Cohen-Macaulay Representations

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

One-dimensional rings of finite Cohen-Macaulay type

We introduce a new invariant for subcategories X of finitely generated modules over a local ring R which we call the radius of X. We show that if R is a complete intersection and X is resolving, then finiteness of the radius forces X to contain only maximal Cohen-Macaulay modules. We also show that the category of maximal Cohen-Macaulay modules has finite radius when R is a Cohen-Macaulay complete local ring with perfect coefficient field. We link the radius to many well-studied notions such as the dimension of the stable category of maximal Cohen-Macaulay modules, finite/countable Cohen-Macaulay representation type and the uniform Auslander condition.

We study the degeneration problem for maximal Cohen-Macaulay modules and give several examples of such degenerations. It is proved that such degenerations over an even-dimensional simple hypersurface singularity of type(An)(A_n)are given by extensions. We also prove that all extended degenerations of maximal Cohen-Macaulay modules over a Cohen-Macaulay complete local algebra of finite representation type are obtained by iteration of extended degenerations of Auslander-Reiten sequences.

The representation theory of curve singularities (more precisely, of their local rings) has turned out to be closely related to their deformation properties. Namely, as was shown in [6],[9],[7], such a ring R is of finite type, that is has only finitely many torsion-free indecomposable modules (up to isomorphism), if and only if it dominates one of the so called simple plane curve singularities in the sense of [1]. In [4] the authors have shown that R is of tame type, that is it has essentially only 1-parameter families of indecomposable torsion-free modules, if and only if it dominates one of the unimodal plane curve singularities of type T pq (T pq2 in the classification of [1]).

Let R be a local ring and let (x 1, …, x r) be part of a system of parameters of a finitely generated R-module M, where r < dimR M. We will show that if (y 1, …, y r) is part of a reducing system of parameters of M with (y 1, …, y r) M = (x 1, …, x r) M then (x 1, …, x r) is already reducing. Moreover, there is such a part of a reducing system of parameters of M iff for all primes P ε Supp M ∩ V R(x 1, …, x r) with dimR R/P = dimR M − r the localization M P of M at P is an r-dimensional Cohen-Macaulay module over R P.Furthermore, we will show that M is a Cohen-Macaulay module iff y d is a non zero divisor on M/(y 1, …, y d−1) M, where (y 1, …, y d) is a reducing system of parameters of M (d:= dimR M).

Two definable subcategories of maximal Cohen-Macaulay modules

Schubert complexes and degeneracy loci

Representations of Lie groups are abstractions of continuous symmetries of linear spaces.By considering their differentials (linear approximations), we obtain representations of Lie algebras.These are regarded as generalizations of the Fourier analysis and important tools not only in mathematics but also in physics.In this article, I present fundamental and important examples on representations of Lie groups and Lie algebras and present some of my latest results.

1.1. Let G be a connected reductive algebraic group defined over a finite (prime) field Fp of p elements. For a given rational G-module M one can consider the restrictions of M to the restricted Lie algebra Lie(G) and to the finite group G(Fp) of Fp-rational points. Even though there is no direct functorial connection between the categories of restricted Lie(G)-modules and G(Fp)-modules, the ambient algebraic group G can serve as a bridge linking these two important module categories. Early results of Curtis in the 1960s provided a one-to-one correspondence between the simple restricted modules (over k = Fp) of Lie(G) and those of G(Fp). This correspondence is given by simply restricting certain classes of simple G-modules. More importantly, these results, along with work of Steinberg, allow one to easily transfer the questions pertaining to the computation of irreducible characters among these three categories. For over thirty years, this approach has been one of the main ideas used to relate the representation and cohomology theory of algebraic groups, Lie algebras, and finite groups of Lie type (in the defining characteristic) (see, e.g., [CPSvan], [Jan2], [Hum3], and [Jan1]). Given the previous results in this area, a natural course of investigation should be to find relationships between canonical modules, like projective modules, for Lie(G) and G(Fp). More precisely,

Some modules with nearly prescribed endomorphism rings

We use the theory of Auslander Buchweitz approximations to classify certain resolving subcategories containing a semidualizing or a dualizing module. In particular, we show that if the ring has a dualizing module, then the resolving subcategories containing maximal Cohen-Macaulay modules are in bijection with grade consistent functions and thus are the precisely the dominant resolving subcategories.

This article consists of an introduction to Iyama's higher Auslander-Reiten theory for Artin algebras from the viewpoint of higher homological algebra. We provide alternative proofs of the basic results in higher Auslander-Reiten theory, including the existence of $d$-almost-split sequences in $d$-cluster-tilting subcategories, following the approach to classical Auslander-Reiten theory due to Auslander, Reiten, and Smal{\o}. We show that Krause's proof of Auslander's defect formula can be adapted to give a new proof of the defect formula for $d$-exact sequences. We use the defect formula to establish the existence of morphisms determined by objects in $d$-cluster-tilting subcategories.

Abstract. Let (R,m) be a Noetherian local ring and M a finitely gener-ated R-module. For an integer s>−1, we say that Mis Cohen-Macaulayin dimension >s if every system of parameters of M is an M-sequencein dimension > s introduced by Brodmann-Nhan [1]. In this paper, wegive some characterizations for Cohen-Macaulay modules in dimension>sin terms of the Noetherian dimension of the local cohomology mod-ules H im (M), the polynomial type of M introduced by Cuong [5] and themultiplicity e(x;M) of M with respect to a system of parameters x. 1. IntroductionThroughoutthis paper, let (R,m) be a Noetherian local ring and Ma finitelygenerated R-module with dimM= d.It is well known that the Cohen-Macaulay modules play an important rolein the theory of Noetherian rings and finitely generated modules. Recall thatM is called Cohen-Macaulay if every system of parameters (s.o.p. for short)of Mis an M-sequence. The structure of Cohen-Macaulay modules are well-known in the multiplicity, local cohomology, m-adic completion, localization,etc (see [3]). There are some extensions of the concepts of M-sequence andCohen-Macaulay modules, among which are the notions of M-sequence in di-mension >sintroduced by Brodmann-Nhan [1] and Cohen-Macaulay modulesin dimension >sdefined by Zamani [21].Definition. Let s> −1 be an integer. A sequence (x

Let A be a (commutative, Noetherian) local ring (with identity) and let a1,…, an be a system of parameters (s.o.p.) for A. A (not necessarily finitely generated) A-module M is said to be a big Cohen–Macaulay.A-module with respect to a1,…, an if a1,…, an is an M-sequence, that is if M ‡ = (a1,…, an) M and, for each i = 1,…, n,One of the main open problems in commutative algebra at the present time is that of establishing the existence of a big Cohen–Macaulay module with respect to a specified s.o.p. in an arbitrary local ring. The work and writings of Hochster, such as (5), show that, if the existence of such modules could be established, then several conjectures in commutative algebra, some of which are quite long-standing, would be settled. Moreover, Hochster has established the existence of such big Cohen–Macaulay modules whenever the local ring A concerned contains a field as a subring, or has dimension not exceeding 2: see ((5), chapters 4, 5) and (4).