The common thread for the projects in this proposal are asymptotic properties of ordinary, Frobenius, and symbolic powers of a given ideal. The background motivation is the still open question of whether tight closure commutes with localization. The main tools that the proposer plans to use and develop further are the Rees-Izumi type comparisons of a large class of valuations, as well as the primary decompositions and the prime filtrations of the cokernels of the powers of an ideal. The Rees-Izumi part would be done in continued collaboration with Reinhold Hubl. The primary decompositions and prime filtrations part is related to Huneke's Artin-Rees lemma and would include collaboration with Anna Guerrieri on primary decompositions of special Jacobian ideals. In the case of affine rings, the asymptotic properties of Frobenius powers of ideals can also be studied via boundedness of the degrees of Grobner bases. This part would be done in collaboration with Susan Hermiller. The part concerned with finding algorithms for calculating adjoints would be undertaken with the proposer's graduate student Mark Rhodes. Serkan Hocsten, Bernd Sturmfels and the proposer plan to work on the stabilization properties of resolutions and of associated primes of powers of monomial ideals.
This proposal is in the area of commutative algebra with applications in algebraic geometry and computational algebra. The main background motivation is proving some basic properties about tight closure, but the proposed techniques would have implications also outside the tight closure theory. Tight closure is a young theory due to Melvin Hochster and Craig Huneke which has in a short time proved many new results in commutative algebra and algebraic geometry and has in addition shortened and simplified several old results. The predominant object of study of this proposal are ideals, and especially sets of ideals which are related to each other in some natural way, such as ordinary, Frobenius, or symbolic powers of a particular ideal. The main goal of this proposal is then establishing certain asymptotic and stabilization properties of these sets of ideals.
