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STACKS OF ALGEBRAS AND THEIR HOMOLOGY Nancy Heinschel and Birge Huisgen-Zimmermann

Nancy Heinschel and Birge Huisgen-Zimmermann
Dedicated to Raymundo Bautista and Roberto Martinez-V´illa on the occasion of their sixtieth birthdays
Abstract. For any increasing function f : N N2 which takes only finitely many dis-
tinct values, a connected finite dimensional algebra is constructed, with the property that
fin dimn = f(n) for all n; here fin dimn is the n-generated finitistic dimension of . The
stacking technique developed for this construction of homological examples permits strong
control over the higher syzygies of -modules in terms of the algebras serving as layers.
1. Introduction and background
The purpose of this paper is twofold. One of our objectives is to introduce a technique
of `stacking' finite dimensional algebras on top of one another so that, on one hand, the ho-
mology of the resulting algebra can be controlled in terms of the layers, while, on the other
hand, this homology differs qualitatively from that of the building blocks. Our second,
principal, goal is to apply such stacks towards realizing new homological phenomena.
Given a finite dimensional algebra over a field K and n N, we denote by fin dimn
the supremum of the finite projective dimensions attained on left -modules with `top
multiplicities n'; in other words, if J denotes the Jacobson radical of , we are focusing
on those left -modules M of finite projective dimension for which the multiplicities of
the simple summands of M/JM are bounded above by n. Since, over a basic algebra, this
condition just means that M can be generated by n elements, we refer to fin dimn as


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


Collections: Mathematics