arXiv subjects
Publications and source records attributed to R. Bautista.
We prove that if ${\Lambda}$ is a tame finite-dimensional algebra over an algebraically closed field and $G$ is a generic ${\Lambda}-$module, then $G$ is a generic brick if and only if it determines a one-parameter family of bricks with the same dimension. In particular, we obtain that ${\Lambda}$ admits a generic brick if and only if ${\Lambda}$ is brick-continuous.
Given an algebra $A$ and an $A-A$-bimodule $U$ with co-algebra structure, a bocs, the algebras of endomorphisms of $A$ as left or right module of the bocs are known as Burt-Butler algebras (up to an appropriate opposite). Here we give a description of these algebras for the bocs associated to a finite partially ordered set in terms of incidence algebras and their balanced versions. We also exhibit their quasi-hereditary structure, provide bound quiver presentations for their Ringel duals, describe the embedding of $A$ as exact Borel subalgebra and characterize the corresponding subcategories of induced and co-induced modules.
We introduce the notion of interlaced weak ditalgebras and apply reduction procedures to their module categories to prove a tame-wild dichotomy for the category ${\cal F}(Δ)$ of $Δ$-filtered modules for an arbitrary finite homological system $({\cal P},\leq,\{Δ_i\}_{i\in {\cal P}})$. This includes the case of standardly stratified algebras. Moreover, in the tame case, we show that given a fixed dimension $d$, for every $d$-dimensional indecomposable module $M\in {\cal F}(Δ)$, with the only possible exception of those lying in a finite number of isomorphism classes, the module $M$ coincides with its Auslander-Reiten translate in ${\cal F}(Δ)$. Our proofs rely on the equivalence of ${\cal F}(Δ)$ with the module category of some special type of ditalgebra.
We introduce and study the category of twisted modules over a triangular differential graded bocs. We show that in this category idempotents split, that it admits a natural structure of a Frobenius category, that a twisted module is homotopically trivial if and only if its underlying complex is acyclic, and that any homotopy equivalence of differential graded bocses determines an equivalence of the corresponding homotopy categories of twisted modules. The category of modules over an $A_{\infty}$-algebra is equivalent to the category of twisted modules over a triangular differential graded bocs, so all preceding statements lift to the former category.