Finite EI Research Articles

Skew group categories, algebras associated to Cartan matrices and folding of root lattices

For an action of a finite group on a finite EI quiver, we construct its ‘orbifold’ quotient EI quiver. The free EI category associated to the quotient EI quiver is equivalent to the skew group category with respect to the given group action. Specializing the result to a finite group action on a finite acyclic quiver, we prove that, under reasonable conditions, the skew group category of the path category is equivalent to a finite EI category of Cartan type. If the ground field is of characteristic $p$ and the acting group is a cyclic $p$ -group, we prove that the skew group algebra of the path algebra is Morita equivalent to the algebra associated to a Cartan matrix, defined in [C. Geiss, B. Leclerc, and J. Schröer, Quivers with relations for symmetrizable Cartan matrices I: Foundations, Invent. Math. 209 (2017), 61–158]. We apply the Morita equivalence to construct a categorification of the folding projection between the root lattices with respect to a graph automorphism. In the Dynkin cases, the restriction of the categorification to indecomposable modules corresponds to the folding of positive roots.

Gorenstein triangular matrix rings and category algebras

We give conditions on when a triangular matrix ring is Gorenstein of a given selfinjective dimension. We apply the result to the category algebra of a finite EI category. In particular, we prove that for a finite EI category, its category algebra is 1-Gorenstein if and only if the given category is free and projective.

Stratifications of Finite Directed Categories and Generalized APR Tilting Modules

A finite directed category is a k-linear category with finitely many objects and an underlying poset structure, where k is an algebraically closed field. This concept unifies structures such as k-linerizations of posets and finite EI categories, quotient algebras of finite-dimensional hereditary algebras, triangular matrix algebras, etc. In this article, we study representations of finite directed categories and discuss their stratification properties. In particular, we show the existence of generalized Auslander-Platzeck-Reiten tilting modules for triangular matrix algebras under some assumptions.

Spectra of tensor triangulated categories over category algebras

Let \({\mathcal{C}}\) be a finite EI category and k be a field. We consider the category algebra \({k\mathcal{C}}\). Suppose \({\sf{K}(\mathcal{C})=\sf{D}^b(k \mathcal{C}-\sf{mod})}\) is the bounded derived category of finitely generated left modules. This is a tensor triangulated category, and we compute its spectrum in the sense of Balmer. When \({\mathcal{C}=G \propto \mathcal{P}}\) is a finite transporter category, the category algebra becomes Gorenstein, so we can define the stable module category \({\underline{\sf{CM}} k(G \propto \mathcal{P})}\), of maximal Cohen–Macaulay modules, as a quotient category of \({{\sf{K}}(G \propto \mathcal{P})}\). Since \({\underline{\sf{CM}} k(G\propto\mathcal{P})}\) is also tensor triangulated, we compute its spectrum as well. These spectra are used to classify tensor ideal thick subcategories of the corresponding tensor triangulated categories.

On the representation types of category algebras of finite EI categories

A finite EI category is a small category with finitely many morphisms such that every endomorphism is an isomorphism. They include finite groups, finite posets and free categories of finite quivers as special cases. In this paper we consider category algebras of finite EI categories with two or three objects, describe some criteria for finite representation type, and classify the representation types for certain classes of finite EI categories with extra properties.

A generalized Koszul theory and its application

Let A A be a graded algebra. In this paper we develop a generalized Koszul theory by assuming that A 0 A_0 is self-injective instead of semisimple and generalize many classical results. The application of this generalized theory to directed categories and finite EI categories is described.

A characterization of finite EI categories with hereditary category algebras

In this paper we give an explicit algorithm to construct the ordinary quiver of a finite EI category for which the endomorphism groups of all objects have orders invertible in the field k. We classify all finite EI categories with hereditary category algebras, characterizing them as free EI categories (in a sense which we define) for which all endomorphism groups of objects have invertible orders. Some applications on the representation types of finite EI categories are derived.

Standard stratifications of EI categories and Alperin's weight conjecture

We characterize the finite EI categories whose representations are standardly stratified with respect to the natural preorder on the simple representations. The orbit category of a finite group with respect to any set of subgroups is always such a category. Taking the subgroups to be the p-subgroups of the group, we reformulate Alperin's weight conjecture in terms of the standard and proper costandard representations of the orbit category. We do this using the properties of the Ringel dual construction and a theorem of Dlab, which have elsewhere been described for standardly stratified algebras where there is a partial order on the simple modules. We indicate that these results hold in the generality of an algebra whose simple modules are preordered, rather than partially ordered.

On the cohomology rings of small categories

Let C be a small category and R a commutative ring with identity. The cohomology ring of C with coefficients in R is defined as the cohomology ring of the topological realization of its nerve. First we give an example showing that this ring modulo nilpotents is not finitely generated in general, even when the category is finite EI. Then we study the relationship between the cohomology ring of a category and those of its subcategories and extensions. The main results generalize certain theorems in group cohomology theory.