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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna


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0933-7741
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Volume 27, Issue 3

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Recollements induced from tilting modules over tame hereditary algebras

Hongxing Chen
  • School of Mathematical Sciences, Capital Normal University, 100048 Beijing, People's Republic of China; and Beijing International Center for Mathematical Research, Peking University, 100871 Beijing, People's Republic of China
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/ Changchang Xi
Published Online: 2013-02-06 | DOI: https://doi.org/10.1515/forum-2011-0077

Abstract

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

Keywords: Adèle ring; recollement; stratification; tame hereditary algebra; tilting module; universal localization

MSC: 18E30; 16G10; 13B30; 16S10; 13E05

About the article

Received: 2011-07-31

Revised: 2012-09-21

Published Online: 2013-02-06

Published in Print: 2015-05-01


Funding Source: China Postdoctoral Science Foundation

Award identifier / Grant number: 2012M510242

Funding Source: Ministry of Education of China

Award identifier / Grant number: Doctor Program Funds


Citation Information: Forum Mathematicum, Volume 27, Issue 3, Pages 1849–1901, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2011-0077.

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