Locally Complete Intersection Maps and the Proxy Small Property. (26th April 2021)

• Record Type: Journal Article
• Authors: Briggs, Benjamin; Iyengar, Srikanth B; Letz, Janina C; Pollitz, Josh
• Is Part Of: International mathematics research notices. Volume 2022:Number 16(2022)
• Page Start: 12625
• Page End: 12652
• Publication Date: 2021-04-26
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnab041
• Languages: English
• ISSNs: 1073-7928

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.