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In this paper, we consider a finite, torsion-free module $E$ over a Gorenstein local ring. We provide sufficient conditions for $E$ to be of linear type and for the Rees algebra $\mathcal{R}(E)$ of $E$ to be Cohen-Macaulay. Our results are obtained by constructing a generic Bourbaki $I$ ideal of $E$ and exploiting properties of the residual intersections of $I$., Comment: 20 pages. Previously uploaded under the title "On the Cohen-Macaulayness and defining ideal of Rees algebras of modules". Section 3 from the first version has been expanded and now occupies Sections 3 and 4. Content of former Section 4 now appears in arXiv:2011.08453

Let $(R, \frak m)$ be a generalized Cohen-Macaulay local ring of dimension $d$, and $f_1, \ldots, f_r$ a part of system of parameters of $R$. In this paper we give explicit numbers $N$ such that the lengths of all lower local cohomology modules and the Hilbert function of $R/(f_1, \ldots, f_r)$ are preserved when we perturbs the sequence $f_1, \ldots, f_r$ by $\varepsilon_1, \ldots, \varepsilon_r \in {\frak m}^N$. The second assertion extends a previous result of Srinivas and Trivedi for generalized Cohen-Macaulay rings., J. Algebra

In this note, we study validity of the original Briancon-Skoda theorem for analytic local rings. By introducing a new concept of multiplier ideal sheaves associated to plurisubharmonic functions on any (reduced) complex space of pure dimension, we establish the original Briancon-Skoda theorem for normal weakly rational singularities (not essentially of finite type over C and Cohen-Macaulay). Moreover, we also characterize two dimensional rational singularities by the original Briancon-Skoda theorem.

Let ( A , m , k ) be a Gorenstein local ring of dimension d ≥ 1 . Let I be an ideal of A with ht ( I ) ≥ d − 1 . We prove that the numerical function n ↦ l ( Ext A i ( k , A / I n + 1 ) ) is given by a polynomial of degree d − 1 in the case when i ≥ d + 1 and curv ( I n ) > 1 for all n ≥ 1 . We prove a similar result for the numerical function n ↦ l ( Tor i A ( k , A / I n + 1 ) ) under the assumption that A is a Cohen-Macaulay local ring. We note that there are many examples of ideals satisfying the condition curv ( I n ) > 1 , for all n ≥ 1 . We also consider more general functions n ↦ l ( Tor i A ( M , A / I n ) for a filtration { I n } of ideals in A. We prove similar results in the case when M is a maximal Cohen-Macaulay A-module and { I n = I n ‾ } is the integral closure filtration, I an m -primary ideal in A.

In this paper, we study the $F$-rationality of the Rees algebra and the extended Rees algebra of $\mathfrak{m}$-primary ideals in excellent local rings $(R, \mathfrak{m})$ of prime characteristic. We partially answer some conjectures and questions raised by N. Hara, K.-i. Watanabe and K.-i. Yoshida (J. Algebra, pp.153--190, vol 247, 2002)., 13 pages; comments welcome

In this paper, we introduce the notion of the strong Rees property (SRP) for m -primary ideals of a Noetherian local ring and prove that any power of the maximal ideal m has its property if the associated graded ring G of m satisfies depth G ≥ 2 . As its application, we characterize two-dimensional excellent normal local domains so that m is a p g -ideal. Finally we ask what m -primary ideals have SRP and state a conjecture which characterizes the case when m n are the only ideals which have SRP.

This paper is a sequel to [8] where we introduced an invariant, called canonical degree, of Cohen–Macaulay local rings that admit a canonical ideal. Here to each such ring R with a canonical ideal, we attach a different invariant, called bi-canonical degree, which in dimension 1 appears also in [12] as the residue of R. The minimal values of these functions characterize specific classes of Cohen–Macaulay rings. We give a uniform presentation of such degrees and discuss some computational opportunities offered by the bi-canonical degree.

Let R be a commutative Noetherian local ring and M , N be finitely generated R-modules. We prove a number of results of the form: if Hom R ( M , N ) has some nice properties and Ext R 1 ≤ i ≤ n ( M , N ) = 0 for some n, then M (and sometimes N) must be close to free. Our methods are quite elementary, yet they suffice to give a unified treatment, simplify, and sometimes extend a number of results in the literature.

The Sally module of an ideal is an important tool to interplay between Hilbert coefficients and the properties of the associated graded ring. In this paper we give new insights on the structure of the Sally module. We apply these results characterizing the almost minimal value of the first Hilbert coefficient in the case of the normal filtration in an analytically unramified Cohen-Macaulay local ring., 15 pages

Let S be a deeply embedded, equicharacteristic, Artinian Gorenstein local ring. We prove that if R is a non-Gorenstein quotient of S of small colength, then every totally reflexive R-module is free. Indeed, the second syzygy of the canonical module of R has a direct summand T which is a test module for freeness over R in the sense that if Tor + R ( T , N ) = 0 , for some finitely generated R-module N, then N is free.

Let $D$ be a 2-dimensional regular local ring and let $Q(D)$ denote the quadratic tree of 2-dimensional regular local overrings of $D$. We examine the Noetherian rings that are intersections of rings in $Q(D)$. To do so, we describe the desingularization of projective models over $D$ both algebraically in terms of the saturation of complete ideals and order-theoretically in terms of the quadratic tree $Q(D)$., Comment: 27 pages, comments welcome

Let ( A , m ) be an analytically unramified formally equidimensional Noetherian local ring with depth A ≥ 2 . Let I be an m -primary ideal and set I ⁎ to be the integral closure of I. Set G ⁎ ( I ) = ⨁ n ≥ 0 ( I n ) ⁎ / ( I n + 1 ) ⁎ be the associated graded ring of the integral closure filtration of I. We prove that depth G ⁎ ( I n ) ≥ 2 for all n ≫ 0 . As an application we prove that if A is also an excellent normal domain containing an algebraically closed field isomorphic to A / m then there exists s 0 such that for all s ≥ s 0 and J is an integrally closed ideal strictly containing ( m s ) ⁎ then we have a strict inequality μ ( J ) μ ( ( m s ) ⁎ ) (here μ ( J ) is the number of minimal generators of J).

Let ( R , m , κ ) be a local ring. We give a characterization of R-modules M whose local cohomology is finite length up to some index in terms of asymptotic vanishing of Koszul cohomology on parameter ideals up to the same index. In particular, we show that a quasi-unmixed module M is asymptotically Cohen-Macaulay if and only if M is Cohen-Macaulay on the punctured spectrum if and only if sup ⁡ { l ( H i ( f 1 , … , f d ; M ) ) | f 1 , … , f d = m , i d } ∞ for d = dim ⁡ ( M ) = dim ⁡ ( R ) .

The dimension of triangulated categories was introduced by Rouquier in 2008 ( [5] ). As an analogue of this concept, Dao and Takahashi defined the dimension of subcategories of abelian categories in 2014 ( [3] ). The dimension of categories is an important invariant for studying the structure of both the category and the ground ring when it is the category of modules. For example, the dimension of the category of maximal Cohen-Macaulay modules is zero if the ground ring has finite Cohen-Macaulay representation type. In this paper, we focus on the category of maximal Cohen-Macaulay modules over a hypersurface of dimension one.

A recent continuous family of multiplicity functions on local rings was introduced by Taylor interpolating between Hilbert-Samuel and Hilbert-Kunz multiplicities. The obvious goal is to use this as a tool for deforming results from one to the other. The values in this family which do not match these classic variants however are not known yet to be well-behaved. This article explores lower bounds for these intermediate multiplicities as well as gives evidence for analogies of the Watanabe-Yoshida minimality conjectures for unmixed singular rings., Comment: 10 pages

We show that for an infinite sequence { ( R n , m n ) } n ≥ 0 of local normalized quadratic transforms of analytically unramified one-fibered Noetherian local domains, the corresponding sequence of Rees valuations converges. This extends a result of [4] , where the authors show that for an infinite sequence of local quadratic transforms of regular local rings, the order valuations converge.

Let ( R , m ) be a Noetherian local ring which is a quotient of a Gorenstein local ring. Let M be a finitely generated R-module and K M the canonical module of M. Denote by nCM ( M ) the non Cohen–Macaulay locus of M. In this paper, the relationship between the two loci nCM ( K M ) and nCM ( M ) is given. The structure of M is clarified in case where dim ⁡ nCM ( K M ) ≤ 0 (i.e. K M is generalized Cohen–Macaulay).

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., 18 pages

Let ( R , m ) be a commutative complete Gorenstein local ring and let Λ be a Gorenstein order, that is to say, Λ is a maximal Cohen–Macaulay R-module and Hom R ( Λ , R ) is a projective Λ-module. The main theme of this paper is to study the representation-theoretic properties of generalized lattices, i.e. those Λ-modules which are Gorenstein projective over R. It is proved that Λ has only finitely many isomorphism classes of indecomposable lattices if and only if every generalized lattice is the direct sum of finitely generated ones. It is also turn out that, if R is one-dimensional, then a generalized lattice M which is not the direct sum of copies of a finite number of lattices, contains indecomposable sublattices of arbitrarily large finite h _ -length, an invariant assigned to each generalized lattice which measures Hom modulo projectives.

Lower bounds on Hilbert-Samuel multiplicity are known for several types of commutative noetherian local rings, and rings with multiplicities which achieve these lower bounds are said to have minimal multiplicity. The first part of this paper gives characterizations of rings of minimal multiplicity in terms of the Ext-algebra of the ring; in particular, we show that minimal multiplicity can be detected via an Ext-algebra which is Gorenstein or Koszul AS-regular. The second part of this paper characterizes rings of minimal multiplicity via a numerical homological invariant introduced by J. Herzog and S. B. Iyengar called linearity defect. Our characterizations allow us to answer in two special cases a question raised by Herzog and Iyengar., Final version; minor changes made according to the referee report; to appear in the Journal of Algebra

We study Cohen–Macaulay non-Gorenstein local rings ( R , m , k ) admitting certain totally reflexive modules. More precisely, we give a description of the Poincare series of k by using the Poincare series of a non-zero totally reflexive module with minimal multiplicity. Our results generalize a result of Yoshino to higher-dimensional Cohen–Macaulay local rings. Moreover, from a quasi-Gorenstein ideal satisfying some conditions, we construct a family of non-isomorphic indecomposable totally reflexive modules having an arbitrarily large minimal number of generators.

Let ( R , m ) be a Noetherian local ring and M a finitely generated R-module. Let P ∈ Spec ( R ˆ ) and p = P ∩ R . Then the natural map R p → R ˆ P is a flat local homomorphism. In this paper, we provide some relations between the two sets of attached primes of the Artinian local cohomology modules H P R ˆ P i + r P ( M p ⊗ R p R ˆ P ) and H p R p i ( M p ) , where i ≥ 0 is an integer and r P = dim ⁡ ( R ˆ P / p R ˆ P ) . Then, we compute the dimension and multiplicity of H P R ˆ P i + r P ( M p ⊗ R p R ˆ P ) in terms of that of H p R p i ( M p ) respectively. As applications, we give connections between the Cohen–Macaulayness in dimension >s of M ˆ P and that of M p , for an integer s ≥ − 1 .

Let A and B be rings and let M be an A–B-bimodule that is finitely generated and projective in A-mod and in mod-B. Also let I be an ideal of A and let J be an ideal of B such that I M = M J . Our main result is a partial converse of a known result: Proposition. Suppose that I ≤ J ( A ) , J ≤ J ( B ) so that M / ( I M ) is an A ‾ = A / I – B ‾ = B / J -bimodule that is finitely generated and projective in A ‾ -mod and in mod- B ‾ and that induces a Morita Equivalence between A ‾ -mod and B ‾ -mod. Then M induces a Morita Equivalence between A-mod and B-mod. This result should be particularly useful in the context that A and B are O -algebras where O is a commutative local ring, I = J ( O ) A and J = I ( O ) B . In which case, A ‾ and B ‾ are finite dimensional algebras over the field k = O / J ( O ) .

The purpose of the paper is to study the annihilators of local cohomology of the extended Rees rings. We will prove that if ( A , m ) is a noetherian local ring with infinite residue field, and a is a uniform local cohomological annihilator of A, then there is an integer n such that ( a t ) n is a uniform local cohomological annihilator of the extended Rees ring A [ t , t − 1 I ] for any m -primary ideal I of A.

This note has three contributions. The first one is to give elementary proofs about finite free resolution using only the notion of latent regular element (regular element which may be obtained after addition of indeterminates) and no notion from homological algebra. The second contribution is to use this to provide a simple proof of Wiebe's Theorem, a fundamental result in the theory of resultants [5] . Finally the third contribution is to explain how to “automatically” prove results or obtain counter-examples about regular sequences of a given length. We obtain in this way a new example of a local ring with a regular sequence b 1 , b 2 such that b 2 , b 1 is not regular.

We study the Poincare series of modules over a fiber product of commutative local rings. We introduce the notion of a weak complete intersection ideal; these are the ideals with the property that every differential in their minimal free resolutions can be represented by a matrix whose entries are in the ideal itself. We show that many properties of the Poincare series that are known to hold for a fiber product over the maximal ideal also hold for those over weak intersection ideals.

For a Noetherian ring R and a cotilting R-module T of injective dimension at least 1, we prove that the derived dimension of R with respect to the category X T is precisely the injective dimension of T by applying Auslander–Buchweitz theory and Ghost Lemma. In particular, when R is a commutative Noetherian Cohen–Macaulay local ring with a canonical module ω R and dim ⁡ R ≥ 1 , the derived dimension of R with respect to the category of maximal Cohen–Macaulay modules is precisely dim ⁡ R .

We study the converse of a theorem of Butler and Auslander–Reiten. We show that a Cohen–Macaulay local ring with an isolated singularity has only finitely many isomorphism classes of indecomposable summands of syzygies of Cohen–Macaulay modules if the Auslander–Reiten sequences generate the relation of the Grothendieck group of finitely generated modules. This extends a recent result of Hiramatsu, which gives an affirmative answer in the Gorenstein case to a conjecture of Auslander.

The purpose of this paper is to introduce new invariants of Cohen–Macaulay local rings. Our focus is the class of Cohen–Macaulay local rings that admit a canonical ideal. Attached to each such ring R with a canonical ideal C , there are integers–the type of R, the reduction number of C –that provide valuable metrics to express the deviation of R from being a Gorenstein ring. We enlarge this list with other integers–the roots of R and several canonical degrees. The latter are multiplicity based functions of the Rees algebra of C .

Let (R,m) and (S,n) be regular local rings of dim⁡(S)=dim⁡(R)≥2 such that S birationally dominates R, and let V be the order valuation ring of S with corresponding valuation ν:=ordS. Assume that IS≠S and ν∈ReesSIS. Let u:=αt with IS=αIS, where α∈S. Then V=W∩Q(R) with W=(R[It]‾)Q=(S[ISu]‾)Q′, where Q∈Min(mR[It]‾) and Q′∈Min(nS[ISu]‾). Let P,P′ be the center of W on R[It] and S[Isu], respectively. We prove that if [Sn:Rm]=1, then R[It]P=S[Isu]P′. Let I be a finitely supported complete m-primary ideal of a regular local ring (R,m) of dimension d≥2. Let T be a terminal base point of I and V be the mT-adic order valuation of T with corresponding valuation v:=ordT. Let n≥1 be an integer. Assume that IT=mTn and [TmT:Rm]=1. Let P∈Min(mR[It]) such that P=Q∩R[It] with V=(R[It]‾)Q∩Q(R), where Q∈Min(mR[It]‾). We prove that the quotient ring R[It]P is d-dimensional normal Cohen–Macaulay standard graded domain over k with the multiplicity nd−1. In particular, R[It]P is regular if and only if n=1. We prove that k:=Rm is relatively algebraically closed in kv:=VmV. Also we determine the multiplicity of R[It]P, and we prove that if IT=mT, then R[It]P is regular if and only if [TmT:Rm]=1.

A right R-module is called R-projective if it is projective relative to the right R-module R R . A ring R is called right testing for projectivity if every right R-projective module is projective. Every right perfect ring is right testing and there are examples of local rings that are not right testing. In this paper an attempt is made to understand the class of right testing rings and several examples are provided to highlight the set theoretic obstacles in characterizing such rings.

Let R be a commutative Noetherian ring, I ⊆ J be ideals of R and M be a finitely generated R-module. In this paper it is shown that q ( J , M ) ≤ q ( I , M ) + cd ( J , M / I M ) . Furthermore, it is shown that, for any ideal I of R and any finitely generated R-module M with q ( I , M ) ≤ 1 , the local cohomology modules H I i ( M ) are I-cofinite for all integers i ≥ 0 . As a consequence of this result it is shown that, if q ( I , R ) ≤ 1 , then for any finitely generated R-module M, the local cohomology modules H I i ( M ) are I-cofinite for all integers i ≥ 0 . Finally, it is shown that the category of all I-cofinite R-modules C ( R , I ) c o f is an Abelian subcategory of the category of all R-modules, whenever ( R , m ) is a complete Noetherian local ring and I is an ideal of R with q ( I , R ) ≤ 1 . These assertions answer affirmatively two questions raised by R. Hartshorne in [16] , in the some special cases.

In J. Herzog and E. Kunz (1973) [6] it was shown that for any pair ( d , t ) ∈ N × N + with ( d , t ) ≠ ( 1 , 1 ) there exists a local Cohen–Macaulay ring R having deviation d ( R ) = d and type t ( R ) = t . By E. Kunz (1974) [7] the case d ( R ) = 1 , t ( R ) = 1 cannot occur. In this paper certain Cohen–Macaulay rings are studied for which there are close relations between deviation, type and embedding dimension. Similar relations for other classes of local rings have been proved in the recent paper by L. Sharifan (2014) [15] . Our relations will be applied to numerical semigroups (or equivalently monomial curves) and lead also to some further cases, generalizing E. Kunz (2016) [8] with ring-theoretic proofs, in which a question of H. Wilf (1978) [16] has a positive answer.

Let M be a finitely generated module over a Noetherian local ring R. In this paper we introduce the notion of sequential polynomial type of M, which is denoted by sp ( M ) , in order to measure how far M is different from the sequential Cohen–Macaulayness. We study the sequential polynomial type under localization and m -adic completion. We investigate an ascent–descent property of sequential polynomial type between M and M / x M for certain parameter x of M. When R is a quotient of a Gorenstein local ring, we describe sp ( M ) in term of the deficiency modules of M.

We classify 1-tilting classes over an arbitrary commutative ring. As a consequence, we classify all resolving subcategories of finitely presented modules of projective dimension at most 1. Both these collections are in 1-1 correspondence with faithful Gabriel topologies of finite type, or equivalently, with Thomason subsets of the spectrum avoiding a set of primes associated in a specific way to the ring. We also provide a generalization of the classical Fuchs and Salce tilting modules, and classify the equivalence classes of all 1-tilting modules. Finally we characterize the cases when tilting modules arise from perfect localizations., Comment: 19 pages, 1 figure

In this paper, we introduce and study the notion of linkage by perfect modules, which we call perfect linkage, for Cohen–Macaulay modules over Cohen–Macaulay local rings. We explore perfect linkage in connection with syzygies, maximal Cohen–Macaulay approximations and Yoshino–Isogawa linkage. We recover a theorem of Yoshino and Isogawa, and analyze the structure of double perfect linkage. Moreover, we establish a criterion for two Cohen–Macaulay modules of codimension one to be perfectly linked, and apply it to the classical linkage theory for ideals. We also construct various examples of linkage of modules and ideals.

Let J be an equimultiple ideal of height a in a formally equidimensional local ring (R,m). If I is a proper ideal that contains J, we show that the degree of the multiplicity function fJ,I(n)=e(In/Jn) is at most a with equality if and only if J is not a reduction of I. As a consequence, we are able to define a unique filtration J⊆J[a]⊆…⊆J[1]⊆J‾ between the ideal J and its integral closure J‾ with J[k] being the largest ideal containing J such that deg⁡fJ,I(n)≤a−k−1. Further results consider the ideal J[1] and its relation to the S2-ification of the Rees algebra R[Jt].

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d > 0$. Let $I_\bullet = \{I_n\}_{n \in \mathbb{N}}$ be a graded family of $\mathfrak{m}$-primary ideals in $R$. We examine how far off from a polynomial can the length function $\ell_R(R/I_n)$ be asymptotically. More specifically, we show that there exists a constant $\gamma > 0$ such that for all $n \ge 0$, $$\ell_R(R/I_{n+1}) - \ell_R(R/I_n) < \gamma n^{d-1}.$$, Comment: 11 pages

This paper studies the structure of special preenvelopes and envelopes with respect to maximal Cohen–Macaulay modules. We investigate the structure of them in terms of their kernels and cokernels. Moreover, using this result, we also study the structure of special proper coresolutions with respect to maximal Cohen–Macaulay modules over a Henselian Cohen–Macaulay local ring.

Let $A$ be a Noetherian standard $\mathbb{N}$-graded algebra over an Artinian local ring $A_0$. Let $I_1,\ldots,I_t$ be homogeneous ideals of $A$ and $M$ a finitely generated $\mathbb{N}$-graded $A$-module. We prove that there exist two integers $k$ and $k'$ such that \[ \mathrm{reg}(I_1^{n_1} \cdots I_t^{n_t} M) \leq (n_1 + \cdots + n_t) k + k' \quad\mbox{for all }~n_1,\ldots,n_t \in \mathbb{N}. \], 9 pages

Let $M$ be a finitely generated module over a Noetherian local ring. This paper reports, for a given parameter ideal $Q$ for $M$, a criterion for the equality ${\mathrm{g}}_s(Q;M)=\operatorname{hdeg}_Q(M)-{\mathrm{e}}_Q^0(M)-{\mathrm{T}}_Q^1(M)$, where ${\mathrm{g}}_s(Q;M)$, ${\mathrm{e}}_Q^0(M)$, ${\mathrm{e}}_Q^1(M)$, and ${\mathrm{T}}_Q^1(M)$ respectively denote the sectional genus, the multiplicity, the first Hilbert coefficient, and the Homological torsion of $M$ with respect to $Q$., Comment: 17 pages. arXiv admin note: substantial text overlap with arXiv:1404.2455

We study the class of almost uniserial rings as a straightforward common generalization of left uniserial rings and left principal ideal domains. A ring R is called almost left uniserial if any two non-isomorphic left ideals of R are linearly ordered by inclusion, i.e., for every pair I, J of left ideals of R either I ⊆ J , or J ⊆ I , or I ≅ J . Also, an R-module M is called almost uniserial if any two non-isomorphic submodules are linearly ordered by inclusion. We give some interesting and useful properties of almost uniserial rings and modules. It is shown that a left almost uniserial ring is either a local ring or its maximal left ideals are cyclic. A Noetherian left almost uniserial ring is a local ring or a principal left ideal ring. Also, a left Artinian principal left ideal ring R is almost left uniserial if and only if R is left uniserial or R = M 2 ( D ) , where D is a division ring. Finally we consider Artinian commutative rings which are almost uniserial and we obtain a structure theorem for these rings.