is then defined as an abelian group-valued functor on HG(X), and the Bredon cohomology of X with twisted coefficients can now be constructed as the cohomology H*(AG(X), M) of the category AG(X). This is invariant under weak G-homotopy equivalence, see Theorem 2.3. (The converse is also true: a map of G-spaces is a weak G-homotopy equivalence whenever it induces an equivalence of fundamental groupoids as well as an

Let ( R , m ) (R,m) be a local ring and I I an m m -primary ideal. A result of Trung shows that if the local cohomology of g r I ( R ) g{r_I}(R) satisfies certain conditions, then the reduction number of I I is independent of the minimal reduction chosen. These conditions consist of t = dim ⁡ R − grade g r I ( R ) + t = \dim R - \operatorname {grade} \;g{r_I}{(R)^ + } inequalities. We show that if R R is Cohen-Macaulay, then one of these inequalities is always satisied, while another can often be easily checked. Applications are then given in two-dimensional Cohen-Macaulay rings. For instance, we show that if the Hilbert function of I I equals the Hilbert polynomial of I I for all integers greater than 1 1 , then the reduction number is independent of the choice of minimal reduction.

In this paper, we consider the cohomology ring of a finite 2 2 -group with coefficients in a field of characteristic two. We show that, for any positive integer n n , there exists a 2 2 -group whose cohomology ring has elements of nilpotency degree n + 1 n + 1 and all smaller degrees.

All Mackey functors over a finite group G G are built up by short exact sequences from Mackey functors arising from modules over the integral group rings of appropriate subquotients WH of G G . The equivariant cohomology theories with coefficients in Mackey functors arising from WH-modules admit particularly simple descriptions.

We apply cohomology of sheaves to arrangements of hyperplanes. In particular we prove an inequality for the depth of cohomology modules of local sheaves on the intersection lattice of an arrangement. This generalizes a result of Solomon-Terao about the cummulative property of local functors. We also prove a characterization of free arrangements by certain properties of the cohomlogy of a sheaf of derivation modules. This gives a condition on the Möbius function of the intersection lattice of a free arrangement. Using this condition we prove that certain geometric lattices cannot afford free arrangements although their Poincaré polynomials factor.

Using only the long exact sequence of local cohomology, we give a brief derivation of Orlik and Solomon’s presentation for the cohomology ring of a complement of hyperplanes in a vector space.

In this paper it is shown that relative Lie algebra cohomology is related to relative cohomology for restricted Lie algebras by a spectral sequence. Also an interpretation of the relative cohomology groups for Lie algebras in terms of the relative right derived functors of a certain relative derivative functor is given.

In 2006 Belkale and Kumar defined a new product on the cohomology of flag varieties and used this new product to give an improved solution to the eigencone problem for complex reductive groups. In this paper, we give a generalization of the Belkale-Kumar product to the branching Schubert calculus setting. The study of branching Schubert calculus attempts to understand the induced map on cohomology of an equivariant embedding of flag varieties. The main application of our work is a compact formulation of the solution to the branching eigencone problem.

Let f : G → K f\colon G\to K be a homomorphism of p p -groups such that f ( n ) : H n ( K , Z / p ) → H n ( G , Z / p ) f^{(n)}\colon H^n(K,\mathbf Z /p)\to H^n(G,\mathbf Z/p) is injective, for 1 ≤ n ≤ 2 1\le n\le 2 . We prove that the non-bijectivity of f f implies the existence of a quotient L L of G G containing K K as a proper direct factor. This gives a refined proof of a result of Evens, which asserts that f f is bijective if f ( 1 ) f^{(1)} is.

In this paper we show how to construct the equivariant bordism class of a closed manifold with differentiable ( Z 2 ) k {({Z_2})^k} -action from its fixed data.

The term "finitely graded" is introduced here to refer to graded modules which are nonzero in only finitely many graded pieces. We consider the question of when the local cohomology modules of a graded module are finitely graded. Using a theorem of Faltings concerning the annihilation of local cohomology, we obtain some partial answers to this question. These results are then used to compare the depths of the Rees algebra and the associated graded ring of an ideal in a local ring.

Let X be an H-space with H* (X; Z2) Z2[xl,..., Xd] ? A(y1, .* I Yd), where degx, = 4 and y, = Sq1 xi . In this article we prove that X cannot be homotopy-commutative. Combining this result with a theorem of Michael Slack results in the following theorem: Let X be a homotopycommutative H-space with mod 2 cohomology finitely generated as an algebra. Then H*(X; Z2) is isomorphic as an algebra over A(2) to the mod 2 cohomology of a torus producted with a finite number of CP(oo)s and K(Z2r, 1)s. 0. INTRODUCTION In this article we prove the following theorem: Theorem A. Let X be an H-space with H* (X; Z2) =Z2 [x,s *@ .. Xd] 9A(ylj SYd) where degxi = 4 and yi = SqI xi . Then X cannot be homotopy-commutative. The significance of Theorem A lies in its relationship to the following theorem, due to Michael Slack: Theorem (Slack). Let X be a homotopy-commutative H-space with mod 2 cohomology finitely generated as an algebra. Then (1) All even-degree generators have infinite height and are in degrees two and four. (2) All odd-degree generators lie in degrees one andfive. The one-dimensional generators have infinite height and the five-dimensional generators are exterior. (3) Sq1: QH4(X; Z2) _+ QH5(X; Z2) is an isomorphism. Received by the editors February 27, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 55P45, 55S40.

A space X that has the cohomology of the one-point union P2(n) 53n or Sn VS2n VS3n is said to have cohomology type (a, 0) . Here we construct examples of free Zp actions (p an odd prime) on certain of these spaces and give a complete description of possible fixed point sets.

The aim of this note is to define the transfer in generalized sheaf cohomology and state its most important properties. Under appropriate conditions the transfer defined here agrees with the transfer defined using different methods by Roush, Kahn, and Priddy. Transfer for finite coverings. In order to formulate our main theorem (1.1) we need some basic definitions involving 1.0 Generalized sheaf cohomology and finite coverings. When speaking of sheaves of spectra we shall adopt the language of Brown as formulated in [4 and 5]. Let & be a sheaf of spectra over B. Unless otherwise stated we shall assume that & is in Sta Ho' (B) or that B has finite cohomological dimension. Let f: X -> B be a continuous map and ( a family of supports on B. We let (i) f*: HO,(B; &) Hf-,,0(X; f '6) be the natural homomorphism in cohomology induced by the natural f-cohomomorphism. (ii) f 46 +& (f 4 denotes pullback). The cohomology group on the right in (i) will be written H,,*(X; S) with f understood. The map f is called a finite covering map if it is a locally trivial bundle projection with discrete finite fiber. If U C B and f-l(U) = W C X, then we call f: (X, W) -(B, U) a finite covering map of pairs. 1.1 EXISTENCE THEOREM. For each finite covering map f there exists an abelian group homomorphism f !: HZ, (X; H) H, H(B; &) called transfer in & cohomology with the following properties: 1.2 Transfer is functorial over pullbacks of coverings. 1.3 Transfer for an identity covering is the identity. 1.4 If a covering splits as a disjoint union of coverings, then transfer over the total covering is the sum of the transfers over the component covering. 1.5 Transfer for a composition of coverings is the composition of the transfers. 1.6 Transfer commutes with homomorphisms induced by a morphism of coefficient sheaves. 1.7 Transfer induces a morphism on the E2 terms of the generalized Atiyah-Hirzebruch spectral sequence [5, Theorem 8, p. 451] and is compatible with the differentials of that sequence. (We do not assume here that & E Sta Ho' (B).) Received by the editors April 23, 1982. Presented to the winter meeting of the AMS in Denver in January 1983. 1980 Mathematics Subject Classification. Primary 55N30. Kev words and phrases. Transfer, sheaf, spectra, Postnikov decomposition, finite covering map, hypercovering, generalized Cech cohomology. It1984 American Mathematical Society 0002-9939/84 $1.00 + $.25 per page

For a non-Abelian group G G , we show that the obstruction to the existence of an extension of G G by Π \Pi that induces ϕ : Π → \phi :\Pi \to Out G G is also the k k -invariant of the classifying space for K ( G , 1 ) K(G,1) -bundles.