Bass Numbers Research Articles

Asymptotic behavior of homological invariants of localizations of modules

Let R be a commutative noetherian ring, I an ideal of R, and M a finitely generated R-module. We consider the asymptotic injective dimensions, projective dimensions, Bass numbers, and Betti numbers of localizations of M/InM at prime ideals of R and prove that these invariants are stable or have polynomial growth for large integers n that do not depend on the prime ideals.

On a generalization of a result of Peskine and Szpiro

Let (R,\mathfrak{m}) be a regular local ring containing a field K . Let I be a Cohen–Macaulay ideal of height g . If \operatorname{char} K = p > 0 then by a result of Peskine and Szpiro the local cohomology modules H^i_I(R) vanish for i > g . This result is not true if \operatorname{char} K = 0 . However, we prove that the Bass numbers of the local cohomology module H^g_I(R) completely determine whether H^i_I(R) vanish for i > g . The result of this paper has been proved more generally for Gorenstein local rings by Hellus and Schenzel (2008) (Theorem 3.2). However, our result is elementary to prove. In particular, we do not use spectral sequences in our proof.

Distributive lattices and Auslander regular algebras

Let L denote a finite lattice with at least two points and let A denote the incidence K-algebra of L over a field K. We prove that L is distributive if and only if A is an Auslander regular ring, which gives a homological characterisation of distributive lattices. In this case, A has an explicit minimal injective coresolution, whose i-th term is given by the elements of L covered by precisely i elements. We give a combinatorial formula of the Bass numbers of A. We apply our results to show that the order dimension of a distributive lattice L coincides with the global dimension of the incidence algebra of L. Also we categorify the rowmotion bijection for distributive lattices using higher Auslander-Reiten translates of the simple modules.

Characterizing certain semidualizing complexes via their Betti and Bass numbers

Two distinguished types of semidualizing complexes are the shifts of the underlying rings and dualizing complexes. Let C be a semidualizing complex for an analytically irreducible local ring R and set and We show that C is quasi-isomorphic to a shift of R if and only if the nth Betti number of C is one. Also, we show that C is a dualizing complex for R if and only if the dth Bass number of C is one.

Extension Functors of Generalized Local Cohomology Modules

Let $R$ be a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ an ideal of $R$, $M$ a finitely generated $R$--module, and $X$ an arbitrary $R$--module. In this paper, for non-negative integers $s, t$ and a finitely generated $R$--module $N$, we study the membership of $\operatorname{Ext}_{R}^{s}(N, \operatorname{H}^{t}_{\mathfrak{a}}(M, X))$ in Serre subcategories of the category of $R$--modules and present some upper bounds for the injective dimension and the Bass numbers of $\operatorname{H}^{t}_{\mathfrak{a}}(M, X)$. We also give some results on cofiniteness and minimaxness of $\operatorname{H}^{t}_{\mathfrak{a}}(M, X)$ and finiteness of $\operatorname{Ass}_R(\operatorname{H}^{t}_{\mathfrak{a}}(M, X)$.

Asymptotic depth of Ext modules over complete intersection rings

Let (A,𝔪) be a local complete intersection ring and let I be an ideal in A. Let M, N be finitely generated A-modules. Then for l=0,1, the values depthExtA2i+l(M,N∕InN) become independent of i, n for i,n≫0. We also show that if 𝔭 is a prime ideal in A then the j-th Bass numbers μj(𝔭,ExtA2i+l(M,N∕InN)) have polynomial growth in (n,i) with rational coefficients for all sufficiently large (n,i).

Local cohomology modules and their properties

UDC 512.5 Let be a complete Noetherian local ring and let be a generalized Cohen-Macaulay -module of dimension We show thatwhere and is the ideal transform functor. Also, assuming that is a proper ideal of a local ring , we obtain some results on the finiteness of Bass numbers, cofinitness, and cominimaxness of local cohomology modules with respect to

On the cofiniteness of generalized local cohomology modules

Let $R$ be a commutative Noetherian ring with non-zero identity and $I$ be an ideal of $R$. Let $M$ and $N$ be two finitely generated $R$-modules. In this paper it is shown that, if ${\rm cd}(I,R)\leq 1$ and ${\rm pd_{R}}(M)<\infty$, then $H_{I}^{i}(M,N)$ are $I$-cofinite for each $i\geq 0$. Moreover, it is shown that if $\mbox{q}(I,R) \leq 1$ and ${\rm pd_{R}}(M)<\infty$, then the Bass numbers of generalized local cohomology modules $H_{I}^{i}(M,N)$ are finite for each $i\geq 0$. We also characterize the greatest integer $i$ such that $H_{I}^{i}(M,N)$ is not Artinian and $I$-cofinite.

Extremal growth of Betti numbers and trivial vanishing of (co)homology

A Cohen–Macaulay local ring R R satisfies trivial vanishing if Tor i R ⁡ ( M , N ) = 0 \operatorname {Tor}_i^R(M,N)=0 for all large i i implies that M M or N N has finite projective dimension. If R R satisfies trivial vanishing, then we also have that Ext R i ⁡ ( M , N ) = 0 \operatorname {Ext}^i_R(M,N)=0 for all large i i implies that M M has finite projective dimension or N N has finite injective dimension. In this paper, we establish obstructions for the failure of trivial vanishing in terms of the asymptotic growth of the Betti and Bass numbers of the modules involved. These, together with results of Gasharov and Peeva, provide sufficient conditions for R R to satisfy trivial vanishing; we provide sharpened conditions when R R is generalized Golod. Our methods allow us to settle the Auslander–Reiten conjecture in several new cases. In the last part of the paper, we provide criteria for the Gorenstein property based on consecutive vanishing of Ext. The latter results improve similar statements due to Ulrich, Hanes–Huneke, and Jorgensen–Leuschke.

Bass numbers of local cohomology of cover ideals of graphs

We develop splitting techniques to study the Lyubeznik numbers of cover ideals of graphs which allow us to describe them for large families of graphs including forests, cycles, wheels and cactus graphs. More generally, we are able to compute all the Bass numbers and the shape of the injective resolution of local cohomology modules by considering the connected components of all the induced subgraphs. Indeed, our method gives us a very simple criterion for the vanishing of these local cohomology modules in terms of the number of connected components of the induced subgraphs.

Complete Intersection Hom Injective Dimension

We introduce and investigate a new injective version of the complete intersection dimension of Avramov, Gasharov, and Peeva. It is like the complete intersection injective dimension of Sahandi, Sharif, and Yassemi in that it is built using quasi-deformations. Ours is different, however, in that we use a Hom functor in place of a tensor product. We show that (a) this invariant characterizes the complete intersection property for local rings, (b) it fits between the classical injective dimension and the G-injective dimension of Enochs and Jenda, (c) it provides modules with Bass numbers that are bounded by polynomials, and (d) it improves a theorem of Peskine, Szpiro, and Roberts (Bass’ conjecture).

Cohen–Macaulayness and canonical module of residual intersections

We show the Cohen–Macaulayness and describe the canonical module of residual intersections J = a : R I J={\mathfrak {a}}\colon _R I in a Cohen–Macaulay local ring R R , under sliding depth type hypotheses. For this purpose, we construct and study, using a recent article of Hassanzadeh and the second author, a family of complexes that contains important information on a residual intersection and its canonical module. We also determine several invariants of residual intersections as the graded canonical module, the Hilbert series, the Castelnuovo–Mumford regularity and the type. Finally, whenever I I is strongly Cohen–Macaulay, we show duality results for residual intersections that are closely connected to results by Eisenbud and Ulrich. It establishes some tight relations between the Hilbert series of some symmetric powers of I / a I/{\mathfrak {a}} . We also provide closed formulas for the types and for the Bass numbers of some symmetric powers of I / a . I/{\mathfrak {a}}.

Minimal resolutions of graded modules over an exterior algebra

Let K be a field, E the exterior algebra of a n --dimensional K -vector space V . We study projective and injective resolutions over E . More precisely, given a category M of finitely generated Z-graded left and right E -modules, we give upper bounds for the graded Betti numbers and the graded Bass numbers of classes of modules in M .

Structural theorem for gr-injective modules over gr-noetherian G-graded commutative rings and local cohomology functors

It is well known that the decomposition of injective modules over noetherian rings is one of the most aesthetic and important results in commutative algebra. Our aim is to prove similar results for graded noetherian rings. In this paper, we will study the structure theorem for $gr$-injective modules over $gr$-noetherian $G$-graded commutative rings, give a definition of the $gr$-Bass numbers, and study their properties. We will show that every $gr$-injective module has an indecomposable decomposition. Let $R$ be a $gr$-noetherian graded ring and $M$ be a $gr$-finitely generated $R$-module, we will give a formula for expressing the Bass numbers using the functor $Ext$. We will define the section functor $\Gamma_{V}$ with support in a specialization-closed subset $V$ of $Spec^{gr}(R)$ and the abstract local cohomology functor. Finally, we will show that a left exact radical functor $F$ is of the form $\Gamma_V$ for a specialization-closed subset $V$.

Finiteness properties and numerical behavior of local cohomology

This manuscript discusses many problems connected with finiteness properties of local cohomology modules, e.g. the associated primes, the minimal primes, Bass numbers, as well as the behavior of injective dimension, Lyubeznik numbers, and content.This article is dedicated to Gennady Lyubeznik on the occasion of his 60th birthday, and celebrates his many contributions to commutative algebra.

Bass numbers of derived local cohomology with respect to a specialization closed subset

Let [Formula: see text] be a specialization closed subset of Spec [Formula: see text] and [Formula: see text] a left homologically bounded complex with finitely generated homologies. We provide some inequalities between the Bass numbers of [Formula: see text] and its local cohomology modules with respect to [Formula: see text]. As an application of these inequalities, we provide a comparison between the injective dimensions of [Formula: see text] and its nonzero local cohomology module [Formula: see text]. Our versions contain variations of some results already known in these areas.

Coherent functors and asymptotic stability

Asymptotic properties of high powers of an ideal related to a coherent functor F are investigated. It is shown that when N is an artinian module the sets of attached prime ideals AttAF(0:Nan) are the same for n large enough. Also it is shown that for an artinian module N if the modules F(0:Nan) have finite length and for a finitely generated module M if the modules F(M/anM) have finite length, their lengths are given by polynomials in n, for large n. When A is local it is shown that, the Betti numbers βi(F(M/anM)) and the Bass numbers μi(F(M/anM)) are given by polynomials in n for large n.

Chains of semidualizing modules

Let [Formula: see text] be a commutative Noetherian local ring. We study the suitable chains of semidualizing [Formula: see text]-modules. We prove that when [Formula: see text] is Artinian, the existence of a suitable chain of semidualizing modules of length [Formula: see text] implies that the Poincar[Formula: see text] series of [Formula: see text] and the Bass series of [Formula: see text] have very specific forms. Also, in this case, we show that the Bass numbers of [Formula: see text] are strictly increasing. This gives an insight into the question of Huneke about the Bass numbers of [Formula: see text].

Bass numbers of generalized local cohomology modules with respect to a pair of ideals

Let [Formula: see text] be a Noetherian local ring, [Formula: see text] and [Formula: see text] are ideals of [Formula: see text], and [Formula: see text] and [Formula: see text] are [Formula: see text]-modules. We study the relationship between the Bass numbers of [Formula: see text] and [Formula: see text]. As a consequence, it follows that if one of the following holds: (a) [Formula: see text] is a principal ideal of [Formula: see text], (b) [Formula: see text], (c) [Formula: see text] (when [Formula: see text] is local and [Formula: see text] is finitely generated), (d) [Formula: see text] (when [Formula: see text] is local), (e) [Formula: see text] (when [Formula: see text] is local), then [Formula: see text] is finite for all [Formula: see text] and [Formula: see text], whenever [Formula: see text] is finitely generated and flat, [Formula: see text] is minimax, and [Formula: see text].

COMINIMAXNESS WITH RESPECT TO IDEALS OF DIMENSION ONE

Let R denote a commutative Noetherian (not necessarily local) ring and let I be an ideal of R of dimension one. The main purpose of this note is to show that the category <TEX>${\mathfrak{M}}(R,\;I)_{com}$</TEX> of I-cominimax R-modules forms an Abelian subcategory of the category of all R-modules. This assertion is a generalization of the main result of Melkersson in [15]. As an immediate consequence of this result we get some conditions for cominimaxness of local cohomology modules for ideals of dimension one. Finally, it is shown that the category <TEX>${\mathcal{C}}^1_B(R)$</TEX> of all R-modules of dimension at most one with finite Bass numbers forms an Abelian subcategory of the category of all R-modules.