Algebras determined by their supports

Abstract

In this paper, we introduce and study a class of algebras which we call ada algebras. An artin algebra is ada if every indecomposable projective and every indecomposable injective module lies in the union of the left and the right parts of the module category. We describe the Auslander-Reiten components of an ada algebra, showing in particular that its representation theory is entirely contained in that of its left and right supports, which are both tilted algebras. Also, we prove that an ada algebra over an algebraically closed field is simply connected if and only if its first Hochschild cohomology group vanishes.

$τ$-Tilting Finite Algebras With Nonempty Left Or Right Parts Are Representation-Finite

“…We also need the following result on the structure of the Auslander-Reiten components of a left supported algebra Λ. We recall a connected component Γ of Γ(mod Λ) is called a postprojective component if Γ does not contain an oriented cycle and each indecomposable module X ∈ Γ is of the form τ −r P for some r ∈ N and an indecomposable projective Λ-module P. Following [3], an algebra Λ is an ada algebra if Λ ⊕ DΛ ∈ add(L Λ ∪ R Λ ). In [2], an algebra Λ is right ada if Λ ∈ add(L Λ ∪R Λ ).…”

The representation dimension of Artin algebras

“…The so-called Brenner Butler Theorem proven in [13] establishes that Hom A (T, −) is an equivalence between (T (T ) and Y(T )) while Ext 1 A (T, −) is an equivalence between F(T ) and X (T ). A particular important case occurs when the tilting module T is splitting, that is, when the torsion pair (X (T ), Y(T )) splits.…”

Dimensão global forte e complexidade na categoria derivada

“…Por exemplo, para a classe das álgebras ada, introduzidas em [ACLV12], que incluem a classe das álgebras shods, podemos construir com ideias semelhantes as aqui apresentadas uma álgebra ada com dimensão global 4 e dimensão global forte n, para cada n ≥ 4. É sabido também que todo módulo indecomponível X sobre uma álgebra ada Λ também satisfaz pd X + id X ≤ d + 1, onde d = gl.dim Λ (vide Corollary 2.6 e Remark 2.7 de [ACLV12]).…”