REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

Jerzy Weyman

ABSTRACT: In this talk I will discuss a recent solution of the problem of the existence of a (Noetherian) generic ring for finite free resolutions of length 3.

Let us recall that for a given format (r_n,\ldots ,r_1) of the free complex 0-->F_n-->F_{n-1}-->\ldots F_0
with the rank of the i-th differential d_i equal to r_i (and thus rank F_i =r_r+r_{i+1}, we say that an acyclic  complex F_{gen} over a given ring R_{gen} is generic if for every complex G of this format over a Noetherian ring S there exists a homomorphism f:R_{gen}--> S such that G=F_{gen}\otimes_{R_{gen}} S.

For length 2 the existence of the generic acyclic complex was established by Hochster and Huneke in the 1980's. It is a normalization of the ring giving a generic complex (two matrices with composition zero and rank conditions).

I prove the following result: Associate to a triple of ranks (r_3, r_2, r_1) a triple (p,q,r)=(r_3+1, r_2-1, r_1+1). Associate to (p,q,r) the graph T_{p,q,r} (three arms of lenghts p-1, q-1, r-1 attached to the central vertex). Then there exists a Noetherian generic ring if and only if T_{p,q,r} is a Dynkin graph. In other cases one can construct in a uniform way a non-Noetherian generic ring, which carries an action of the corresponding Kac-Moody Lie algebra.

I will provide the background on finite free resolutions.