Modules and Cohomology over Group Algebras: One Commutative Algebraist's Perspective

Three-Dimensional Manifolds, Skew-Gorenstein Rings and their Cohomology

Graded skew-commutative rings occur often in practice. Here are two examples: 1) The cohomology ring of a compact three-dimensional manifold. 2) The cohomology ring of the complement of a hyperplane arrangement (the Orlik-Solomon algebra). We present some applications of the homological theory of these graded skew-commutative rings. In particular we find compact oriented 3-manifolds without boundary for which the Hilbert series of the Yoneda Ext-algebra of the cohomology ring of the fundamental group is an explicit transcendental function. This is only possible for large first Betti numbers of…

Thick subcategories of modules over commutative rings

For a commutative noetherian ring A, we compare the support of a complex of A-modules with the support of its cohomology. This leads to a classification of all full subcategories of A-modules which are thick (that is, closed under taking kernels, cokernels, and extensions) and closed under taking direct sums.

The Jacobian ideal of a commutative ring and annihilators of cohomology

It is proved that for a ring $R$ that is either an affine algebra over a field, or an equicharacteristic complete local ring, some power of the Jacobian ideal of $R$ annihilates $\mathrm{Ext}^{d+1}_{R}(-,-)$, where $d$ is the Krull dimension of $R$. Sufficient conditions are identified under which the Jacobian ideal itself annihilates these Ext-modules, and examples are provided that show that this is not always the case. A crucial new idea is to consider a derived version of the Noether different of an algebra.

An Auslander-Buchsbaum identity for semidualizing modules

Of the many interesting insights in the Auslander-Bridger Memoir of 1969, the theory of Gorenstein dimension has most often been taken up by commutative algebraists. Over a local ring, it deals with resolutions by modules which are totally reflexive with regard to free modules. For some years now, one has been trying to replace frees by semidualizing modules. This has met with success, and more is to follow. To me however, it makes sense to first try and understand semidualizing modules themselves better. This note shows in a direct way that they share many properties with free modules. One ho…

On higher order analogues of de Rham cohomology

If K is a commutative ring and A is a K-algebra, for any sequence $σ$ of positive integers there exists an higher order analogue dR($σ$) of the standard de Rham complex dR(1,...,1,...), which can also be defined starting from suitable ("differentially closed") subcategories of (A-mod). The main result of this paper is that the cohomology of dR($σ$) does not depend on $σ$, under some smoothness assumptions on the ambient category. Before proving the main theorem we give a rather detailed exposition of all relevant (to our present purposes) functors of differential calculus on commutative algebr…

Gorenstein dimension of modules

In these expository notes I discuss several concepts and results in the theory of modules over commutative rings, revolving around the Gorenstein dimension of modules. Some of the related notions are the Auslander dual, k-torsionless modules, and k-th syzygies. Essentially everything in these notes can be found, in one form or another, in the memoir "Stable module theory" by M. Auslander and M. Bridger (Mem. A.M.S., no. 94, 1969). The only difference is in presentation. In the Auslander-Bridger memoir many of the results are proved in the most general setting, e.g. over possibly non-commutativ…

A note on some top local cohomology modules

Let $\fa$ be an ideal of a $d$-dimensional commutative Noetherian ring $R$. In this paper we give some information on some last non-zero local cohomology modules known as top local cohomology modules in particular, $H^{d-1}_{\fa}(R)$.

A unified approach to formal local cohomology and local Tate cohomology

Let R be a commutative Noetherian ring. We introduce a theory of formal local cohomology for complexes of R-modules. As an application, we establish some relations between formal local cohomology, local homology, local cohomology and local Tate cohomology through some natural isomorphisms. We investigate vanishing of formal local cohomology modules. Also, we give a characterization of Cohen-Macaulay complexes.

Derived Hochschild functors over commutative adic algebras

Let $\k$ be a commutative ring, and let $(A,\mfrak{a})$ be an adic ring which is a $\k$-algebra. We study complete and torsion versions of the derived Hochschild homology and cohomology functors of $A$ over $\k$. To do this, we first establish weak proregularity of certain ideals in flat base changes of noetherian rings. Next, we develop a theory of DG-affine formal schemes, extending the Greenlees-May duality and the MGM equivalence to this setting. Finally, we define complete and torsion derived Hochschild homology and cohomology functors in this setting, and show that if $\k$ is noetherian …

A Degree Formula for Equivariant Cohomology Rings

This paper generalizes a result of Lynn on the "degree" of an equivariant cohomology ring $H^*_G(X)$. The degree of a graded module is a certain coefficient of its Poincaré series, and is closely related to multiplicity. In the present paper, we study these commutative algebraic invariants for equivariant cohomology rings. The main theorem is an additivity formula for degree: $$deg(H^*_G(X)) = \sum_{[A,c] \in \mathcal{Q'}_{max}(G,X)}\frac{1}{|W_G(A,c)|} °(H^*_{C_G(A,c)}(c)).$$ We also show how this formula relates to the additivity formula from commutative algebra, demonstrating both the algeb…

Attached and Assoiciated Primes Of Local Cohomology Modules Via Linkage

Let $R$ be a commutative Noetherian ring and $M$ be a finitely generated $R$-module. Considering the new concept of linkage of ideals over a module, we study associated prime ideals, cofiniteness and Artinianness of local cohomology modules of $M$ with respect to some linked ideals over it.

Associated primes of formal local cohomology modules

Let $\mathfrak{a}$ be an ideal of a commutative Noetherian ring $R$ and $M$ a finitely generated $R$-module. In this paper we proved that if $\operatorname{Supp}\mathfrak{F}_\mathfrak{a}^i(M)$ is finite for all $i<t$, then so is $\operatorname{Ass}(\mathfrak{F}_{\mathfrak{a}}^t(M))$.

Local cohomology modules supported at determinantal ideals

We provide new results on the vanishing of local cohomology modules supported at ideals of minors of matrices over arbitrary commutative Noetherian rings. In the process, we compute the local cohomology of rings of polynomials with integer coefficients---supported at generic determinantal ideals---and also obtain results on F-modules and D-modules that are likely to be of independent interest.

Socle finiteness of local cohomology modules and Gorenstein ideals

The purpose of this paper is to give some equivalent conditions to the socle and Bass numbers' conjectures which raised by C. Huneke in (Problems on local cohomology, Free resolutions in commutative algebra and algebraic geometry, Res. Notes Math. 1992, pp. 93-108). In addition, some results about certain Gorenstein ideals are included.

The negative side of cohomology for Calabi-Yau categories

We study integer-graded cohomology rings defined over Calabi-Yau categories. We show that the cohomology in negative degree is a trivial extension of the cohomology ring in non-negative degree, provided the latter admits a regular sequence of central elements of length two. In particular, the product of elements of negative degrees are zero. As corollaries we apply this to Tate-Hochschild cohomology rings of symmetric algebras, and to Tate cohomology rings over group algebras. We also prove similar results for Tate cohomology rings over commutative local Gorenstein rings.

Cofiniteness of modules and local cohomology

Let $A$ be a commutative noetherian ring, let $\mathfrak a$ be an ideal of $A$ and let $n$ be a non-negative integer. In this paper, we study $\mathcal{S}_{n}(\mathfrak{a})$, a certain class of $A$-modules and we find some sufficient conditions so that a module belongs to $\mathcal{S}_{n}(\mathfrak{a})$. Moreover, we study the cofiniteness of local cohomology modules when $\dim A/\frak a\geq 3$.

The deformation theory of sheaves of commutative rings

We define a sheaf of abelian groups whose cohomology is represented by the cotangent complex. We show how obstructions to some standard deformation problems arise as the classes of torsors under and gerbes banded by this sheaf.

Cohomology over complete intersections via exterior algebras

A general method for establishing results over a commutative complete intersection local ring by passing to differential graded modules over a graded exterior algebra is described. It is used to deduce, in a uniform way, results on the growth of resolutions of complexes over such local rings.

Regularity of FI-modules and local cohomology

We resolve a conjecture of Li and Ramos that relates the regularity of an FI-module to its local cohomology groups. This is an analogue of the familiar relationship between regularity and local cohomology in commutative algebra.

Annihilation of cohomology and strong generation of module categories

The cohomology annihilator of a noetherian ring that is finitely generated as a module over its center is introduced. Results are established linking the existence of non-trivial cohomology annihilators and the existence of strong generators for the category of finitely generated modules. Exploiting this link, results of Popescu and Roczen, and Wang concerning cohomology annihilators of commutative rings, and also results of Aihara and Takahashi, Keller and Van den Bergh, and Rouquier on strong finite generation of the corresponding bounded derived category, are generalized to cover excellent …

The Multiplicative Structure on Hochschild Cohomology of a Complete Intersection

We determine the product structure on Hochschild cohomology of commutative algebras in low degrees, obtaining the answer in all degrees for complete intersection algebras. As applications, we consider cyclic extension algebras as well as Hochschild and ordinary cohomology of finite abelian groups.

Reduction of derived Hochschild functors over commutative algebras and schemes

We study functors underlying derived Hochschild cohomology, also called Shukla cohomology, of a commutative algebra S essentially of finite type and of finite flat dimension over a commutative noetherian ring K. We construct a complex of S-modules D, and natural reduction isomorphisms Ext^*_{S\otimes^L_{K}S}(S|K;M\otimes^L_{K}N) ~ Ext^*_S(RHom_S(M,D),N) for all complexes of S-modules N and all complexes M of finite flat dimension over K whose homology H(M) is finitely generated over S; such isomorphisms determine D up to derived isomorphism. Using Grothendieck duality theory we establish analo…

Annihilators of local cohomology modules via a classification theorem of the dominant resolving subcategories

This paper investigates when local cohomology modules have an annihilator that does not depend on the choice of an ideal. Takahashi classified the dominant resolving subcategories of the category of finitely generated modules over a commutative Noetherian ring. We show that his classification theorem describes annihilation results of local cohomology modules over a finite-dimensional ring with certain assumptions or a Cohen-Macaulay ring.

Cohomology of finite graded group varieties

We prove that, if $A$ is a positively graded, graded commutative, local, finite Hopf algebra, its cohomology is finitely generated, thus unifying classical results of Wilkerson and Hopkins-Smith, and of Friedlander-Suslin. We do this by showing the existence of conormal elementary quotients.

Tate cohomology with respect to semidualizing modules

We investigate Tate cohomology of modules over a commutative noetherian ring with respect to semidualizing modules. We identify classes of modules admitting Tate resolutions and analyze the interaction between the corresponding relative and Tate cohomology modules. As an application of our approach, we prove a general balance result for Tate cohomology. Our results are based on an analysis of Tate cohomology in abelian categories.

On some local cohomology invariants of local rings

Let A be a commutative Noetherian local ring containing a field of characteristic p>0. The integer invariants $λ_{i,j}(A)$ have been introduced in an old paper of ours. In this paper we completely describe $λ_{d,d}(A)$, where d=dimA, in terms of the topology of SpecA.

Introduction to uniformity in commutative algebra

These notes are based on three lectures given by the first author as part of an introductory workshop at MSRI for the program in Commutative Algebra, 2012-13. The notes follow the talks, but there are extra comments and explanations, as well as a new section on the uniform Artin-Rees theorem. The notes deal with the theme of uniform bounds, both absolute and effective, as well as uniform annihilation of cohomology

Faltings' local-global principle for the minimaxness of local cohomology modules

The concept of Faltings' local-global principle for the minimaxness of local cohomology modules over a commutative Noetherian ring $R$ is introduced, and it is shown that this principle holds at level 2. We also establish the same principle at all levels over an arbitrary commutative Noetherian ring of dimension not exceeding 3. These generalize the main results of Brodmann et al. in \cite{BRS}. Moreover, it is shown that if $M$ is a finitely generated $R$-module, $\frak a$ an ideal of $R$ and $r$ a non-negative integer such that $\frak a^tH^i_{\frak a}(M)$ is skinny for all $i<r$ and for some…

关于Siegfried Bosch《代数几何与交换代数》的笔记（献给格罗滕迪克）

Notes on Siegfried Bosch's Algebraic Geometry and Commutative Algebra (dedicated to Grothendieck)

本文系统整理交换代数与代数几何的核心内容，涵盖环模层概形及平展上同调等进阶主题，作为对格罗滕迪克的致敬。

Local cohomology modules of a smooth Z-algebra have finitely many associated primes

Let $R$ be a commutative Noetherian ring that is a smooth $\mathbb Z$-algebra. For each ideal $I$ of $R$ and integer $k$, we prove that the local cohomology module $H^k_I(R)$ has finitely many associated prime ideals. This settles a crucial outstanding case of a conjecture of Lyubeznik asserting this finiteness for local cohomology modules of all regular rings.

Cofiniteness of generalized local cohomology modules

Let $\fa$ denote an ideal of a commutative Noetherian ring $R$ and $M$ and $N$ two finitely generated $R$-modules with $\pd M< \infty$. It is shown that if $\fa$ is principal or $R$ is complete local and $\fa$ a prime ideal with $\dim R/\fa=1$, then the generalized local cohomology module $H^i_{\fa}(M,N)$ is $\fa$-cofinite for all $i \geq 0$. This provides an affirmative answer for the above ideal $\fa$ to a question proposed in [{\bf 13}].

关于相对于理想系统的广义局部上同调模的余有限性

Cofiniteness of generalized loca cohomology modules with respect to a system of ideals

本文研究了交换Noether环上相对于理想系统的广义局部上同调模$H^i_Φ(M,X)$的Artin性与余有限性，该模扩展了Herzog的广义局部上同调模。

On generalized Hartshorne's conjecture and local cohomology modules

Let $\mathfrak{a}$ denote an ideal of a commutative Noetherian ring $R$. Let $M$ and $N$ be two $R$-modules. In this paper, we give partial answers on the extension of Hartshorne's conjecture about the cofiniteness of torsion and extension functors. For this purpose, we study the cofiniteness of the generalized local cohomology module $H^{i}_{\mathfrak{a}}(M, N)$ for a new class of modules, called $\mathfrak{a}$-weakly finite modules, in the local and non-local case. Furthermore, we derive some results on attached primes of top generalized local cohomology modules.

First steps in brave new commutative algebra

This is an expository account of completion and local cohomology in brave new commutative algebra, especially as it applies to completion theorems and their duals in equivariant topology. The first part is fairly direct, but the second considers Morita theory and Gorenstein ring spectra. This draws on work of the author with Benson, Dwyer, Iyengar and May, amongst others.

Finite injective dimension over rings with Noetherian cohomology

We study rings which have Noetherian cohomology under the action of a ring of cohomology operators. The main result is a criterion for a complex of modules over such a ring to have finite injective dimension. This criterion generalizes, by removing finiteness conditions, and unifies several previous results. In particular we show that for a module over a ring with Noetherian cohomology, if all higher self-extensions of the module vanish then it must have finite injective dimension. Examples of rings with Noetherian cohomology include commutative complete intersection rings and finite dimension…

Length and Multiplicities in Graded Commutative Algebra

This paper is a review of concepts from graded commutative algebra with specific attention given to length and multiplicity. The author's motivation for this paper comes from the study of equivariant cohomology in algebraic topology where the graded commutative algebra of the module is intimately connected to topological properties of the space (as shown by Quillen). Results (and their proofs) which are often left as exercises, or considered 'folklore' in the commutative algebra community are included in this paper, as are references to relevant applications in topology. As such, this paper is…

Annihilating the cohomology of group schemes

Our goal in this note is to show that cohomology classes with coefficients in finite flat group schemes can be killed by finite covers of the base scheme, and similarly for abelian schemes with "finite covers" replaced by "proper covers." We apply this result to commutative algebra by giving a new and more conceptual proof of Hochster-Huneke's theorem on the existence of big Cohen-Macaulay algebras in positive characteristic; all previous proofs of this result were equational or cocycle-theoretic in nature.

Some results on generalized local cohomology modules

Let $R$ be a commutative Noetherian ring with non-zero identity, $\fa$ an ideal of $R$, $M$ a finite $R$--module and $X$ an arbitrary $R$--module. Here, we show that, in the Serre subcategories of the category of $R$--modules, how the generalized local cohomology modules, the ordinary local cohomology modules and the extension modules behave similarly at the initial points. We conclude some Artinianness and cofiniteness results for $\lc^{n}_{\fa}(M, X)$, and some finiteness results for $\Supp_R(\lc^{n}_{\fa}(M, X))$ and $\Ass_R(\lc^{n}_{\fa}(M, X))$.

Lie-Rinehart cohomology and integrable connections on modules of rank one

Let $k$ be an algebraically closed field of characteristic 0, let $R$ be a commutative $k$-algebra, and let $M$ be a torsion free $R$-module of rank one with a connection $\nabla$. We consider the Lie-Rinehart cohomology with values in $End_{R}(M)$ with its induced connection, and give an interpretation of this cohomology in terms of the integrable connections on $M$. When $R$ is an isolated singularity of dimension $d\geq2$, we relate the Lie-Rinehart cohomology to the topological cohomology of the link of the singularity, and when $R$ is a quasi-homogenous hypersurface of dimension two, we g…