Contracting Endomorphisms and Gorenstein Modules

On the (non)rigidity of the Frobenius Endomorphism over Gorenstein Rings

It is well-known that for a large class of local rings of positive characteristic, including complete intersection rings, the Frobenius endomorphism can be used as a test for finite projective dimension. In this paper, we exploit this property to study the structure of such rings. One of our results states that the Picard group of the punctured spectrum of such a ring $R$ cannot have $p$-torsion. When $R$ is a local complete intersection, this recovers (with a purely local algebra proof) an analogous statement for complete intersections in projective spaces first given in SGA and also a specia…

赋值域的收缩自同态

Contracting Endomorphisms of Valued Fields

本文证明了配备 Frobenius 自同态的可分代数闭赋值域类的可判定性，通过构建模型完备理论 $\widetilde{\mathrm{VFE}}$ 融合已有理论，并在特征零下证明其 NTP$_2$ 性质及稳定型分类。

Almost Gorenstein rings

The notion of almost Gorenstein ring given by Barucci and Fr{ö}berg \cite{BF} in the case where the local rings are analytically unramified is generalized, so that it works well also in the case where the rings are analytically ramified. As a sequel, the problem of when the endomorphism algebra $\m : \m$ of $\m$ is a Gorenstein ring is solved in full generality, where $\m$ denotes the maximal ideal in a given Cohen-Macaulay local ring of dimension one. Characterizations of almost Gorenstein rings are given in connection with the principle of idealization. Examples are explored.

Silting complexes and Gorenstein projective modules

We introduce Gorenstein silting modules (resp. complexes), and give the relation with the usual silting modules (resp. complexes). We show that Gorenstein silting modules are the module-theoretic counterpart of 2-term Gorenstein silting complexes; and partial Gorenstein silting modules are in bijection with τ_{G}-rigid modules for finite dimensional algebras of finite CM-type. We also give the relation between 2-term Gorenstein silting complexes, t-structures and torsion pair in module categories; and generalise the classical Brenner-Butler theorem to this setting; and characterise the global …

Some homological criteria for regular, complete intersection and Gorenstein rings

Regularity, complete intersection and Gorenstein properties of a local ring can be characterized by homological conditions on the canonical homomorphism into its residue field (Serre, Avramov, Auslander). It is also known that in positive characteristic, the Frobenius endomorphism can also be used for these characterizations (Kunz, ...), and more generally any contracting endomorphism. We introduce here a class of local homomorphisms, in some sense larger than all above, for which these characterizations still hold, providing an unified treatment for this class of homomorphisms.

G-dimension over local homomorphisms. Applications to the Frobenius endomorphism

We develop a theory of G-dimension for modules over local homomorphisms which encompasses the classical theory of G-dimension for finite modules over local rings. As an application, we prove that a local ring R of characteristic p is Gorenstein if and only if it possesses a nonzero finite module of finite projective dimension that has finite G-dimension when considered as an R-module via some power of the Frobenius endomorphism of R. We also prove results that track the behavior of Gorenstein properties of local homomorphisms under (de)composition.

关于广义戈林斯坦局部环

On generalized Gorenstein local rings

本文提出广义戈林斯坦局部环的概念，并探讨其与Cohen-Macaulay环相关代数结构的联系，同时提供了多种实例。

Gorenstein内射模的Bass数与自同态环

Bass numbers and endomorphism rings of Gorenstein injective modules

本文针对带对偶复形的交换诺特环，研究了Gorenstein内射模相关的模结构判定、Bass数有限性条件，还定义并刻画了一类自同态环的诺特子环及其结构。

Notes on endomorphisms, local cohomology and completion

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…

The tiny trace ideals of the canonical modules in Cohen-Macaulay rings of dimension one

We study one-dimensional Cohen-Macaulay rings whose trace ideal of the canonical module is as small as possible. In this paper we call such rings far-flung Gorenstein rings. We investigate far-flung Gorenstein rings in relation with the endomorphism algebras of the maximal ideals and numerical semigroup rings. We show that the solution of the Rohrbach problem in additive number theory provides an upper bound for the multiplicity of far-flung Gorenstein numerical semigroup rings. Reflexive modules over far-flung Gorenstein rings are also studied.

Quasihomogeneity of curves and the Jacobian endomorphism ring

We give a quasihomogeneity criterion for Gorenstein curves. For complete intersections, it is related to the first step of Vasconcelos' normalization algorithm. In the process, we give a simplified proof of the Kunz-Ruppert criterion.

On Invariants and Endomorphism Rings of Local Cohomology Modules

Let $(R,\mathfrak{m})$ denote an $n$-dimensional Gorenstein ring. For an ideal $I \subset R$ with $\grade I = c$ we define new numerical invariants $τ_{i,j}(I)$ as the socle dimensions of $H^i_{\mathfrak{m}}(H^{n-j}_I(R))$. In case of a regular local ring containing a field these numbers coincide with the Lyubeznik numbers $λ_{i,j}(R/I)$. We use $τ_{d,d}(I), d = \dim R/I,$ to characterize the surjectivity of the natural homomorphism $f : \hat{R} \to \Hom_{\hat{R}}(H^c_{I\hat{R}}(\hat{R}),H^c_{I\hat{R}}(\hat{R}))$. As a technical tool we study several natural homomorphisms. Moreover we prove a …

Auslander-Reiten sequences for Gorenstein rings of dimension one

Let $R$ be a complete local Gorenstein ring of dimension one, with maximal ideal m. We show that if $M$ is a Cohen-Macaulay $R$-module which begins an AR-sequence, then this sequence is produced by a particular endomorphism of m corresponding to a minimal prime ideal. We apply this result to determining the shape of some components of Auslander-Reiten quivers.

Endomorphism rings of finite global dimension

For a commutative local ring $R$, consider (noncommutative) $R$-algebras $Λ$ of the form $Λ= End_R(M)$ where $M$ is a reflexive $R$-module with nonzero free direct summand. Such algebras $Λ$ of finite global dimension can be viewed as potential substitutes for, or analogues of, a resolution of singularities of $Spec R$. For example, Van den Bergh has shown that a three-dimensional Gorenstein normal $C$-algebra with isolated terminal singularities has a crepant resolution of singularities if and only if it has such an algebra $Λ$ with finite global dimension and which is maximal Cohen--Macaulay…

Local rings with self-dual maximal ideal

Let R be a Cohen-Macaulay local ring possessing a canonical module. In this paper we consider when the maximal ideal of R is self-dual, i.e. it is isomorphic to its canonical dual as an R-module. local rings satisfying this condition are called Teter rings, and studied by Teter, Huneke-Vraciu, Ananthnarayan-Avramov-Moore, and so on. On the positive dimensional case, we show such rings are exactly the endomorphism rings of the maximal ideals of some Gorenstein local rings of dimension one. We also provide some connection between the self-duality of the maximal ideal and near Gorensteinness.

On Lyubeznik's invariants and endomorphisms of local cohomology modules

Let $(R, \mathfrak m)$ denote an $n$-dimensional Gorenstein ring. For an ideal $I \subset R$ of height $c$ we are interested in the endomorphism ring $B = \Hom_R(H^c_I(R), H^c_I(R)).$ It turns out that $B$ is a commutative ring. In the case of $(R,\mathfrak m)$ a regular local ring containing a field $B$ is a Cohen-Macaulay ring. Its properties are related to the highest Lyubeznik number $l = \dim_k \Ext_R^d(k,H^c_I(R)).$ In particular $R \simeq B$ if and only if $l = 1.$ Moreover, we show that the natural homomorphism $\Ext_R^d(k, H^c_I(R)) \to k$ is non-zero.

Torsion in tensor products over one-dimensional domains

Over a one-dimensional Gorenstein local domain $R$, let $E$ be the endomorphism ring of the maximal of $R$, viewed as a subring of the integral closure $\overline R$. If there exist finitely generated $R$-modules $M$ and $N$, neither of them free, whose tensor product is torsion-free, we show that $E$ must be local with the same residue field as $R$.

On endomorphism rings and dimensions of local cohomology modules

Let $(R,\mathfrak m)$ denote an $n$-dimensional complete local Gorenstein ring. For an ideal $I$ of $R$ let $H^i_I(R), i \in \mathbb Z,$ denote the local cohomology modules of $R$ with respect to $I.$ If $H^i_I(R) = 0$ for all $i \not= c = \height I,$ then the endomorphism ring of $H^c_I(R)$ is isomorphic to $R$ (cf. \cite{HSt} and \cite{HS}). Here we prove that this is true if and only if $H^i_I(R) = 0, i = n, n -1$ provided $c \geq 2$ and $R/I$ has an isolated singularity resp. if $I$ is set-theoretically a complete intersection in codimension at most one. Moreover, there is a vanishing resu…

Trace ideals, normalization chains, and endomorphism rings

In this paper we consider reduced (non-normal) commutative noetherian rings $R$. With the help of conductor ideals and trace ideals of certain $R$-modules we deduce a criterion for a reflexive $R$-module to be closed under multiplication with scalars in an integral extension of $R$. Using results of Greuel and Knörrer this yields a characterization of plane curves of finite Cohen--Macaulay type in terms of trace ideals. Further, we study one-dimensional local rings $(R,\mathfrak{m})$ such that that their normalization is isomorphic to the endomorphism ring $\mathrm{End}_R(\mathfrak{m})$: we gi…

Syzygies of Cohen-Macaulay modules over one dimensional Cohen-Macaulay local rings

We study syzygies of (maximal) Cohen-Macaulay modules over one dimensional Cohen-Macaulay local rings. We compare these modules to Cohen-Macaulay modules over the endomorphism ring of the maximal ideal. After this comparison, we give several characterizations of almost Gorenstein rings in terms of syzygies of Cohen-Macaulay modules.

On connectedness and indecomposibility of local cohomology modules

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…

"Infinite" properties of certain local cohomology modules of determinantal rings

For given integers $m,n \geq 2$ there are examples of ideals $I$ of complete determinantal local rings $(R,\mathfrak{m}), \dim R = m+n-1, \operatorname{grade} I = n-1,$ with the canonical module $ω_R$ and the property that the socle dimensions of $H^{m+n-2}_I(ω_R)$ and $H^m_{\mathfrak{m}}(H^{n-1}_I(ω_R))$ are not finite. In the case of $m = n$, i.e. a Gorenstein ring, the socle dimensions provide further information about the $τ$-numbers as studied in \cite{MS}. Moreover, the endomorphism ring of $H^{n-1}_I(ω_R)$ is studied and shown to be an $R$-algebra of finite type but not finitely generat…

有限内射维数模的注记

Notes on modules of finite injective dimension

受Bass猜想的启发，研究有限生成且具有有限内射维数的模及其对环的额外约束，包括约化性、正规性、整环性、完全交或Gorenstein等性质，并探讨模的自反性、无挠性、张量积和内射维数等。

Homology over local homomorphisms

The notions of Betti numbers and of Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S, for some local homomorphism f: R --> S. Various techniques are developed to study the new invariants and to establish their basic properties. In several cases they are computed in closed form. Applications go in several directions. One is to identify new classes of finite R-modules whose classical Betti numbers or Bass numbers have extremal growth. Another is to transfer ring theoretical properties between R and S in situations where S may…

One-dimensional stable rings

A commutative ring $R$ is stable provided every ideal of $R$ containing a nonzerodivisor is projective as a module over its ring of endomorphisms. The class of stable rings includes the one-dimensional local Cohen-Macaulay rings of multiplicity at most $2$, as well as certain rings of higher multiplicity, necessarily analytically ramified. The former are important in the study of modules over Gorenstein rings, while the latter arise in a natural way from generic formal fibers and derivations. We characterize one-dimensional stable local rings in several ways. The characterizations involve the …

关于无限积分扩域的一些同调问题的注记

Remarks on some Homological Problems regarding Infinite Integral Extensions

本文应用收缩自同态的相关结果，在NBIM环正则性等同调问题上取得进展，解决了F-纯环的猜想，还在混合特征下对完美纯环证明了相关结论。

Auslander-Reiten的Cohen-Macaulay代数与收缩预投射代数

Auslander-Reiten's Cohen-Macaulay algebras and contracted preprojective algebras

本文通过证明所有Dynkin类型的收缩预投射代数均为Cohen-Macaulay代数，给出了首个非平凡的Cohen-Macaulay代数类，并否定了Auslander-Reiten关于CM模范畴与$d$次合冲范畴是否一致的猜想。