Ext Modules Research Articles

On André-Quillen homology and complete intersection ideals

We establish new characterizations of complete intersection ideals in Cohen-Macaulay local rings as a byproduct of our investigation on a certain André-Quillen homology vector space in connection to the vanishing of (finitely many) Ext modules over a suitable quasi-deformation. Along the way, by means of (co)homology vanishing, we also detect criteria for quasi-complete intersection ideals.

Ext modules related to syzygies of the residue field

Let R be a commutative noetherian ring. In this paper, we find out close relationships between the module M being embedded in a module of projective dimension at most n and the (n+1)-torsionfreeness of the nth syzygy of M. As an application, when R is local with residue field k, we compute the dimensions as k-vector spaces of Ext modules related to syzygies of k.

The Auslander-Reiten conjecture for certain non-Gorenstein Cohen-Macaulay rings

The Auslander-Reiten conjecture is a notorious open problem about the vanishing of Ext modules. In a Cohen-Macaulay complete local ring R with a parameter ideal Q, the Auslander-Reiten conjecture holds for R if and only if it holds for the residue ring R/Q. In the former part of this paper, we study the Auslander-Reiten conjecture for the ring R/Qℓ in connection with that for R, and prove the equivalence of them for the case where R is Gorenstein and ℓ≤dim⁡R. In the latter part, we generalize the result of the minimal multiplicity by J. Sally. Due to these two of our results, we see that the Auslander-Reiten conjecture holds if there exists an Ulrich ideal whose residue ring is a complete intersection. We also explore the Auslander-Reiten conjecture for determinantal rings.

Morphisms represented by monomorphisms with n-torsionfree cokernel

We introduce and study a new class of morphisms which includes morphisms represented by monomorphisms in the sense of Auslander and Bridger. As an application, we give not only an extension of Kato’s theorem on morphisms represented by monomorphisms, but also a common generalization of several results due to Auslander and Bridger that describe relationships between torsionfreeness and the grades of Ext modules.

Trace ideals of semidualizing modules and two generalizations of nearly Gorenstein rings

In this article, we study the trace ideal of semidualizing modules. Also, we introduce two new classes of rings, namely quasi nearly Gorenstein and weakly nearly Gorenstein, which are generalizations of nearly Gorenstein rings. Furthermore, we show that the trace ideal of a semidualizing module is annihilators of certain Ext modules. Moreover, we give new versions of Tachikawa conjecture using quasi and weakly nearly Gorenstein rings.

Maximal Cohen–Macaulay tensor products and vanishing of Ext modules

In this paper, we investigate the maximal Cohen-Macaulay property of tensor products of modules, and then give criteria for projectivity of modules in terms of vanishing of Ext modules. One of the applications shows that the Auslander-Reiten conjecture holds for Cohen-Macaulay normal rings.

Cohomological characterizations of smoothness and the case of rational surface singularities

We establish characterizations of smoothness for a complex affine algebraic variety at a given Cohen-Macaulay point. Our main result treats the case of surfaces with (at most) rational singularities, by a technique that requires the vanishing of suitable Ext modules rather than of sheaf cohomology groups. We also prove results in arbitrary dimension which, in addition to smoothness, detect curves and surfaces as a global feature.

English

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.

Comparisons Between Annihilators of Tor and Ext

In this paper, we compare annihilators of Tor and Ext modules of finitely generated modules over a commutative noetherian ring. For local Cohen–Macaulay rings, one of our results refines a theorem of Dao and Takahashi.

The (ir)regularity of Tor and Ext

We investigate the asymptotic behavior of Castelnuovo-Mumford regularity of Ext and Tor, with respect to the homological degree, over complete intersection rings. We derive from a theorem of Gulliksen a linearity result for the regularity of Ext modules in high homological degrees. We show a similar result for Tor, under the additional hypothesis that high enough Tor modules are supported in dimension at most one; we then provide examples showing that the behavior could be pretty hectic when the latter condition is not satisfied.

Criteria for prescribed bound on projective dimension

We establish a prescribed upper bound for the projective dimension of a finitely generated module over a Cohen-Macaulay local ring (with canonical module) satisfying certain cohomological conditions. The central hypothesis is the vanishing of finitely many suitable Ext modules. We then derive corollaries which provide freeness criteria and a characterization of regular local rings. Finally, we raise two main questions related to our results; one generalizes a problem due to D. Jorgensen, and the other retrieves the long-standing Auslander-Reiten conjecture in the Gorenstein case.

Trace ideals of canonical modules, annihilators of Ext modules, and classes of rings close to being Gorenstein

In this note we study trace ideals of canonical modules. Characterizations of the trace ideals in terms of annihilators of certain Ext modules are given. We apply our results to study many classes of rings close to being Gorenstein that appear in recent literature. We discuss the behavior of near Gorensteinness, introduced by Herzog, Hibi and Stamate, under reductions by regular sequences. A nearly Gorenstein version of the Tachikawa conjecture is posed, and an affirmative answer is given for type 2 rings. We also introduce the class of weakly almost Gorenstein rings using the canonical module, and show that they are almost Gorenstein rings in dimension one, and are closely related to Teter rings in the artinian case. We explain their basic properties and give some examples.

Decomposition of local cohomology tables of modules with large E-depth

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 second author and Smirnov. Finally, we obtain a non-Artinian version of a socle-lemma proved by Kustin and Ulrich.

Homology of twisted quiver bundles with relations

We study the Ext modules in the category of left modules over a twisted algebra of a finite quiver over a ringed space (X,OX), allowing for the presence of relations. We introduce a spectral sequence which relates the Ext modules in that category with the Ext modules in the category of OX-modules. Contrary to what happens in the absence of relations, this spectral sequence in general does not degenerate at the second page. We also consider local Ext sheaves. Under suitable hypotheses, the Ext modules are represented as hypercohomology groups.

Two generalizations of Auslander–Reiten duality and applications

This paper extends Auslander–Reiten duality in two directions. As an application, we obtain various criteria for freeness of modules over local rings in terms of vanishing of Ext modules, which recover a lot of known results on the Auslander–Reiten conjecture.

FLAT RING EPIMORPHISMS OF COUNTABLE TYPE

AbstractLet R→U be an associative ring epimorphism such that U is a flat left R-module. Assume that the related Gabriel topology $\mathbb{G}$ of right ideals in R has a countable base. Then we show that the left R-module U has projective dimension at most 1. Furthermore, the abelian category of left contramodules over the completion of R at $\mathbb{G}$ fully faithfully embeds into the Geigle–Lenzing right perpendicular subcategory to U in the category of left R-modules, and every object of the latter abelian category is an extension of two objects of the former one. We discuss conditions under which the two abelian categories are equivalent. Given a right linear topology on an associative ring R, we consider the induced topology on every left R-module and, for a perfect Gabriel topology $\mathbb{G}$, compare the completion of a module with an appropriate Ext module. Finally, we characterize the U-strongly flat left R-modules by the two conditions of left positive-degree Ext-orthogonality to all left U-modules and all $\mathbb{G}$-separated $\mathbb{G}$-complete left R-modules.

Local homology, Koszul homology and Serre classes

Given a Serre class $\mathcal {S}$ of modules, we compare the containment of Koszul homology, Ext modules, Tor modules, local homology and local cohomology in $\mathcal {S}$ up to a given bound $s \geq 0$. As applications, we give a full characterization of Noetherian local homology modules. Furthermore, we establish a comprehensive vanishing result which readily leads to formerly known descriptions of the numerical invariants' width and depth in terms of Koszul homology, local homology and local cohomology. In addition, we recover a few renowned vanishing criteria scattered throughout the literature.

Regularity and cohomology of determinantal thickenings

We consider the ring S = C [ x i j ] of polynomial functions on the vector space C m × n of complex m × n matrices. We let GL = GL m ( C ) × GL n ( C ) and consider its action via row and column operations on C m × n (and the induced action on S). For every GL-invariant ideal I ⊆ S and every j ⩾ 0 , we describe the decomposition of the modules Ext S j ( S / I , S ) into irreducible GL-representations. For any inclusion I ⊇ J of GL-invariant ideals we determine the kernels and cokernels of the induced maps Ext S j ( S / I , S ) ⟶ Ext S j ( S / J , S ) . As a consequence of our work, we give a formula for the regularity of the powers and symbolic powers of generic determinantal ideals, and in particular we determine which powers have a linear minimal free resolution. As another consequence, we characterize the GL-invariant ideals I ⊆ S for which the induced maps Ext S j ( S / I , S ) ⟶ H I j ( S ) are injective. In a different direction we verify that Kodaira vanishing, as described in work of Bhatt–Blickle–Lyubeznik–Singh–Zhang, holds for determinantal thickenings.

On the Auslander–Reiten conjecture for Cohen–Macaulay local rings

This paper studies vanishing of Ext modules over Cohen–Macaulay local rings. The main result of this paper implies that the Auslander–Reiten conjecture holds for maximal Cohen–Macaulay modules of rank one over Cohen–Macaulay normal local rings. It also recovers a theorem of Avramov–Buchweitz–Şega and Hanes–Huneke, which shows that the Tachikawa conjecture holds for Cohen–Macaulay generically Gorenstein local rings.

The syzygies of some thickenings of determinantal varieties

The vector space of m x n complex matrices (m >= n) admits a natural action of the group GL = GL_m x GL_n via row and column operations. For positive integers a,b, we consider the ideal I_{a x b} defined as the smallest GL-equivariant ideal containing the b-th powers of the a x a minors of the generic m x n matrix. We compute the syzygies of the ideals I_{a x b} for all a,b, together with their GL-equivariant structure, generalizing earlier results of Lascoux for the ideals of minors (b=1), and of Akin-Buchsbaum-Weyman for the powers of the ideals of maximal minors (a=n). Our methods rely on a nice connection between commutative algebra and the representation theory of the superalgebra gl(m|n), as well as on our previous calculation of Ext modules done in the context of describing local cohomology with determinantal support. Our results constitute an important ingredient in the proof by Nagpal-Sam-Snowden of the first non-trivial Noetherianity results for twisted commutative algebras which are not generated in degree one.