Bounds on Injective Dimension and Exceptional Complete Intersection Maps

TL;DR

This paper establishes conditions under which a local homomorphism between noetherian local rings is an exceptional complete intersection map, based on bounds on the injective dimension of modules.

Contribution

It proves that certain bounds on injective dimension imply the map is an exceptional complete intersection, linking module properties to ring homomorphism classification.

Findings

Injective dimension bounds imply modules are injective over S

Such bounds characterize exceptional complete intersection maps

Provides criteria for identifying exceptional complete intersection maps

Abstract

We prove that if $f:R \rightarrow S$ is a local homomorphism of noetherian local rings, and $M$ is a non-zero finitely generated or artinian $S$-module whose injective dimension over $R$ is bounded by the difference of the embedding dimensions of $R$ and $S$, then $M$ is an injective $S$-module and $f$ is an exceptional complete intersection map.