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...

This paper develops a duality theory for connected cochain DG algebras, with particular emphasis on the non-commutative aspects. One of the main items is a dualizing DG module which induces a duality between the derived categories of DG left-modules and DG right-modules with finitely generated cohomology. As an application, it is proved that if the canonical module A/A has a semi-free resolutio...

We also give partial results on computation of cohomology groups on U for a locally constant sheaf of general rank and on computation of H(C \ Z,C) where Z is a general algebraic set. Our algorithm is based on computations of Gröbner bases in the ring of differential operators with polynomial coefficients, algorithms for functors in the theory of D-modules ([24] and [25]), and Grothendieck-Deli...

0. Introduction. This paper continues the study of the noncommutative innn-itesimal cohomology we introduced in 3]. This is the cohomology of sheaves on a noncommutative version of the commutative innnitesimal site of Grothendieck ((8]). Grothendieck showed that, for a smooth scheme X of characteristic zero, the cohomology of the structure sheaf on the innnitesimal site gives de Rham cohomol-og...

Let G be a finite group with integral group ring Λ = Z[G]. The syzygies Ωr(Z) are the stable classes of the intermediate modules in a free Λ-resolution of the trivial module. They are of significance in the cohomology theory of G via the ‘co-represention theorem’ Hr(G,N) = HomDer(Ωr(Z), N). We describe the Ωr(Z) explicitly for the dihedral groups D4n+2, so allowing the construction of free reso...

Freyd’s generating hypothesis, interpreted in the stable module category of a finite p-group G, is the statement that a map between finite-dimensional kG-modules factors through a projective if the induced map on Tate cohomology is trivial. We show that Freyd’s generating hypothesis holds for a non-trivial p-group G if and only if G is either Z/2 or Z/3. We also give various conditions which ar...

One of the purposes in the computation of cohomology groups is to establish invariants which may be helpful in the classification of the objects under consideration. In the theory of continuous Hochschild cohomology for operator algebras R. V. Kadison and J. R. Ringrose proved [10] that for any hyperfinite von Neumann algebra M and any dual normal M-bimodule S, all the continuous cohomology gro...

Let M be a finitely generated module over an Artin algebra. By considering the lengths of each module in the minimal projective resolution of M , we obtain its Betti sequence. This sequence must be bounded if M is eventually periodic, but the converse fails to hold in general. We give conditions under which this holds, using techniques from Hochschild cohomology. We also provide a result which ...

On a symplectic manifold M , the quantum product defines a complex, one parameter family of flat connections called the A-model or Dubrovin connections. Let ~ denote the parameter. Associated to them is the quantum D module D/I over the Heisenberg algebra of first order differential operators on a complex torus. An element of I gives a relation in the quantum cohomology of M by taking the limit...