Mathematics – Commutative Algebra
Scientific paper
2012-02-16
Mathematics
Commutative Algebra
Scientific paper
Given a finitely generated module $M$ over a local ring $A$ of characteristic $p$ with $\pd M < \infty$, we study the asymptotic intersection multiplicity $\chi_\infty(M, A/\underline{x})$, where $\underline{x} = (x_1,..., x_r)$ is a system of parameters for $M$. We give a necessary and sufficient condition for the existence of a system of parameters such that $\chi_\infty$ is positive. 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 $\pd M < \infty$.
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