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

TL;DR

This paper investigates the minimal number of generators of the first Koszul homology module in Artinian rings, relating it to the complete intersection defect and providing a lower bound based on ring quotients.

Contribution

It establishes a new lower bound for the number of generators of the first Koszul homology module in terms of the complete intersection defect of the ring and its quotients.

Findings

Number of generators ≥ n + cid(A) - cid(A/(x_1,...,x_n)A)

Connects Koszul homology generators to complete intersection defect

Provides bounds for Artinian local rings' homology modules

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 $A$.