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Generalized Koszul resolution. (English) Zbl 1309.19004
Summary: In this paper, we generalize a notion of Koszul resolutions and characterize modules which admits such resolutions. It turns out that for a noetherian ring $$A$$ and a coherent $$A$$-module $$M$$, $$M$$ has a two-dimensional generalized Koszul resolution if and only if $$M$$ is a pure weight two module in the sense of [T. Hiranouchi and S. Mochizuki, in: Deformation spaces. Perspectives on algebro-geometric moduli. Including papers from the workshops held at the Max-Planck-Institut für Mathematik, Bonn, Germany, July 2007 and August 2008. Wiesbaden: Vieweg+Teubner. 75–89 (2010; Zbl 1220.19001)]. As an application, we attack the Gersten conjecture for weight two case.
##### MSC:
 19D35 Negative $$K$$-theory, NK and Nil 18E10 Abelian categories, Grothendieck categories
##### Keywords:
Gersten conjecture; Koszul resolution; matrix factorization
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##### References:
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