Automorphisms of the Koszul homology of a local ring

TL;DR

This paper investigates the automorphisms of the Koszul complex of a local ring, showing conditions under which these automorphisms induce the identity on homology and exploring the structure of the automorphism groups involved.

Contribution

It establishes when dg algebra automorphisms of the Koszul complex induce the identity on homology and describes the automorphism groups for various rings.

Findings

Automorphisms inducing identity on homology under certain conditions.

Existence of rings with non-identity automorphisms on homology.

Automorphism group of homology induced by automorphisms of the complex is abelian.

Abstract

This work concerns the Koszul complex $K$ of a commutative noetherian local ring $R$, with its natural structure as differential graded $R$-algebra. It is proved that under diverse conditions, involving the multiplicative structure of $H(K)$, any dg $R$-algebra automorphism of $K$ induces the identity map on $H(K)$. In such cases, it is possible to define an action of the automorphism group of $R$ on $H(K)$. On the other hand, numerous rings are described for which $K$ has automorphisms that do not induce the identity on $H(K)$. For any $R$, it is shown that the group of automorphisms of $H(K)$ induced by automorphisms of $K$ is abelian.