Examples of degenerations of Cohen-Macaulay modules

This book is a comprehensive treatment of the representation theory of maximal Cohen-Macaulay (MCM) modules over local rings. This topic is at the intersection of commutative algebra, singularity theory, and representations of groups and algebras. Two introductory chapters treat the Krull-Remak-Schmidt Theorem on uniqueness of direct-sum decompositions and its failure for modules over local rings. Chapters 3-10 study the central problem of classifying the rings with only finitely many indecomposable MCM modules up to isomorphism, i.e., rings of finite CM type. The fundamental material--ADE/simple singularities, the double branched cover, Auslander-Reiten theory, and the Brauer-Thrall conjectures--is covered clearly and completely. Much of the content has never before appeared in book form. Examples include the representation theory of Artinian pairs and Burban-Drozd's related construction in dimension two, an introduction to the McKay correspondence from the point of view of maximal Cohen-Macaulay modules, Auslander-Buchweitz's MCM approximation theory, and a careful treatment of nonzero characteristic. The remaining seven chapters present results on bounded and countable CM type and on the representation theory of totally reflexive modules.

Let k be an algebraically closed field of characteristic p and G a finite group such that p divides the order of G. We compute all almost split sequences over kG when kG is of finite representation type, or more generally, for a finite dimensional k-algebra Λ \Lambda given by a Brauer tree. We apply this to show that if Λ \Lambda and Λ ′ \Lambda ’ are stably equivalent k-algebras given by Brauer trees, then they have the same number of simple modules.

M. C. R. Butler has shown that if Λ \Lambda is an artin algebra of finite representation type, then the almost split sequences generate the relations for the Grothendieck group of Λ \Lambda . This paper is primarily devoted to proving the converse of Butler’s result, i.e. Λ \Lambda is of finite representation type if the almost split sequences generate the relations for the Grothendieck group of Λ \Lambda .

This paper describes the Hochschild cohomology ring of a selfinjective algebra Λ \Lambda of finite representation type over an algebraically closed field K K , showing that the quotient HH ∗ ⁡ ( Λ ) / N \operatorname {HH}^*(\Lambda )/\mathcal {N} of the Hochschild cohomology ring by the ideal N {\mathcal N} generated by all homogeneous nilpotent elements is isomorphic to either K K or K [ x ] K[x] , and is thus finitely generated as an algebra. We also consider more generally the property of a finite dimensional algebra being selfinjective, and as a consequence show that if all simple Λ \Lambda -modules are Ω \Omega -periodic, then Λ \Lambda is selfinjective.

For monomial special multiserial algebras, which in general are of wild representation type, we construct radical embeddings into algebras of finite representation type. As a consequence, we show that the representation dimension of monomial and self-injective special multiserial algebras is less than or equal to three. This implies that the finitistic dimension conjecture holds for all special multiserial algebras.

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In this paper, we show that generalized Nakajima Categories provide a framework to construct a desingularization of quiver Grassmannians for self-injective algebras of finite representation type. Furthermore, we show that all standard Frobenius models of orbit categories of the bounded derived category considered in Keller, Documenta Math. 10: 551–581, 2005 are equivalent to proj 𝓡, the finitely generated projective modules of the regular Nakajima category 𝓡.

The minimal projective bimodule resolution of Möbius algebras that form a class of self-injective algebras of finite representation type is constructed. Bibliography: 7 titles.

INFINITESIMAL SCHUR ALGEBRAS OF FINITE REPRESENTATION TYPE Get access STEPHEN R. DOTY, STEPHEN R. DOTY Search for other works by this author on: Oxford Academic Google Scholar DANIEL K. NAKANO, DANIEL K. NAKANO Search for other works by this author on: Oxford Academic Google Scholar KARL M. PETERS KARL M. PETERS Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 48, Issue 3, September 1997, Pages 323–345, https://doi.org/10.1093/qmath/48.3.323 Published: 01 September 1997 Article history Received: 13 May 1995 Revision received: 11 November 1995 Published: 01 September 1997

One-dimensional rings of finite Cohen-Macaulay type

Schubert complexes and degeneracy loci

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 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.

Two definable subcategories of maximal Cohen-Macaulay modules

Cohen–Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras