# Mathematical Sciences Research Institute

Home » Derived representation schemes and non-commutative geometry

# Seminar

Derived representation schemes and non-commutative geometry February 19, 2013 (02:00 PM PST - 03:00 PM PST)
Parent Program: -- MSRI: Simons Auditorium
Speaker(s) Yuri Berest
Description No Description
Video
If k is a field, the set of all representations of an associative k-algebra A on a finite-dimensional vector space V can be given the structure of an affine k-scheme, called the representation scheme Rep_V(A). According to a heuristic principle proposed by M.Kontsevich and A.Rosenberg, the family of schemes {Rep_V(A)} for a given non-commutative algebra A should be thought of as a substitute or approximation' for Spec(A)'. The idea is that every property or non-commutative geometric structure on A should naturally induce a corresponding geometric property or structure on Rep_V(A) for all V.
Rep_V(A) fails to have a corresponding property in the usual algebro-geometric sense. The reason for this seems to be that the representation functor Rep_V is not exact' and should be replaced by its derived functor DRep_V (in the sense of non-abelian homological algebra).