Cartan’s method and its applications in sheaf cohomology

This paper aims to use Cartan’s original method in proving Theorems A and B on closed cubes to provide a different proof of the vanishing of sheaf cohomology over a closed cube if either (i) the degree exceeds its real dimension or (ii) the sheaf is (locally) constant and the degree is positive. In the first case, we can further use Godement’s argument to show the topological dimension of a paracompact topological manifold is less than or equal to its real dimension.     

Halphen pencils on quartic threefolds

For any smooth quartic threefold in P⁴ we classify pencils on it whose general element is an irreducible surface birational to a surface of Kodaira dimension zero.     

Motivic cohomology and unramified cohomology of quadrics

This is the last of a series of three papers where we compute the unramified cohomology of quadrics in degree up to 4. Complete results were obtained in the two previous papers for quadrics of dimension ≤4 and ≥11. Here we deal with the remaining dimensions between 5 and 10. We also prove that the unramified cohomology of Pfister quadrics with divisible coefficients always comes from the ground field, and that the same holds for their unramified Witt rings. We apply these results to real quadrics. For most of the paper we have to assume that the ground field has characteristic 0, because we use Voevodsky’s motivic cohomology. Received August 18, 1999 / final version received December 10, 1999?Published online April 19, 2000     

Complete cohomology for complexes with finite Gorenstein AC-projective dimension

We study complete cohomology of complexes with finite Gorenstein AC-projective dimension. We show first that the class of complexes admitting a complete level resolution is exactly the class of complexes with finite Gorenstein AC-projective dimension. This lets us give some general techniques for computing complete cohomology of complexes with finite Gorenstein AC-projective dimension. As a consequence, the classical relative cohomology for modules of finite Gorenstein AC-projective dimension is extended. Finally, the relationships between projective dimension and Gorenstein AC-projective dimension for complexes are given.     

Local cohomology with support in a parameter ideal

Motivated in part by an attempt to understand better the notion of a parameter-like sequence introduced in Hochster (2003) [7, (2.2)], we study results concerning the heights of the annihilators and the finiteness of the dimension of the socle in certain local cohomology modules with support in a parameter ideal. We obtain positive results under certain hypotheses of low dimension or codimension, but we also find examples that show, in the general case, that support in a parameter ideal does not restrict the behavior of local cohomology much more than support in an arbitrary ideal. The results obtained here strongly suggest that it would be worthwhile to seek a modification of the notion of a parameter-like sequence introduced in [7].     

The Brauer and Witt Groups of Real Ruled Surfaces

The Brauer group and the Witt group of a real irrational ruled surface are determined by using equivariant cohomology.     

On some moduli spaces of stable vector bundles on cubic and quartic threefolds

We study certain moduli spaces of stable vector bundles of rank 2 on cubic and quartic threefolds. In many cases under consideration, it turns out that the moduli space is complete and irreducible and a general member has vanishing intermediate cohomology. In one case, all except one component of the moduli space has such vector bundles.     

On the Algebraic Cohomology of Real Algebraic M-Varieties

For a real algebraic M-variety X, the canonical homomorphism of the algebraic cohomology group of the set of real points into the algebraic cohomology group of the set of complex points $$\varrho _k :H_{alg}^k (X(\mathbb{R}),\mathbb{Z}/2) \to H_{alg}^{2k} (X(\mathbb{C}),\mathbb{Z}/2)$$ is considered. This homomorphism agrees with the cycle mappings. Estimates for the dimension of the kernel of this homomorphism are given.     

A modification of the Hodge star operator on manifolds with boundary

For smooth compact oriented Riemannian manifolds M of dimension 4k+2, k≥0, with or without boundary, and a vector bundle F on M with an inner product and a flat connection, we construct a modification of the Hodge star operator on the middle-dimensional (parabolic) cohomology of M twisted by F. This operator induces a canonical complex structure on the middle-dimensional cohomology space that is compatible with the natural symplectic form given by integrating the wedge product. In particular, when k=0 we get a canonical almost complex structure on the non-singular part of the moduli space of flat connections on a Riemann surface, with monodromies along boundary components belonging to fixed conjugacy classes when the surface has boundary, that is compatible with the standard symplectic form on the moduli space.     

Gorenstein projective dimension for complexes

We define and study a notion of Gorenstein projective dimension for complexes of left modules over associative rings. For complexes of finite Gorenstein projective dimension we define and study a Tate cohomology theory. Tate cohomology groups have a natural transformation to classical Ext groups. In the case of module arguments, we show that these maps fit into a long exact sequence, where every third term is a relative cohomology group defined for left modules of finite Gorenstein projective dimension.

     

Equivariant cohomology of K-contact manifolds

We investigate the equivariant cohomology of the natural torus action on a K-contact manifold and its relation to the topology of the Reeb flow. Using the contact moment map, we show that the equivariant cohomology of this action is Cohen–Macaulay, the natural substitute of equivariant formality for torus actions without fixed points. As a consequence, generic components of the contact moment map are perfect Morse-Bott functions for the basic cohomology of the orbit foliation ${{\mathcal F}}$ of the Reeb flow. Assuming that the closed Reeb orbits are isolated, we show that the basic cohomology of ${{\mathcal F}}$ vanishes in odd degrees, and that its dimension equals the number of closed Reeb orbits. We characterize K-contact manifolds with minimal number of closed Reeb orbits as real cohomology spheres. We also prove a GKM-type theorem for K-contact manifolds which allows to calculate the equivariant cohomology algebra under the nonisolated GKM condition.     

Sur la cohomologie des super algèbres de Lie étranges

In this paper we compute the cohomology with trivial coefficients for the Lie superalgebraspsl(n, n), p (n) andq(2n); we show that the cohomology ring ofq(2n+1) is of Krull dimension 1 and we calculate the ring forq(3) andq(5). The last section is devoted to a result about the cohomology of a Lie superalgebra with reductive even part with coefficients in a finite dimensional moduleM.     

Non-Abelian Cohomology of Groups

Following Guin's approach to non-abelian cohomology [4] and, using the notion of a crossed bimodule, a second pointed set of cohomology is defined with coefficients in a crossed module, and Guin's six-term exact cohomology sequence is extended to a nine-term exact sequence of cohomology up to dimension 2     

Real Algebraically Maximal Varieties

For real algebraic varieties whose real algebraic cohomology group is maximal, a canonical homomorphism is constructed from the cohomology group of the set of complex points into the cohomology group of the set of real points, and then it is proved that this homomorphism is an isomorphism.     

Complete Cohomology for Extriangulated Categories

Let (C,E,s) be an extriangulated category with a proper class ξ of E-triangles.We study complete cohomology of objects in (C,E,s) by applying ξ-projective resolutions and ξ-injective coresolutions constructed in (C,E,s).Vanishing of complete cohomology detects objects with finite ξ-projective dimension and finite ξ-injective dimension.As a consequence,we obtain some criteria for the validity of the Wakamatsu tilting conjecture and give a necessary and sufficient condition for a virtually Gorenstein algebra to be Gorenstein.Moreover,we give a general technique for computing complete cohomology of objects with finite ξ-Gprojective dimension.As an application,the relations between ξ-projective dimension and ξ-Gprojective dimension for objects in (C,E,s) are given.     

Transitive actions on Hopf homogeneous spaces

In this paper we investigate transitive actions of compact connected Lie groups on certain spaces X which are not spheres, whose dimension is not too small and whose rational cohomology algebra is an exterior algebra on homogeneous generators of odd degree. In case X is a simply connected classical group, a 3-connected real or a 5-connected complex or a quaternionic Stiefel manifold, we obtain (in principle) the classification of the transitive actions on X up to equivariant homeomorphism.     

Four-cocycles in commutative semigroup cohomology

Semigroup Forum - The symmetric cohomology group of a commutative semigroup is defined in dimension 4 and is shown to equal its triple cohomology group.     

Asymptotic Behaviour and Artinian Property of Graded Local Cohomology Modules

In this article, considering the difference between the finiteness dimension and cohomological dimension of a finitely generated graded module, we investigate the asymptotic behaviour of grades of components of a graded local cohomology module with respect to the irrelevant ideal. To this end, we study some Artinian and tameness properties of certain graded local cohomology modules.