31 may 2016 -- 12:00
Aula Tricerri, DiMaI, Università di Firenze
Abstract.
Let us recall that a format $(r_{n},\ldots ,r_{1})$ of a free complex $0 \to F_{n} \to F_{n-1} \to \ldots \to F_0$ over a commutative Noetherian ring is the sequence of ranks $r_{i}$ of the $i$-th differential $d_{i}$. We will assume that rank $F_{i} = r_{i}+r_{i+1}$. We say that an acyclic complex $F_{gen}$ of a given format 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 $f:R_{gen} \to S$ such that $G=F_{gen}\otimes_{R_{gen}} S$. For complexes of 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 will discuss the ideas going into the proof of 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 lengths $p-1, q-1, r-1$ attached to the central vertex). Then there exists a Noetherian generic ring for this format 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 deforms to a ring carrying an action of the Kac-Moody Lie algebra corresponding to the graph $T_{p,q,r}$.