Tight closure and localization, and the Multiplicity Conjecture

Holger Brenner and Paul Monsky have posted a preprint entitled “Tight closure does not commute with localization” to the arXiv. This answers possibly the biggest question in tight closure theory.
In similar news, David Eisenbud and Frank-Olaf Schreyer have posted a preprint entitled “Betti Numbers of Graded Modules and Cohomology of Vector Bundles”. In particular, this paper contains a proof of the characteristic-zero case of a famous conjecture of Huneke and Srinivasan on the relation between multiplicities and Betti numbers.
Finally, we’ve posted some photos from the conference in honor of Juergen Herzog in Cortona, Italy, last September. Thanks to Winfried Bruns and Tim Roemer for sending us the links! Go see!