SMRI Algebra and Geometry Online: Greenlees -- The singularity category of C^*(BG) for a finite group G

SMRI Algebra and Geometry Online 
'The singularity category of C^*(BG) for a finite group G' 
(Report on joint work with G.Stevenson and D.Benson) 
John Greenlees (University of Warwick) 

Thursday Sep 16, 2021 4:00-5:30PM 
Register:
https://uni-sydney.zoom.us/meeting/register/tZYtf-iorT0iGdWQdjCCm3U7UmqW9jzrpyf0 

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).

Biography: John Greenlees obtained his PhD from Cambridge in 1986, and spent 3 years 
at the National University of Singapore and a year at the University of Chicago 
before moving to Sheffield in 1990.  He spent 28 years there, serving as Head of 
Pure Maths and then Head of the School of Mathematics and Statistics before moving 
to Warwick as Head of Department in 2018.  He has published over 90 papers and 4 
books on algebra and topology, and serves on the editorial boards of 3 algebraic 
topology journals.  He has acted as undergraduate External Examiner for 5 
institutions, and conducted external reviews for 7 departments.  

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

After registering, you will be sent a confirmation email ~24 hours prior to the
seminar.