Balance in complete cohomology

We describe degree zero mod-p complete cohomology modulo its radical in purely group-theoretic terms, for members of a class of groups that includes all groups of finite virtual cohomological dimension. We give examples to show that our description does not apply to all discrete groups. We also give examples of discrete groups to which Quillen's description of the ordinary mod-p cohomology ring (up to F-isomorphism) does not apply.

A group $G$ is said to have periodic cohomology with period $q$ after $k$ steps, if the functors $H^{i}(G,-)$ and $H^{i+q}(G,-)$ are naturally equivalent for all $i>k$ . Mislin and the author have conjectured that periodicity in cohomology after some steps is the algebraic characterization of those groups $G$ that admit a finite-dimensional, free $G$ -CW-complex, homotopy equivalent to a sphere. This conjecture was proved by Adem and Smith under the extra hypothesis that the periodicity isomorphisms are given by the cup product with an element in $H^q (G,\mathbb{Z})$ . It is expected that the periodicity isomorphisms will always be given by the cup product with an element in $H^q (G,\mathbb{Z})$ ; this paper shows that this is the case if and only if the group $G$ admits a complete resolution and its complete cohomology is calculated via complete resolutions. It is also shown that having the periodicity isomorphisms given by the cup product with an element in $H^q (G,\mathbb{Z})$ is equivalent to silp $G$ being finite, where silp $G$ is the supremum of the injective lengths of the projective $\mathbb{Z}G$ -modules.

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">$ (\mathcal{C}, \mathbb{E}, \mathfrak{s}) $</tex-math></inline-formula> be an extriangulated category with a proper class <inline-formula><tex-math id="M2">$ \xi $</tex-math></inline-formula> of <inline-formula><tex-math id="M3">$ \mathbb{E} $</tex-math></inline-formula>-triangles. In this paper, we study the balance of complete cohomology in <inline-formula><tex-math id="M4">$ (\mathcal{C}, \mathbb{E}, \mathfrak{s}) $</tex-math></inline-formula>, which is motivated by a result of Nucinkis that complete cohomology of modules is not balanced in the way the absolute cohomology Ext is balanced. As an application, we give some criteria for identifying a triangulated catgory to be Gorenstein and an Artin algebra to be <inline-formula><tex-math id="M5">$ F $</tex-math></inline-formula>-Gorenstein.</p>

Let (R,𝔪) be a Noetherian local ring, M a finitely generated R-module with finite projective dimension n, N an arbitrary R-module, and 𝔞 be an ideal of R which is generated by s elements. In this article, we provide a surjective homomorphism from ordinary local cohomology module to top generalized local cohomology module , where P n, M is an nth syzygy of a projective resolution of M. Also, by using this epimorphism, we prove some results about the attached primes, coassociated primes, the Betti numbers, and Artinian properties of certain generalized local cohomology modules.

Let R be a Noetherian ring, and I and J be two ideals of R. Let S be a Serre subcategory of the category of R-modules satisfying the condition CI and M be a ZD-module. As a generalization of the S-depth(I, M) and depth(I, J, M), the S-depth of (I, J) on M is defined as $$S{\rm{ - depth}}\left( {I,J,M} \right) = \inf \left\{ {S{\rm{ - depth}}\left( {\mathfrak{a},M} \right): \mathfrak{a}\in \widetilde{W}\left( {I,J} \right)} \right\}$$ , and some properties of this concept are investigated. The relations between S-depth(I, J, M) and H (M) are studied, and it is proved that S-depth(I, J, M) = inf{i: H (M) ∉ S}, where S is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology modules with respect to a pair of ideals coincide with ordinary local cohomology modules under these conditions. Let SuppRH (M) be a finite subset of Max(R) for all i < t, where M is an arbitrary R-module and t is an integer. It is shown that there are distinct maximal ideals $$\mathfrak{m}_1,\mathfrak{m}_2,...\mathfrak{m}_k\in{W(I,J)}$$ such that $$H_{I,J}^i\left( M \right) \cong H_{\mathfrak{m}{_1}}^i\left( M \right) \oplus H_{\mathfrak{m}{_2}}^i\left( M \right) \oplus \ldots \oplus H_{\mathfrak{m}_{_k}}^i\left( M \right)$$ for all i < t.

Local cohomology based on a nonclosed support defined by a pair of ideals

Duality and de Rham cohomology for graded [formula omitted]-modules