Stratifications of derived categories from tilting modules over tame hereditary algebras

2011/07/03 by Chen, Hongxing, Changchang Xi, Xi, Changchang · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1107.0444

openalex publication_date 2011/07/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider the endomorphism algebras of infinitely generated tilting modules of the form R_{\mathcal U}⊕ R_{\mathcal U}/R over tame hereditary k-algebras R with k an arbitrary field, where R_U is the universal localization of R at an arbitrary set U of simple regular R-modules, and show that the derived module category of \End_R(R_{\mathcal U}⊕ R_{\mathcal U}/R) is a recollement of the derived module category \DR of R and the derived module category \D\mathbb A_U of the adèle ring \mathbb A_U associated with U. When k is an algebraically closed field, the ring \mathbb A_U can be precisely described in terms of Laurent power series ring k((x)) over k. Moreover, if \mathcal U is a union of finitely many cliques, we give two different stratifications of the derived category of \End_R(R_{\mathcal U}⊕ R_{\mathcal U}/R) by derived categories of rings, such that the two stratifications are of different finite lengths.

Citations

• On the structure and ideal theory of complete local rings

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