Cohomological support varieties under local homomorphisms

TL;DR

This paper investigates how cohomological support varieties of modules over local rings behave under local homomorphisms, revealing new invariance properties and recovering classical results in the context of complete intersections.

Contribution

It extends the understanding of support varieties under local maps, especially for complete intersections and finite flat dimension maps, connecting to known theorems.

Findings

Support variety dimension is preserved or increases under certain local maps.

Recovers Bergh and Jorgensen's theorem for complete intersection surjective maps.

Shows the complete intersection property is stable under localization.

Abstract

Given a bounded complex of finitely generated modules $M$ over a commutative noetherian local ring $R$, one assigns to it a variety, $\mathcal V_R(M)$, called the cohomological support variety of $M$ over $R$. The variety $\mathcal V_R(M)$ holds important homological information about the complex and the ring. In this paper, we study the behavior of cohomological support varieties under restriction of scalars along local maps. In the case where the rings involved are complete intersections and the map is a surjective complete intersection, this recovers a theorem of Bergh and Jorgensen. Additionally, we show that if $R\to S$ is a local map of finite flat dimension, then the dimension of $\mathcal V_R(R)$ is less than or equal to that of $\mathcal V_S(S)$. This allows us to recover Avramov's result that the complete intersection property is preserved under localization.