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TRUNCATED PATH ALGEBRAS ARE HOMOLOGICALLY TRANSPARENT
 

Summary: TRUNCATED PATH ALGEBRAS ARE
HOMOLOGICALLY TRANSPARENT
A. Dugas, B. Huisgen-Zimmermann, and J. Learned
To the memory of Tony Corner
1. Introduction and notation
The problem of opening up general access roads to the finitistic dimensions of a finite
dimensional algebra , given through quiver and relations, is quite challenging. This is
witnessed, for instance, by the fact that the longstanding question "Is the (left) little
finitistic dimension of ,
fin dim = sup{p dim M | M P<
(-mod)},
always finite?" (Bass 1960) has still not been settled. Here p dim M is the projective
dimension of a module M, and P<
(-mod) denotes the category of finitely generated
(left) -modules of finite projective dimension.
In [1], Babson, the second author, and Thomas showed that truncated path algebras of
quivers are particularly amenable to geometric exploration, while nonetheless displaying
a wide range of interesting phenomena. This led the authors of the present paper to the
serendipitous discovery that the same is true for the homology of such algebras. By a
truncated path algebra we mean an algebra of the form KQ/I, where KQ is the path

  

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

 

Collections: Mathematics