 
Summary: THE HOMOLOGY OF STRING ALGEBRAS I
B. HuisgenZimmermann 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], DonovanFreislich [12], ButlerRingel [8],
