Blowup formulas for nilpotent sensitive cohomology theories

Abstract: This is joint work in progress with Shuji Saito. Many cohomology theories of interest (l-adic cohomology, de Rham cohomology, motivic cohomology, K-theory...) have long exact sequences associated to blowups. Such a property can be neatly encoded in a Grothendieck topology such as the cdh-topology or the h-topology. These topologies appeared in Voevodsky's proof of the Bloch-Kato conjecture, and more recently in Beilinson's simple proof of Fontaine's CdR conjecture, and in Bhatt and Scholze's work on projectivity of the affine Grassmanian.

A feature of these topologies which often turns out to be a bug is that the associated sheaves cannot see nilpotents. In this talk I will discuss a modification which can see nilpotents, and which still has long exact sequences for many blowups.

commutative algebraalgebraic geometryanalysis of PDEsalgebraic topologydifferential geometrygeneral topologygeometric topologymetric geometryoperator algebrasquantum algebrarings and algebrassymplectic geometry

Audience: researchers in the topic

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Algebra Seminar (presented by SMRI)

Series comments: Algebra Seminar:

'Homological comparison of resolution and smoothing'

Will Donovan (Tsinghua University)

Friday Sep 23, 12:00-1:00PM

Online via Zoom

Register here: uni-sydney.zoom.us/meeting/register/tZEpd-isqD0iGdLoqtmNEEQKxmg0xlakSdCq

Abstract: A singular space often comes equipped with (1) a resolution, given by a morphism from a smooth space, and (2) a smoothing, namely a deformation with smooth generic fibre. I will discuss work in progress on how these may be related homologically, starting with the threefold ordinary double point as a key example.

Biography: Will Donovan is currently an Associate professor at Yau MSC, Tsinghua University, Beijing. He is also a member of the adjunct faculty at BIMSA, Yanqi Lake, Huairou, Beijing and a visiting associate scientist at Kavli IPMU, University of Tokyo. He received his PhD in Mathematics in 2011 from Imperial College London. His interests are algebraic geometry, noncommutative geometry, representation theory, string theory and symplectic geometry.

www.maths.usyd.edu.au/u/AlgebraSeminar/

Note: These seminars will be recorded, including participant questions (participants only when asking questions), and uploaded to the SMRI YouTube Channel www.youtube.com/c/SydneyMathematicalResearchInstituteSMRI

Other upcoming SMRI events can be found here: mathematical-research-institute.sydney.edu.au/news-events/

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