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# Mathematics

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for events the day of Friday, March 11, 2022.

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Friday, March 11, 2022

3:00 pm in 341 Altgeld Hall,Friday, March 11, 2022

#### Applications of Dualizing Complex in Commutative Algebra

###### Likun Xie (UIUC Math)

Abstract: Starting with a finitely generated module of finite projective dimension over the completion of a Noetherian local ring A, a natural question is when does this module descend, i.e. when is this module the completion of a finite A-module of finite projective dimension? We will need the theorem of local duality to show it happens over some “good” rings. We will introduce dualizing complex both in the language of derived categories, and in an explicit form for commutative Noetherian ring. We will apply them to study some descent problems for module of finite projective dimension. The techniques employed also allow one to recover a theorem of Horrocks about vector bundles over a punctured spectrum of a local ring. We follow Section I.5 in Dimension projective finie et cohomologie locale by Peskine and Szpiro.