Locally Complete Intersection Maps and the Proxy Small Property

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