Maximal Cohen Macaulay R-module Research Articles

Grothendieck groups, convex cones and maximal Cohen–Macaulay points

Let R be a commutative noetherian ring. Let $$\mathsf {H}(R)$$ be the quotient of the Grothendieck group of finitely generated R-modules by the subgroup generated by pseudo-zero modules. Suppose that the $$\mathbb {R}$$ -vector space $$\mathsf {H}(R)_\mathbb {R}=\mathsf {H}(R)\otimes _\mathbb {Z}\mathbb {R}$$ has finite dimension. Let $$\mathsf {C}(R)$$ (resp. $$\mathsf {C}_r(R)$$ ) be the convex cone in $$\mathsf {H}(R)_\mathbb {R}$$ spanned by maximal Cohen–Macaulay R-modules (resp. maximal Cohen–Macaulay R-modules of rank r). We explore the interior, closure and boundary, and convex polyhedral subcones of $$\mathsf {C}(R)$$ . We provide various equivalent conditions for R to have only finitely many rank r maximal Cohen–Macaulay points in $$\mathsf {C}_r(R)$$ in terms of topological properties of $$\mathsf {C}_r(R)$$ . Finally, we consider maximal Cohen–Macaulay modules of rank one as elements of the divisor class group $${\text {Cl}}(R)$$ .

A Study of Strongly Cohen–Macaulay Ideals by Delta Invariant

Let R be a Cohen–Macaulay local ring, I a strongly Cohen–Macaulay ideal of R. We show that $$R/\text{ Ann }\,_R(I)$$ is a maximal Cohen–Macaulay R-module by means of the delta-invariant. Also it is shown that there exists a Cohen–Macaulay ideal J of R such that the $$\delta _{R/J}$$ -invariant of all Koszul homologies of R with respect I is zero.

On the existence of maximal Cohen-Macaulay modules over Noetherian complete local rings

Let be a Noetherian complete local ring of dimension d and be a system of parameters for R. Assume that M is a big Cohen-Macaulay module with respect to In this paper it is shown that if the R-module is of finite length then is a maximal Cohen-Macaulay R-module, where D denotes the Matlis dual functor.

On the Degree of Hilbert Polynomials of Derived Functors

Given a d-dimensional Cohen–Macaulay local ring (R,m), let I be an m-primary ideal, and let J be a minimal reduction ideal of I. If M is a maximal Cohen–Macaulay R-module, then, for n large enough and 1 ≤ i ≤ d, the lengths of the modules ExtRi(R/J,M/InM) and ToriR(R/J,M/InM) are polynomials of degree d − 1. It is also shown that $$\deg \beta _i^R(M/{I^n}M) = \deg \mu _R^i(M/{I^n}M) = d - 1,$$ where β i R (·) and μ R i (·) are the ith Betti number and the ith Bass number, respectively.

Specifying the Auslander transpose in submodule category and its applications

Let (R,m) be a d-dimensional commutative Noetherian local ring. Let M denote the morphism category of finitely generated R-modules, and let S be the full subcategory of M consisting of monomorphisms, known as the submodule category. This article reveals that the Auslander transpose in the category S can be described explicitly within modR, the category of finitely generated R-modules. This result is exploited to study the linkage theory as well as the Auslander–Reiten theory in S. In addition, motivated by a result of Ringel and Schmidmeier, we show that the Auslander–Reiten translations in the subcategories H and G, consisting of all morphisms which are maximal Cohen–Macaulay R-modules and Gorenstein projective morphisms, respectively, may be computed within modR via G-covers. The corresponding result for the subcategory of epimorphisms in H is also obtained.

Ulrich modules over Cohen–Macaulay local rings with minimal multiplicity

Abstract Let R be a Cohen–Macaulay local ring. In this paper, we study the structure of Ulrich R-modules mainly in the case where R has minimal multiplicity. We explore generation of Ulrich R-modules and clarify when the Ulrich R-modules are precisely the syzygies of maximal Cohen–Macaulay R-modules. We also investigate the structure of Ulrich R-modules as an exact category.

Gorenstein orders of finite lattice type

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 HomR(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.

The Dual Hilbert–Samuel Function of a Maximal Cohen-Macaulay Module

Let R be a Cohen–Macaulay local ring with a canonical module ω R . Let I be an 𝔪-primary ideal of R and M, a maximal Cohen–Macaulay R-module. We call the function n⟼ℓ(Hom R (M, ω R /I n+1ω R )) the dual Hilbert–Samuel function of M with respect to I . By a result of Theodorescu, this function is of polynomial type. We study its first two normalized coefficients. In particular, we analyze the case when R is Gorenstein.

On the Limit of Frobenius in the Grothendieck Group

Considering the Grothendieck group of finitely generated modules modulo numerical equivalence, we obtain the finitely generated lattice \(\overline {G_{0}(R)}\) for a Noetherian local ring R. Let CCM(R) be the cone in \(\overline {G_{0}(R)}_{\mathbb { R}}\) spanned by cycles of maximal Cohen-Macaulay R-modules. We shall define the fundamental class \(\overline {\mu _{R}}\) of R in \(\overline {G_{0}(R)}_{\mathbb { R}}\), which is the limit of the Frobenius direct images (divided by their rank) [eR]/pde in the case ch(R)=p>0. The homological conjectures are deeply related to the problems whether \(\overline {\mu _{R}}\) is in the Cohen-Macaulay cone CCM(R) or the strictly nef cone SN(R) defined below. In this paper, we shall prove that \(\overline {\mu _{R}}\) is in CCM(R) in the case where R is FFRT or F-rational.

Ranks of Indecomposable Modules over Rings of Infinite Cohen-Macaulay Type

Let (R, 𝔪, k) be a one-dimensional analytically unramified local ring with minimal prime ideals P 1, …, P s . Our ultimate goal is to study the direct-sum behavior of maximal Cohen-Macaulay modules over R. Such behavior is encoded by the monoid ℭ(R) of isomorphism classes of maximal Cohen-Macaulay R-modules: the structure of this monoid reveals, for example, whether or not every maximal Cohen-Macaulay module is uniquely a direct sum of indecomposable modules; when uniqueness does not hold, invariants of this monoid give a measure of how badly this property fails. The key to understanding the monoid ℭ(R) is determining the ranks of indecomposable maximal Cohen-Macaulay modules. Our main technical result shows that if R/P 1 has infinite Cohen-Macaulay type and the residue field k is infinite, then there exist |k| pairwise non-isomorphic indecomposable maximal Cohen-Macaulay R-modules of rank (r 1, ⋅, r s ) provided r 1 ≥ r i for all i ∈ {1,…, s}. This result allows us to describe the monoid ℭ(R) when has infinite Cohen-Macaulay type for every minimal prime ideal Q of the 𝔪-adic completion .

On the structure of Cohen–Macaulay modules over hypersurfaces of countable Cohen–Macaulay representation type

Let R be a complete local hypersurface over an algebraically closed field of characteristic different from two, and suppose that R has countable Cohen–Macaulay (CM) representation type. In this paper, it is proved that the maximal Cohen–Macaulay (MCM) R-modules which are locally free on the punctured spectrum are dominated by the MCM R-modules which are not locally free on the punctured spectrum. More precisely, there exists a single R-module X such that the indecomposable MCM R-modules not locally free on the punctured spectrum are X and its syzygy ΩX and that any other MCM R-modules are obtained from extensions of X and ΩX.

Deformations of maximal Cohen-Macaulay modules

Let (R,m) be a complete Cohen-Macaulay isolated singularity over a field K which is either perfect or [K : K p] 0. According to Dieterich [Di] and [Yo] Ch. 6 (see also [Pol], [PR] or, more generally, [CHP]) there exists a system of parameters x = (xl . . . . . xr) of R such that the base change functor R/(x)®Rdefines an injection v preserving the indecomposability from the set of isomorphism classes of maximal Cohen-Macaulay R-modules to the set of isomorphism classes of R/(x)-modules (in this case the ideal generated by x is called (after [Di]) a reduction ideal. Trying to describe the image of v we noticed in [Po2] that a finitely generated R/(x)module is in Imv if and only if it has the form Xl.. .x~P for a finitely generated R2 := R/(x 2 . . . . . x~ )-module satisfying

On indices of local rings along local homomorphisms

Let φ:R → Sbe a local homomorphism of Gorenstein local rings of finite flat dimension. We prove if M is a stable maximal Cohen-Macaulay R-module, then M ⊗R Sis also a stable maximal Cohen-Macaulay S-module. By using it, we also prove that index(R) ≤ index(S).

Tensor products of modules and the rigidity of Tor

Let R be a hypersurface domain, that is, a ring of the form S/(f), where S is a regular local ring and f is a prime element. Suppose M and N are finitely generated R- modules. We prove two rigidity theorems on the vanishing of Tor. In the first theorem we assume that the regular local ring S is unramified, that M R N has finite length, and that dim(M)+dim(N) dim(R). With these assumptions, if Tor R (M,N) = 0 for some j 0, then Tor R (M,N) = 0 for all i j. The second rigidity theorem states that if M R N is reflexive, then Tor R (M,N) = 0 for all i 1. We use these theorems to prove the following theorem (valid even if S is ramified): If M R N is a maximal Cohen-Macaulay R-module, then both M and N are maximal Cohen-Macaulay modules, and at least one of them is free.