Locally complete intersection maps and the proxy small property
File(s)
Accepted version
Author(s)
Briggs, Benjamin
Iyengar, Srikanth B
Letz, Janina C
Pollitz, Josh
Type
Journal Article
Abstract
It is proved that a map ϕ : R → S of commutative Noetherian rings that is essentially of
finite type and f lat 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 ⊗R S contains a perfect complex supported fully on the diagonal
ideal. This is in the spirit of the classical result that ϕ 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.
finite type and f lat 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 ⊗R S contains a perfect complex supported fully on the diagonal
ideal. This is in the spirit of the classical result that ϕ 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.
Date Issued
2022-08
Date Acceptance
2021-02-04
Citation
International Mathematics Research Notices, 2022, 2022 (16), pp.12625-12652
ISSN
1073-7928
Publisher
Oxford University Press
Start Page
12625
End Page
12652
Journal / Book Title
International Mathematics Research Notices
Volume
2022
Issue
16
Copyright Statement
Copyright © 2021 Oxford University Press. This is a pre-copy-editing, author-produced version of an article accepted for publication in International Mathematics Research Notices following peer review. The definitive publisher-authenticated version Benjamin Briggs, Srikanth B Iyengar, Janina C Letz, Josh Pollitz, Locally Complete Intersection Maps and the Proxy Small Property, International Mathematics Research Notices, Volume 2022, Issue 16, August 2022, Pages 12625–12652, https://doi.org/10.1093/imrn/rnab041 is available online at: https://doi.org/10.1093/imrn/rnab041
Identifier
http://dx.doi.org/10.1093/imrn/rnab041
Publication Status
Published
Date Publish Online
2021-04-26