On a theorem of Gulliksen on the homology of local rings

TL;DR

This paper provides an elementary proof of a key theorem in homological algebra, showing that under certain conditions, the induced map in Andre9-Quillen homology is injective, simplifying previous proofs.

Contribution

It offers a simplified, elementary proof of Avramov's theorem on the injectivity of homology maps for noetherian local rings with finite flat dimension.

Findings

Elementary proof of Avramov's theorem established

Injectivity of the induced homology map demonstrated

Simplifies understanding of homology in local rings

Abstract

We show that a modification of the proof of a result of Gulliksen gives an elementary proof of the following important theorem by Avramov: if $(A,k) \to (B,l)$ is a homomorphism of noetherian local rings and $B$ is of finite flat dimension over $A$, then the homomorphism induced in Andr\'e-Quillen homology modules $H_2(A,l,l)\to H_2(B,l,l)$ is injective.