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TILTING MODULES OVER TAME HEREDITARY ALGEBRAS LIDIA ANGELERI HUGEL AND JAVIER SANCHEZ
 

Summary: TILTING MODULES OVER TAME HEREDITARY ALGEBRAS
LIDIA ANGELERI H¨UGEL AND JAVIER S´ANCHEZ
Abstract. We give a complete classification of the infinite dimensional tilting modules over a
tame hereditary algebra R. We start our investigations by considering tilting modules of the form
T = RU RU /R where U is a union of tubes, and RU denotes the universal localization of R at U
in the sense of Schofield and Crawley-Boevey. Here RU /R is a direct sum of the Pr¨ufer modules
corresponding to the tubes in U. Over the Kronecker algebra, large tilting modules are of this
form in all but one case, the exception being the Lukas tilting module L whose tilting class Gen L
consists of all modules without indecomposable preprojective summands. Over an arbitrary tame
hereditary algebra, T can have finite dimensional summands, but the infinite dimensional part of
T is still built up from universal localizations, Pr¨ufer modules and (localizations of) the Lukas
tilting module. We also recover the classification of the infinite dimensional cotilting R-modules
due to Buan and Krause.
In this paper, we continue our study of tilting modules arising from universal localization started in
[5]. More precisely, we consider tilting modules over a ring R that have the form RU RU /R where
U is a set of finitely presented R-modules of projective dimension one, and RU denotes the universal
localization of R at U in the sense of Schofield. We have seen in [5] that over certain rings this
construction leads to a classification of all tilting modules. For example, over a Dedekind domain,
every tilting module is equivalent to a tilting module of the form RU RU /R for some set of simple
R-modules U. Aim of this paper is to prove a similar result for finite dimensional tame hereditary

  

Source: Angeleri Hügel, Lidia - Dipartimento di Informatica, Universita` di Verona

 

Collections: Mathematics