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The grade conjecture and asymptotic intersection multiplicity


Author: Jesse Beder
Journal: Proc. Amer. Math. Soc. 142 (2014), 4065-4077
MSC (2010): Primary 13A35, 13H15
DOI: https://doi.org/10.1090/S0002-9939-2014-12183-6
Published electronically: August 14, 2014
MathSciNet review: 3266978
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Abstract: Given a finitely generated module $ M$ over a local ring $ A$ of characteristic $ p$ with $ \operatorname {pd} M < \infty $, we study the asymptotic intersection multiplicity $ \chi _\infty (M, A/\underline {x})$, where $ \underline {x} = (x_1, \ldots , x_r)$ is a system of parameters for $ M$. We show that there exists a system of parameters such that $ \chi _\infty $ is positive if and only if $ \dim \operatorname {Ext}^{d-r}(M, A) = r$, where $ d = \dim A$ and $ r = \dim M$. We use this to prove several results relating to the grade conjecture, which states that $ \operatorname {grade} M + \dim M = \dim A$ for any module $ M$ with $ \operatorname {pd} M < \infty $.


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Additional Information

Jesse Beder
Affiliation: Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, Illinois 61801

DOI: https://doi.org/10.1090/S0002-9939-2014-12183-6
Received by editor(s): February 7, 2012
Received by editor(s) in revised form: January 27, 2013
Published electronically: August 14, 2014
Communicated by: Irena Peeva
Article copyright: © Copyright 2014 American Mathematical Society