On a new invariant of finitely generated modules over local rings

Abstract

Let $M$ be a finitely generated module on a local ring $R$ and $\F: M_0\subset M_1\subset...\subset M_t=M$ a filtration of submodules of $M$ such that $d_o<d_1<...<d_t=d$, where $d_i=\dim M_i$. This paper is concerned with a non-negative integer $p_{\mathcal{F}}(M)$ which is defined as the least degree of all polynomials in $n_1,...,n_d$ bounding above the function $$\ell (M/(x_1^{n_1},...,x_d^{n_d})M)-\sum _{i=0}^t{n}_1...n_{d_i}e(x_1,...,x_{d_i};M_i).$$ We prove that $p_{\mathcal{F}}(M)$ is independent of the choices of good systems of parameters $\underline{x}=x_1,...,x_d$. When $\F$ is the dimension filtration of $M$ we also present some relations between $p_\F(M)$ and the polynomial type of each $M_i/M_{i-1}$ and the dimension of the non-sequentially Cohen-Macaulay locus of $M$.

arXiv commentTo appear in the Journal of Algebra and its Applications