Locally complete intersection maps and the proxy small property

TL;DR

This paper characterizes locally complete intersection maps of noetherian rings via the proxy small property of the bimodule, extending classical smoothness results and providing geometric analogues and applications.

Contribution

It establishes that a flat, essentially finite type map is locally complete intersection if and only if the bimodule is proxy small, linking algebraic and geometric properties.

Findings

Characterization of locally complete intersection maps via proxy smallness.

Extension of classical smoothness results to the proxy small context.

Applications to factorization theorems for such maps.

Abstract

It is proved that a map $\varphi\colon R\to S$ of commutative noetherian rings that is essentially of finite type and flat is locally complete intersection if and only $S$ is proxy small as a bimodule. This means that the thick subcategory generated by $S$ as a module over the enveloping algebra $S\otimes_RS$ contains a perfect complex supported fully on the diagonal ideal. This is in the spirit of the classical result that $\varphi$ is smooth if and only if $S$ is small as a bimodule, that is to say, it is itself equivalent to a perfect complex. The geometric analogue, dealing with maps between schemes, is also established. Applications include simpler proofs of factorization theorems for locally complete intersection maps.