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Mathematics Research Report MRR95-077
Homomorphisms In Higher Complexity Quotient Categories
Jon F. Carlson and Wayne W. Wheeler
Abstract:
In previous work the authors together with Peter Donovan initiated a study
of complexity quotient categories for the modular group algebra of a finite
group G. Let Mc denote the full subcategory of
the stable
category consisting of all finitely generated modules having complexity at
most c. The previous work studied quotients of the form
Mc/Mc-1; the current paper is
devoted to studying quotients
Mc/Md when c-d\ge 2. The main
tool used here, a chain complex
called the modular Koszul complex, gives rise to a spectral
sequence
that provides a description of homomorphisms in these higher complexity
quotient categories. This description is then used to decompose the
endomorphism ring of the trivial module by producing a complete set of
primitive orthogonal idempotents determined by certain subvarieties of
VG(k). The paper also considers connections
with Cech cohomology and
with the idempotent modules studied by Rickard.
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