The Grade Conjecture and Asymptotic Intersection Multiplicity

2012/02/16 by Jesse S. Beder, Beder, Jesse S. · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Rings, Modules, and Algebras #math.AC

paper · pdf · doi:10.48550/arxiv.1202.3513

arxiv created 2013/03/06 · arxiv updated 2013/03/07

Abstract

Given a finitely generated module M over a local ring A of characteristic p with \pd M < ∞, we study the asymptotic intersection multiplicity χ_∞(M, A/\underlinex), where \underlinex = (x₁, …, x_r) is a system of parameters for M. We show that there exists a system of parameters such that χ_∞ is positive if and only if dim \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 \grade M + dim M = dim A for any module M with \pd M < ∞.

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