The singularity category of C^*(BG) for a finite group G

Abstract: Abstract: The cohomology ring H^*(BG) (with coefficients in a field k of characteristic p) is a very special graded commutative ring, but this comes out much more clearly if one uses the cochains C^*(BG), which can be viewed as a commutative ring up to homotopy. For example C^*(BG) is always Gorenstein (whilst this is not quite true for H^*(BG)).

This leads one to study C^*(BG) as if it was a commutative local Noetherian ring, though of course one has to use homotopy invariant methods. The ring C^*(BG) is regular if and only if G is p-nilpotent and so it is natural to look for ways of deciding where C^*(BG) lies on the spectrum between regular and Gorenstein rings. For a commutative Noetherian ring, one considers the singularity category D_{sg}(R) (the quotient of finite complexes of finitely generated modules by finitely generated projectives). This is trivial if and only if R is regular, so is the appropriate tool. The talk will describe how to define this for C^*(BG), show it has good basic properties and describe the singularity category in the simplest case it is not trivial (when G has a cyclic Sylow p-subgroup).

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

Audience: researchers in the topic

---

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/

---