Some new invariants of Noetherian local rings related to square of the maximal ideal

Abstract

We introduce two new invariants of a Noetherian (standard graded) local ring $(R,\mathfrak{m})$ that measure the number of generators of certain kinds of reductions of $\mathfrak{m},$ and we study their properties. Explicitly, we consider the minimum among the number of generators of ideals $I$ such that either $I^2={\mathfrak{m}}^2$ or $I\supseteq {\mathfrak{m}}^2$ holds. We investigate subsequently the case that $R$ is the quotient of a polynomial ring $k[x_1,\dots ,x_n]$ by an ideal $I$ generated by homogeneous quadratic forms, and we compute these invariants. We devote specific attention to the case that $R$ is the quotient of a polynomial ring $k[x_1,\dots ,x_n]$ by the edge ideal of a finite simple graph $G.$

First draft. 40 pages. Comments are Welcome!