The number of generators of the first Koszul homology of an Artinian ring

Abstract

We study the conjecture that if $I,J$ are $\mu-$primary in a regular local ring $(R,\mu)$ with dim$(R)=n$, then $\frac{I \cap J}{IJ} \cong \Tor_1(R/I, R/J)$ needs at least $n$ generators, and a related conjecture about the number of generators of the first Koszul homology module of an Artinian local ring $(A,m)$. In this manuscript, we focus our attention on the complete intersection defect of the Artinian ring and its quotient by the Koszul elements. We prove that the number of generators of the first Koszul homology module of $x_1,...,x_n \in m$ on an Artinian local ring $(A,m)$ is at least $n+ \cid(A)- \cid(\frac{A}{(x_1,...,x_n)A})$, where $\cid A$ denotes the complete intersection defect of the Artinian local ring $A$.