Leírás
This is joint work with Craig Huneke and Srikanth Iyengar. The work is motivated by a (still open) conjecture Craig and I made a quarter-century ago: If $M$ is a finitely generated module over a local domain $R$ and $M \otimes_R M^*$ is maximal Cohen-Macaulay, must $M$ be free? For Gorenstein rings its truth would imply the truth of the Auslander-Reiten Conjecture; moreover, it reduces to the one-dimensional case, where it is equivalent to the conjecture that $\Ext^1_R(M,M) = 0$ implies $M$ is free. Modules with no self-extensions are called ``rigid", and we study the case of ideals (in a one-dimensional Gorenstein domain, often assumed to be a complete intersection). Even in this restricted context the problem is still open, but partial results have nice connections with multiplicative ideal theory and the Eisenbud-Goto-Horrocks Conjecture (recently proved by Mark Walker).