Tilting quivers for BB‐tilted algebras

Abstract

Let Λ be a hereditary algebra, 𝐵 0 = End Λ (𝑇 0 ) be a tilted algebra. We will construct tilting 𝐵 0 -modules from tilting Λ-modules and use this result to show how tilting quivers of BB-tilted algebras can be obtained from those of Λ.M S C 2 0 2 0 16G20 (primary) Tilting theory, initiated in [3,5], further developed in [4,11] and then generalized in [18], is an interesting and important topic in representation theory (for more information about tilting theory, we refer to [9]). The construction of tilting modules has been a central topic with emphasis on finding complements to almost complete tilting modules (see [6,13]). In this article, we will give a way of constructing tilting modules from a given one. Our main result reads as follows.Theorem 1. Let 𝐵 0 = End Λ (𝑇 0 ) be a tilted algebra of type Λ. If 𝑇 = 𝑌 ⊕ 𝑋 is a tilting Λ-module with 𝑌 ∈  (𝑇 0 ), 𝑋 ∈  (𝑇 0 ) such that no indecomposable summand of 𝑌 is generated by 𝑋, then Hom Λ (𝑇 0 , 𝑌∕𝑇𝑟 𝑌 𝑋) ⊕ Ext 1 Λ (𝑇 0 , 𝑋) is a tilting 𝐵 0 -module.Here Λ is a hereditary algebra, ( (𝑇 0 ),  (𝑇 0 )) is the torsion pair in mod Λ associated to 𝑇 0 , and 𝑇𝑟 𝑌 𝑋 denotes the trace of 𝑋 in 𝑌.Let 𝐴 be a finite-dimensional 𝑘 algebra over an algebraically closed field 𝑘, and  𝐴 be the set of equivalence classes of tilting 𝐴-modules (tilting modules 𝑇 1 and 𝑇 2 are supposed to be in the same class, provided 𝖺𝖽𝖽(𝑇 1 ) = 𝖺𝖽𝖽(𝑇 2 )). Here 𝖺𝖽𝖽(𝑇) ⊂ mod 𝐴 denotes the additive closure of 𝑇. The tilting quiver ⃖⃖⃗  𝐴 was introduced in [14] and [21]: the vertices are elements of  𝐴 , and for two basic tilting 𝐴-modules, there exists an arrow from 𝑇 1 to 𝑇 2 if and only if 𝑇 1 = 𝑀 ⊕ 𝑋, 𝑇 2 = 𝑀 ⊕ 𝑌 with 𝑋, 𝑌 being indecomposable summands belonging to a nonsplit exact sequence 0 → 𝑋 → 𝑀 → 𝑌 → 0 with 𝑀 ∈ 𝖺𝖽𝖽𝑀. Given 𝑇 1 , 𝑇 2 ∈  𝐴 , a partial order ≤ on  𝐴 is defined via: 𝑇 1 ⩽ 𝑇 2 if and only if 𝖳 𝟣 ⟂ ⊂ 𝖳 𝟤 ⟂ . Here, 𝖳 𝗂 ⟂ are full subcategories of mod 𝐴 (see below). It was proved in [14] that the Hasse quiver of the partially ordered set ( 𝐴 , ⩽) coincides with the tilting quiver ⃖⃖⃗  𝐴 . ⃖⃖⃗  𝐴 turned out to be interesting and useful, because it provides information about 𝐴.