Algebraic Geometry Seminar
Uli Walther
Purdue
Bockstein morphisms and remarks on a conjecture of Lyubeznik
Abstract: If R is a regular ring containing a field then by results of
Huneke-Sharp and Lyubeznik, the local cohomology modules H^i_I(R)
have, for all ideals
I of R and for all integers i, a finite set of associate primes. The
reasons are quite different: D-modules in characteristic 0, the Frobenius
morphism in characteristic p.
Lyubeznik conjectured that these results can be extended to regular rings
of mixed characteristics, and proved it in the unramified local case. The
talk is concerned with the case of a polynomial ring over ZZ.
In the presence of singularities finiteness can be absent, as examples by
Singh, Katzman, and Swanson show. Inspired by the first example of a local
cohomology module with infinitely many associated primes due to Singh, we
give a theorem that shows that if finiteness fails in polynomial rings
over ZZ then it must fail in very strange ways.
Our main tool is an adaptation of the Bockstein morphism from algebraic
topology to local cohomology.
Monday March 16, 2009 at 3:00 PM in SEO 712