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THE HOMOLOGY OF STRING ALGEBRAS I B. Huisgen-Zimmermann and S. O. Smal

B. Huisgen-Zimmermann and S. O. Smalų
Dedikert til v°ar venn og kollega Idun Reiten i andledning hennes seksti°arsdag
Abstract. We show that string algebras are `homologically tame' in the following sense: First, the syzygies of
arbitrary representations of a finite dimensional string algebra are direct sums of cyclic representations, and the
left finitistic dimensions, both little and big, of can be computed from a finite set of cyclic left ideals contained
in the Jacobson radical. Second, our main result shows that the functorial finiteness status of the full subcategory
P<( -mod) consisting of the finitely generated left -modules of finite projective dimension is completely
determined by a finite number of, possibly infinite dimensional, string modules ­ one for each simple -module ­
which are algorithmically constructible from quiver and relations of . Namely, P<( -mod) is contravariantly
finite in -mod precisely when all of these string modules are finite dimensional, in which case they coincide
with the minimal P<( -mod)-approximations of the corresponding simple modules. Even when P<( -mod)
fails to be contravariantly finite, these `characteristic' string modules encode, in an accessible format, all desirable
homological information about -mod.
1. Introduction
The representation theory of the Lorentz group is intimately linked to that of a certain string algebra
(for a definition of string algebras see Section 2), as was observed and exploited by Gelfand and Ponomarev
in [17]. In particular, it was proved there that this algebra ­ along with a class of close relatives ­ has tame
representation type; in fact, its finite dimensional indecomposable representations were explicitly pinned
down. In a sequence of articles by Ringel [29], Bondarenko [6], Donovan-Freislich [12], Butler-Ringel [8],


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


Collections: Mathematics