Locally Complete Intersection Maps and the Proxy Small Property

Abstract

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 if $S$ 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.