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STRONGLY TILTING TRUNCATED PATH ALGEBRAS A. Dugas and B. Huisgen-Zimmermann
 

Summary: STRONGLY TILTING TRUNCATED PATH ALGEBRAS
A. Dugas and B. Huisgen-Zimmermann
Abstract. For any truncated path algebra , we give a structural description of the modules
in the categories P<(-mod) and P<(-Mod), consisting of the finitely generated (resp.
arbitrary) -modules of finite projective dimension. We deduce that these categories are
contravariantly finite in -mod and -Mod, respectively, and determine the corresponding
minimal P<-approximation of an arbitrary -module from a projective presentation. In
particular, we explicitly construct based on the Gabriel quiver Q and the Loewy length
of the basic strong tilting module T (in the sense of Auslander and Reiten) which is
coupled with P<(-mod) in the contravariantly finite case. A main topic is the study
of the homological properties of the corresponding tilted algebra e = End(T)op, such as
its finitistic dimensions and the structure of its modules of finite projective dimension. In
particular, we characterize, in terms of a straightforward condition on Q, the situation where
the tilting module Te
is strong over e as well. In this -e-symmetric situation, we obtain
sharp results on the submodule lattices of the objects in P<(Mod-e), among them a certain
heredity property; it entails that any module in P<(Mod-e) is an extension of a projective
module by a module all of whose simple composition factors belong to P<(mod-e).
1. Introduction and terminology
We let = KQ/I be a truncated path algebra of Loewy length L + 1 for some positive

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics