Character Modules
Cohomological Properties of Modules with Secondary Representations.
Combinatorial topology and the global dimension of algebras arising in combinatorics
In a highly influential paper, Bidigare, Hanlon and Rockmore showed that a number of popular Markov chains are random walks on the faces of a hyperplane arrangement. Their analysis of these Markov chains took advantage of the monoid structure on the set of faces. This theory was later extended by Brown to a larger class of monoids called left regular bands. In both cases, the representation theory of these monoids played a prominent role. In particular, it was used to compute the spectrum of the...
Complexity and periodicity
Let M be a finitely generated module over an Artin algebra. By considering the lengths of the modules in the minimal projective resolution of M, we obtain the Betti sequence of M. This sequence must be bounded if M is eventually periodic, but the converse fails to hold in general. We give conditions under which it holds, using techniques from Hochschild cohomology. We also provide a result which under certain conditions guarantees the existence of periodic modules. Finally, we study the case when...
Construction of Auslander-Gorenstein local rings as Frobenius extensions
Starting from an arbitrary ring R we provide a systematic construction of ℤ/nℤ-graded rings A which are Frobenius extensions of R, and show that under mild assumptions, A is an Auslander-Gorenstein local ring if and only if so is R.
Correction to: Homological Dimension and Stationary Sets. Math. Z. 180, 1-9 (1982).
Correction to: Maximal orders of global dimension and Krull dimension two.
Cycle-finite algebras with almost all indecomposable modules of projective or injective dimension at most one
We describe the structure of artin algebras for which all cycles of indecomposable modules are finite and almost all indecomposable modules have projective or injective dimension at most one.
Derived dimension via -tilting theory
Comparing the bounded derived categories of an algebra and of the endomorphism algebra of a given support -tilting module, we find a relation between the derived dimensions of an algebra and of the endomorphism algebra of a given -tilting module.
Dimension homologique de modules simples
Dimension injective des produits croisés.
Dimensions globales des extensions de Ore et des algèbres de Weyl
Directing components for quasitilted algebras
We show here that a directing component of the Auslander-Reiten quiver of a quasitilted algebra is either postprojective or preinjective or a connecting component.
Dominant Dimension of Double Centralizers.
Double Centralizers and Dominant Dimensions.
Dual dimension of modules over normalizing extensions.
Let S = Σi=1n Rai be a finite normalizing extension of R and suppose that SM is a left S-module. Denote by crk(A) the dual Goldie dimension of the module A. We show that crk(RM) ≤ n · crk(SM) if either SM is artinian or the group homomorphism M → aiM given by x → aix is an isomorphism.
Duality over Auslander-Gorenstein rings.
Estimates of global dimension
In this note we show that for a -module, in particular, an almost -tilting module, over a ring with such that has finite flat dimension, the upper bound of the global dimension of can be estimated by the global dimension of and hence generalize the corresponding results in tilting theory and the ones in the theory of -modules. As an application, we show that for a finitely generated projective module over a VN regular ring , the global dimension of its endomorphism ring is not more...
FC-modules with an application to cotorsion pairs
Let be a ring. A left -module is called an FC-module if is a flat right -module. In this paper, some homological properties of FC-modules are given. Let be a nonnegative integer and the class of all left -modules such that the flat dimension of is less than or equal to . It is shown that is a complete cotorsion pair and if is a ring such that and is closed under direct sums, then is a perfect cotorsion pair. In particular, some known results are obtained as corollaries....
Field-Dependent Homological Behavior of finite dimensional algebras.