Mordechai Katzman
This paper applies G. Lyubeznik's notion of $F$-finite modules to describe in a very down-to-earth manner certain annihilator submodules of some top local cohomology modules over Gorenstein rings. As a consequence we obtain an explicit description of the test ideal of Gorenstein rings in terms of ideals in a regular ring.
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Let $R$ be the power series ring or the polynomial ring over a field $k$ and let $I $ be an ideal of $R.$ Macaulay proved that the Artinian Gorenstein $k$-algebras $R/I$ are in one-to-one correspondence with the cyclic $R$-submodules of the divided power series ring $Γ. $ The result is effective in the sense that any polynomial of degree $s$ produces an Artinian Gorenstein $k$-algebra of socle degree $s.$ In a recent paper, the authors extended Macaulay's correspondence characterizing the $R$-submodules of $Γ$ in one-to-one correspondence with Gorenstein d-dimensional $k$-algebras. However, th…
Gorenstein injective filtrations over rings with dualizing complexes
本文将Gorenstein环上Gorenstein内射模的有限Gorenstein内射子模滤过结论，推广至带对偶复形的交换诺特环，并用Auslander范畴和截面函子实现了这类滤过。
We investigate symmetry in the vanishing of Tate cohomology for finitely generated modules over local Gorenstein rings. For finitely generated R-modules M and N over Gorenstein local ring R, it is shown that $\widehat{Ext}^i_R(M,N)=0$ for all $i\in\mathbb{Z}$ if and only if $\widehat{Ext}^i_R(N,M)=0$ for all $i\in\mathbb{Z}$.
We define a notion of Gorenstein flat dimension for unbounded complexes over left GF-closed rings. Over Gorenstein rings we introduce a notion of Gorenstein cohomology for complexes; we also define a generalized Tate cohomology for complexes over Gorenstein rings, and we show that there is a close connection between the absolute, the Gorenstein and the generalized Tate cohomology.
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.
We describe the graded characters and Hilbert functions of certain graded artinian Gorenstein quotients of the polynomial ring which are also representations of the symmetric group. Specifically, we look at those algebras whose socles are trivial representations and whose principal apolar submodules are generated by the sum of the orbit of a power of a linear form.
Annihilation of cohomology over one dimensional almost Gorenstein rings
本文研究一维几乎Gorenstein环的上同调湮灭理想，给出其与稳定范畴湮灭理想相等的充分条件，并证明其与导子理想一致。
We prove that if M, N are finite modules over a Gorenstein local ring R of codimension at most 4, then the vanishing of Ext^n_R(M,N) for n\gg 0 is equivalent to the vanishing of Ext^n_R(N,M) for n\gg 0. Furthermore, if the completion of $R$ has no embedded deformation, then such vanishing occurs if and only if M or N has finite projective dimension.
We give counterexamples to the following conjecture of Auslander: given a finitely generated module $M$ over an Artin algebra $Λ$, there exists a positive integer $n_M$ such that for all finitely generated $Λ$-modules $N$, if $\Ext_Λ^i(M,N)=0$ for all $i\gg 0$, then $\Ext_Λ^i(M,N)=0$ for all $i\geq n_M$. Some of our examples moreover yield homologically defined classes of commutative local rings strictly between the class of local complete intersections and the class of local Gorenstein rings.
Direct products of Gorenstein flat modules over coherent rings
本文构造了零Krull维数的交换凝聚局部代数及相关模，否定回答了右凝聚环上Gorenstein平坦模是否对直积封闭的问题，给出了相关等价条件并揭示了抽象初等类的性质。
Let $R$ be a Gorenstein local ring with maximal ideal $\mathfrak{m}$ satisfying $\mathfrak{m}^3=0\ne\mathfrak{m}^2$. Set $k=R/\mathfrak{m}$ and $e=\text{rank}_{k}(\mathfrak{m}/\mathfrak{m}^2)$. If $e>2$ and $M$, $N$ are finitely generated $R$-modules, we show that the formal power series $\sum_{i=0}^\infty\text{rank}_{k}\left(\text{Ext}^i_R(M,N)\otimes_R k \right)t^i$ and $\sum_{i=0}^\infty\text{rank}_{k}\left(\text{Tor}_i^R(M,N)\otimes_R k \right)t^i$ are rational, with denominator $1-et+t^2$.
The aim of this paper is to study the theory of cohomology annihilators over commutative Gorenstein rings. We adopt a triangulated category point of view and study the annihilation of stable category of maximal Cohen-Macaulay modules. We prove that in dimension one the cohomology annihilator ideal and the conductor ideal coincide under mild assumptions. We present a condition on a ring homomorphism between Gorenstein rings which allows us to carry the cohomology annihilator of the domain to that of the codomain. As an application, we generalize the Milnor-Jung formula for algebraic curves to t…
Let $\fa$ be an ideal of a Noetherian local ring $R$ and let $C$ be a semidualizing $R$-module. For an $R$-module $X$, we denote any of the quantities $\fd_R X$, $\Gfd_R X$ and $\GCfd_RX$ by $\T(X)$. Let $M$ be an $R$-module such that $\H_{\fa}^i(M)=0$ for all $i\neq n$. It is proved that if $\T(X)<\infty$, then $\T(\H_{\fa}^n(M))\leq\T(M)+n$ and the equality holds whenever $M$ is finitely generated. With the aid of these results, among other things, we characterize Cohen-Macaulay modules, dualizing modules and Gorenstein rings.
Let $R$ be a Gorenstein local ring, $\frak{a}$ an ideal in $R$, and $M$ an $R$-module. The local cohomology of $M$ supported at $\frak{a}$ can be computed by applying the $\frak{a}$-torsion functor to an injective resolution of $M$. Since $R$ is Gorenstein, $M$ has a complete injective resolution, so it is natural to ask what one gets by applying the $\frak{a}$-torsion functor to it. Following this lead, we define stable local cohomology for modules with complete injective resolutions. This gives a functor to the stable category of Gorenstein injective modules. We show that in many ways this b…
Over an arbitrary commutative ring, correspondences among three sets, the set of trace ideals, the set of stable ideals, and the set of birational extensions of the base ring, are studied. The correspondences are well-behaved, if the base ring is a Gorenstein ring of dimension one. It is shown that with one extremal exception, the surjectivity of one of the correspondences characterizes the Gorenstein property of the base ring, provided it is a Cohen-Macaulay local ring of dimension one. Over a commutative Noetherian ring, a characterization of modules in which every submodule is a trace modul…
Let $(R,\fm)$ be a local ring and let $C$ be a semidualizing $R$--module. In this paper, we are concerned in $C$--injective and $G_{C}$--injective dimensions of certain local cohomology modules of $R$. Firstly, the injective dimension of $C$ and the above quantities of dimensions is compared. Then, as an application of the above comparisons, a characterization of a dualizing module of $R$ is given. Finally, it is shown that if $R$ is Cohen-Macaulay of dimension $d$ such that $\H_{\fm}^{d}(C)$ is $C$--injective, then $R$ is Gorernstein. This is an answer to the question which was recently prese…
Quasi-Gorenstein extended Rees algebras associated with filtrations
本文研究诺特局部环上Hilbert滤链对应的扩张Rees代数的拟Gorenstein性质，利用局部上同调模的Matlis对偶的Cohen-Macaulay性给出变形条件，并借助局部上同调长度给出拟Gorenstein性质的刻画。
Direct limits of Gorenstein injective modules
本文研究Gorenstein内射模类对任意正向极限封闭的条件，证明其与左noetherian性、特征模Gorenstein平坦性、覆盖性及纯子模或纯商封闭性等价。
Macaulay's Inverse System gives an effective method to construct Artinian Gorenstein k-algebras. To date a general structure for Gorenstein k-algebras of any dimension (and codimension) is not understood. In this paper we extend Macaulay's correspondence characterizing the submodules of the divided power ring in one-to-one correspondence with Gorenstein d-dimensional k-algebras. We discuss effective methods for constructing Gorenstein graded rings. Several examples illustrating our results are given.
In this paper, we show that for an $F$-pure local ring $(R,\m)$, all local cohomology modules $H_{\m}^i(R)$ have finitely many Frobenius compatible submodules. This answers positively an open question raised by F.Enescu and M.Hochster. We also prove that if $(R,\m)$ is excellent and is $F$-pure on the punctured spectrum, then all local cohomology modules have finite length in the category of $R$-modules with Frobenius action. Finally, we show that the property that all $H_{\m}^(R)$ have finitely many Frobenius compatible submodules passes to localizations.
In this paper several quasi-Gorenstein counterparts to some known properties of Gorenstein rings are given. We, furthermore, give an explicit description of the attach prime ideals of certain local cohomology modules.
We find the defining equations of Rees rings of linearly presented height three Gorenstein ideals. To prove our main theorem we use local cohomology techniques to bound the maximum generator degree of the torsion submodule of symmetric powers in order to conclude that the defining equations of the Rees algebra and the special fiber ring have the same image in the symmetric algebra. We show that this image is the unmixed part of the ideal generated by the maximal minors of a matrix of linear forms which is annihilated by a vector of indeterminates, and otherwise has maximal possible grade. An i…
We investigate relative cohomology functors on subcategories of abelian categories via Auslander-Buchweitz approximations and the resulting strict resolutions. We verify that certain comparison maps between these functors are isomorphisms and introduce a notion of perfection for this context. Our main theorem is a balance result for relative cohomology that simultaneously recovers theorems of Holm and the current authors as special cases.
A generalization of Grothendieck's non-vanishing theorem is proved for a module which is finite over a local homomorphism. It is also proved that the Gorenstein injective dimension of such a module, if finite, is bounded below by its Krull dimension and is equal to the supremum of the depths of the localizations of the ring over primes in the support of the module.
For homomorphism K-->S of commutative rings, where K is Gorenstein and S is essentially of finite type and flat as a K-module, the property that all non-trivial fiber rings of K-->S are Gorenstein is characterized in terms of properties of the cohomology modules Ext_n^{S\otimes_KS}S{S\otimes_KS}.
Let $(R, \m)$ be a $d$-dimensional commutative noetherian local ring. Let $\M$ denote the morphism category of finitely generated $R$-modules and let $\Sc$ be the submodule category of $\M$. In this paper, we specify the Auslander transpose in submodule category $\Sc$. It will turn out that the Auslander transpose in this category can be described explicitly within ${\rm mod}R$, the category of finitely generated $R$-modules. This result is exploited to study the linkage theory as well as the Auslander-Reiten theory in $\Sc$. Indeed, a characterization of horizontally linked morphisms in terms…
Let $(R,\fm)$ be a commutative Noetherian local ring and let $M$ be an $R$-module which is a relative Cohen-Macaulay with respect to a proper ideal $\fa$ of $R$ and set $n:=\h_{M}\fa$. We prove that $\ind M<\infty$ if and only if $\ind\H^{n}_\fa(M)<\infty$ and that $\ind\H^{n}_\fa(M)=\ind M-n$. We also prove that if $R$ has a dualizing complex and $\Gid_{R} M<\infty$, then $\Gid_{R}\H^{n}_\fa(M)<\infty$ and $\Gid_{R}\H^{n}_\fa(M)=\Gid_{R} M-n$. Moreover if $R$ and $M$ are Cohen-Macaulay, then it is proved that $\Gid_{R} M<\infty$ whenever $\Gid_{R}\H^{n}_\fa(M)<\infty$. Next, for a finitely ge…
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.
Given a local ring of positive prime characteristic there is a natural Frobenius action on its local cohomology modules with support at its maximal ideal. In this paper we study the local rings for which the local cohomology modules have only finitely many submodules invariant under the Frobenius action. In particular we prove that F-pure Gorenstein local rings as well as the face ring of a finite simplicial complex localized or completed at its homogeneous maximal ideal have this property. We also introduce the notion of an anti-nilpotent Frobenius action on an Artinian module over a local ri…
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…
In this paper we present a condition on a local Cohen-Macaulay F-injective ring of positive characteristic $p > 2$ which implies that its top local cohomology module with support in the maximal ideal has finitely many Frobenius compatible submodules.
Local Gorenstein duality in chromatic group cohomology
本文研究了紧 Lie 群分类空间上链谱的局部 Gorenstein 对偶性，证明其在多种环谱（如 Lubin-Tate 理论、拓扑 K-理论和拓扑模形式）中系统成立，并通过下降结果扩展了适用范围。
Let \fa be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let \cd_{\fa}(M,N) denote the supremum of the i's such that H^i_{\fa}(M,N)\neq 0. First, by using the theory of Gorenstein homological dimensions, we obtain several upper bounds for \cd_{\fa}(M,N). Next, over a Cohen-Macaulay local ring (R,\fm), we show that \cd_{\fm}(M,N)=\dim R-\grade(\Ann_RN,M), provided that either projective dimension of M or injective dimension of N is finite. Finally, over such rings, we establish an analogue of the Hartshorne-Lichtenbaum Vanishing Theorem in the context…
The paper describes a duality phenomenon for cohomology theories with the character of Gorenstein rings. For a connective cohomology theory with the p-local integers in degree 0, and coefficient ring R_* Gorenstein of shift 0, this states that for X with R_*(X) torsion, we have R^*(X)=Σ^a Hom( R_*(X), Z/p^{\infty}). A corresponding statement for modules over a commutative Gorenstein ring spectrum is also proved. [Minor typographical and bibliographic changes to the last version.]
For a horizontally linked module, over a commutative semiperfect Noetherian ring $R$, the connections of its invariants reduced grade, Gorenstein dimension and depth are studied. It is shown that under certain conditions the depth of a horizontally linked module is equal to the reduced grade of its linked module. The connection of the Serre condition $(S_n)$ on an $R$--module of finite Gorenstein dimension with the vanishing of the local cohomology groups of its linked module is discussed.
Let $(R,\fm)$ be a local ring and $(-)^{\vee}$ denote the Matlis duality functor. We investigate the relationship between Foxby equivalence and local duality through generalized local cohomology modules. Assume that $R$ possesses a normalized dualizing complex $D$ and $X$ and $Y$ are two homologically bounded complexes of $R$-modules with finitely generated homology modules. We present several duality results for $\fm$-section complex ${\bf R}Γ_{\fm}({\bf R}\Hom_R(X,Y))$. In particular, if G-dimension of $X$ and injective dimension of $Y$ are finite, then we show that $${\bf R}Γ_{\fm}({\bf R}\…
Gorenstein rings via homological dimensions, and symmetry in vanishing of Ext and Tate cohomology
本文通过张量-同构伴随谱序列研究交换Noetherian局部环的Gorenstein性，给出用有限Gorenstein维数CM模、内射维数有限性的刻画，还证明CM环上Ext与Tor消失的对称性及Gorenstein环上Tate上同调的类似对称结果。
Bockstein cohomology of Maximal Cohen-Macaulay modules over Gorenstein isolated singularities
本文研究了Gorenstein孤立奇点上极大Cohen-Macaulay模的Bockstein上同调，证明了在特定理想条件下其相伴分次模的Bockstein上同调具有有限长度。
It is proved that if one of the finite modules M and N, over a local ring R, has reducible complexity and has finite Gorenstein dimension then the depth formula holds, provided TorR_i(M,N) = 0 for i>>0. We also study the vanishing of cohomology of a module of finite complete intersection dimension.
We study commutative algebras with Gorenstein duality, i.e. algebras $A$ equipped with a non-degenerate bilinear pairing such that $\langle ac,b\rangle=\langle a,bc\rangle$ for any $a,b,c\in A$. If an algebra $A$ is Artinian, such pairing exists if and only if $A$ is Gorenstein. We give a description of algebras with Gorenstein duality as a quotients of the ring of differential operators by the annihilator of an explicit polynomial (or more generally formal polynomial series). This provides a calculation of Macaulay's inverse systems for (not necessarily Artinian) algebras with Gorenstein dual…
Homological dimensions of Burch ideals, submodules and quotients
本文利用Burch理想、Burch子模及其商的同调不变量刻画局部环，并给出关于复杂度、曲率、Gorenstein性质与Cohen-Macaulay维数的判定结果。
Let $\mathfrak a$ be an ideal of a commutative Noetherian ring $R$ and $t$ be a non-negative integer. Let $M$ and $N$ be two finitely generated $R$-modules. In certain cases, we give some bounds under inclusion for the annihilators of $\operatorname{Ext}^t_R(M, N)$ and $\operatorname{H}^t_{\mathfrak a}(M)$ in terms of minimal primary decomposition of the zero submodule of $M$ which are independent of the choice of minimal primary decomposition. Then, by using those bounds, we compute the annihilators of local cohomology and Ext modules in certain cases.
Let $\fa$ denote an ideal of a $d$-dimensional Gorenstein local ring $R$ and $M$ and $N$ two finitely generated $R$-modules with $\pd M< \infty$. It is shown that $H^d_{\fa}(M,N)=0$ if and only if $\dim \hat{R}\big/ \fa\hat{R}+\fp>0$ for all $\fp\in\Ass_{\hat{R}}\hat{M}\cap\Supp_{\hat{R}}\hat{N}$.
Lefschetz duality for local cohomology
本文通过拟 Gorenstein 簇发展联络理论，推广了 Gorenstein 联络，并应用于证明拟 Gorenstein 子空间排列的连通性及经典拓扑 Lefschetz 对偶。
Given a local domain $(R,m)$ of prime characteristic that is a homomorphic image of a Gorenstein ring, Huneke and Lyubeznik proved that there exists a module-finite extension domain $S$ such that the induced map on local cohomology modules $H^i_m(R)\to H^i_m(S)$ is zero for each $i<\dim R$. We prove that the extension $S$ may be chosen to be generically Galois, and analyze the Galois groups that arise.
Let $R = \bigoplus_{n \in \mathbb{N}_{0}} R_{n}$ be a standard graded ring, $M$ be a finite graded $R$-module and $J$ be a homogenous ideal of $R$. In this paper we study the graded structure of the $i$-th local cohomology module of $M$ defined by a pair of ideals $(R_{+},J)$, i.e. $H^{i}_{R_{+},J}(M)$. More precisely, we discuss finiteness property and vanishing of the graded components $H^{i}_{R_{+},J}(M)_{n}$. Also, we study the Artinian property and tameness of certain submodules and quotient modules of $H^{i}_{R_{+},J}(M)$.
This paper studies Frobenius maps on injective hulls of residue fields of complete local rings with a view toward providing constructive descriptions of objects originating from the theory of tight closure. Specifically, the paper describes algorithms for computing parameter test ideals, and tight closure of certain submodules of the injective hull of residue fields of a class of well-behaved rings which includes all quasi-Gorenstein complete local rings.
Auslander conjectured that every Artin algebra satisfies a certain condition on vanishing of cohomology of finitely generated modules. The failure of this conjecture - by a 2003 counterexample due to Jorgensen and Sega - motivates the consideration of the class of rings that do satisfy Auslander's condition. We call them AC rings and show that an AC Artin algebra that is left-Gorenstein is also right-Gorenstein. Furthermore, the Auslander-Reiten Conjecture is proved for AC rings, and Auslander's G-dimension is shown to be functorial for AC rings that are commutative or have a dualizing complex…
Gorenstein rings are important to mathematical areas as diverse as algebraic geometry, where they encode information about singularities of spaces, and homotopy theory, through the concept of model categories. In consequence, the study of Gorenstein rings has led to the advent of a whole branch of homological algebra, known as Gorenstein homological algebra. This paper solves one of the open problems of Gorenstein homological algebra by showing that so-called Gorenstein projective resolutions exist over quite general rings, thereby enabling the definition of a Gorenstein version of derived fun…
We introduce the notion of E-depth of graded modules over polynomial rings to measure the depth of certain Ext modules. First, we characterize graded modules over polynomial rings with (sufficiently) large E-depth as those modules whose (sufficiently) partial general initial submodules preserve the Hilbert function of local cohomology modules supported at the irrelevant maximal ideal, extending a result of Herzog and Sbarra on sequentially Cohen-Macaulay modules. Second, we describe the cone of local cohomology tables of modules with sufficiently high E-depth, building on previous work of the …
Let $M$ denote a finitely generated module over a Noetherian ring $R$. For an ideal $I \subset R$ there is a study of the endomorphisms of the local cohomology module $H^g_I(M), g = \operatorname{grade} (I,M),$ and related results. Another subject is the study of left derived functors of the $I$-adic completion $Λ^I_i(H^g_I(M))$, motivated by a characterization of Gorenstein rings given in the book by Simon and the author. This provides another Cohen-Macaulay criterion. The results are illustrated by several examples. There is also an extension to the case of homomorphisms of two different loc…
Let $R$ be a commutative Noetherian local ring with residue field $k$. Using the structure of Vogel cohomology, for any finitely generated module $M$, we introduce a new dimension, called $ζ$-dimension, denoted by $ζ-dim_R M$. This dimension is finer than Gorenstein dimension and has nice properties enjoyed by homological dimensions. In particular, it characterizes Gorenstein rings in the sense that: a ring $R$ is Gorenstein if and only if every finitely generated $R$-module has finite $ζ$-dimension. Our definition of $ζ$-dimension offer a new homological perspective on the projective dimensio…
Graded-Injective Modules and Bass Numbers of Veronese Submodules
本文研究了标准分次代数上分次模的Bass数，利用Veronese函子计算了Veronese子模在非齐次极大素理想处的Bass数，并将其应用于确定局部上同调模的Bass数。
Let $I$ denote an ideal of a local Gorenstein ring $(R, \mathfrak m)$. Then we show that the local cohomology module $H^c_I(R), c = \height I,$ is indecomposable if and only if $V(I_d)$ is connected in codimension one. Here $I_d$ denotes the intersection of the highest dimensional primary components of $I.$ This is a partial extension of a result shown by Hochster and Huneke in the case $I$ the maximal ideal. Moreover there is an analysis of connectedness properties in relation to various aspects of local cohomology. Among others we show that the endomorphism ring of $H^c_I(R)$ is a local Noet…
We establish a criterion for the strong $F$-regularity of a (non-Gorenstein) Cohen-Macaulay reduced complete local ring of dimension at least $2$, containing a perfect field of prime characteristic $p$. We also describe an explicit generating morphism (in the sense of Lyubeznik) for the top local cohomology module with support in certain ideals arising from an $n\times (n-1)$ matrix $X$ of indeterminates. For $p\geq 5$, these results led us to derive a simple, new proof of the well-known fact that the generic determinantal ring defined by the maximal minors of $X$ is strongly $F$-regular.
Some characterizations of Gorenstein Rees Algebras
本文利用扩展Rees代数的分次典范模，在不假设基环为Cohen-Macaulay的条件下刻画了Rees代数的Gorenstein性质，并应用于Kawasaki算术Cohen-Macaulay化。
Given a commutative Noetherian ring $R$ with total ring of fractions $Q(R)$, and a finitely generated $R$-submodule $M$ of $Q(R)$, we prove an equality between trace ideal, and certain annihilator of Ext and Tor of $M$. As a consequence, we answer in one-dimensional local analytically unramified case, a question raised by the present author and R. Takahashi. As another application, we give an alternative proof of a recent result of Ö. Esentepe that for one-dimensional analytically unramified Gorenstein local rings, the cohomology annihilator of Iyengar and Takahashi coincides with the conducto…
This paper is concerned about the relation between local cohomology modules defined by a pair of ideals and Serre classes of R-modules, as a generalization of results of J. Azami, R. Naghipour and B. Vakili (2009) and M. Asgharzadeh and M.Tousi (2010). Let R be a commutative Noetherian ring, I, J be two ideals of R and M be an R-module. Let a\in W(I; J) and t \in N_0 be such that Ext^t_R(R/a,M)\in S and Ext^j_R(R/a,H^i_I;J(M))\inS for all i < t and all j>=0. Then for any submodule N of H^t_I;J(M) such that Ext^1_R(R/a;N)\in,we obtain HomR(R=a;H^t_I;J(M)/N)\inS.
Given a commutative noetherian non-positive DG-ring with bounded cohomology which has a dualizing DG-module, we study its regular, Gorenstein and Cohen-Macaulay loci. We give a sufficient condition for the regular locus to be open, and show that the Gorenstein locus is always open. However, both of these loci are often empty: we show that no matter how nice $\mathrm{H}^0(A)$ is, there are examples where the Gorenstein locus of $A$ is empty. We then show that the Cohen-Macaulay locus of a commutative noetherian DG-ring with bounded cohomology which has a dualizing DG-module always contains a de…
Stable cohomology is a generalization of Tate cohomology to associative rings, first defined by Pierre Vogel. For a commutative local ring $R$ with residue field $k$, stable cohomology modules $\widehat{\mathrm{Ext}}{\vphantom E}^{n}_R\;(k,k)$, defined for $n\in\mathbb{Z}$, have been studied by Avramov and Veliche. Stable cohomology carries a structure of $\mathbb{Z}$-graded $k$-algebra. One of the main goals of this paper is to prove that, for a class of Gorenstein rings, this algebra is a trivial extension of absolute cohomology $\mathrm{Ext}_R(k,k)$ and a shift of $\mathrm{Hom}_k(\mathrm{Ex…
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