Date & Time: Wednesday, April 16, 2014, 17:05-18:05.
Venue: Ramanujan Hall
Speaker: Roger Wiegand, University of Nebraska
Title: Non-uniqueness of direct-sum decompositions
Abstract: Let $R$ be a local domain of dimension one. For example, $R$ might be the local ring of a singular point on an algebraic curve, or a localization of an order in an algebraic number field. We are particularly interested in the situation where the completion of the ring is not an integral domain. Examples include the nodal cubic curve $C[x,y]/(y^2-x^3-x^2)$ and rings such as $Z[\sqrt{-5}]$ (suitably localized).
Let $C[R]$ be the set of isomorphism classes of finitely generated torsion-free $R$-modules. We make $C[R]$ an additive semigroup, using the direct-sum relation: $[M] + [N] = [M \oplus N]$, where $[ ]$ denotes the isomorphism class of a module. This semigroup encodes all direct- sum relations among finitely generated torsion-free modules.
The structure of this semigroup has been worked out in two antipodal cases: (1) when all branches of the completion have infinite representation type, and (2) when $R$ has finite representation type. The intermediate case, where $R$ has infinite representation type but at least one branch has finite representation type seems to be much more difficult, but some progress has been made.
In this talk I will survey some of these results and give concrete examples to show spectacular failure of uniqueness of direct-sum decompositions. For example, given any integer $n \geq 2$ one can find a ring $R$ (as above) and indecomposable modules $M,N,V$ such that $M \oplus N$ is isomorphic to the direct sum of $n$ copies of $V$.