Let $R$ be a commutative Noetherian ring with non-zero identity, $fa$ an ideal of $R$, and $X$ an $R$--module. Here, for fixed integers $s, t$ and a finite $fa$--torsion $R$--module $N$, we first study the membership of $Ext^{s+t}_{R}(N, X)$ and $Ext^{s}_{R}(N, H^{t}_{fa}(X))$ in the Serre subcategories of the category of $R$--modules. Then, we present some conditions which ensure the exi...

A free bimodule resolution is constructed for the integral group ring of the dihedral group of order 4m. This resolution is applied for a description, in terms of generators and defining relations, of the Hochschild cohomology algebra of this group ring. Introduction Let K be a commutative ring with unity, let R be an associative K-algebra that is a finitely generated projective K-module, let Λ...

In this paper, we study the (small) quantum cohomology ring of the partial flag manifold. We give proofs of the presentation of the ring and of the quantum Giambelli formula for Schubert varieties. These are known results, but our proofs are more natural and direct than the previous ones. One of our goals is to give evidence of a relationship between universal Schubert polynomials, which give t...

We give some general results on the ring structure of Hochschild cohomology of smash products of algebras with Hopf algebras. We compute this ring structure explicitly for a large class of finite dimensional Hopf algebras of rank one.

In this note we describe aspects of the cohomology of coherent sheaves on a complete toric variety X over a field k and, more generally, the local cohomology, with supports in a monomial ideal, of a finitely generated module over a polynomial ring S. This leads to an efficient way of computing such cohomology, for which we give explicit algorithms. The problem is finiteness. The ith local cohom...

We study Ruan’s cohomological crepant resolution conjecture [39] for orbifolds with transversal ADE singularities. In the An-case we compute both the orbifold cohomology ring H ∗ orb([Y ]) and the quantum corrected cohomology ring H(Z)(q1, ..., qn). The former is achieved in general, the later up to some additional, technical assumptions. We construct an explicit isomorphism between H orb([Y ])...

A large chunk of the intersection theory of the moduli stacks of n-marked, genus 0 stable maps to a specified target variety X is encoded by the quantum cohomology ring of X. In particular, one can use the associativity of the ring structure to calculate Gromov-Witten invariants and, thus, in favorable circumstances, to calculate the number of rational curves in X passing through an appropriate...

Consider the ring R := Q[τ, τ−1] of Laurent polynomials in the variable τ . The Artin’s Pure Braid Groups (or Generalized Pure Braid Groups) act over R, where the action of every standard generator is the multiplication by τ . In this paper we consider the cohomology of these groups with coefficients in the module R (it is well known that such cohomology is strictly related to the untwisted int...

Chen and Ruan [6] defined a very interesting cohomology theory-Chen-Ruan orbifold cohomology. The primary objective of this paper is to compute the Chen-Ruan orbifold cohomology of weighted projective space, one of the most important space used in physics. The classical tools (orbifold cohomology defined by Chen and Ruan,toric varieties, the localization formula) which have been proved to be su...

In this paper, we introduce the notion of $G_\infty$-ring spectra. These are globally equivariant homotopy types with a structured multiplication, giving rise to power operations on their and cohomology groups. We illustrate structure by analysing when Moore spectrum can be endowed structure. Such $G_\infty$-structures correspond underlying ring, indexed Burnside ring. exhibit close relation be...