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- Title
Locally Complete Intersection Maps and the Proxy Small Property.
- Authors
Briggs, Benjamin; Iyengar, Srikanth B; Letz, Janina C; Pollitz, Josh
- 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.
- Subjects
NOETHERIAN rings; COMMUTATIVE rings; ALGEBRA; FACTORIZATION
- Publication
IMRN: International Mathematics Research Notices, 2022, Vol 2022, Issue 16, p12625
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnab041