Mel Hochster

My research is primarily concerned with commutative Noetherian rings, especially local rings (i.e., rings with a unique maximal ideal). I have been particularly interested in Cohen-Macaulay rings and modules (a finitely generated graded algebra over a field is Cohen-Macaulay if and only if it is a finitely generated free module over a polynomial ring). One of the main techniques in my work has been that of attacking problems, even problems that seem to arise primarily over a field of characteristic 0, by using positive characteristic methods (the action of the Frobenius endomorphism). One example of a theorem that can be proved this way is that, given a linear algebraic group (i.e., a group of matrices) which is linearly reductive (every representation is completely reducible) over the complex numbers acting on a polynomial ring, the ring of invariants (or fixed ring) is Cohen-Macaulay.

A great deal of my research, much of it joint with Craig Huneke, has been aimed at developing the notion of tight closure. The result mentioned above on the Cohen-Macaulay property for rings of invariants and several other apparently unrelated results can be proved using this technique. Tight closure is a closure operation defined on ideals (and submodules of Noetherian modules) first over Noetherian rings of positive prime characteristic p, but it can be extended to all Noetherian rings containing a field. Other applications include the Briancon-Skoda theorem on integral closures of ideals, and the result that regular rings are direct summands of their module-finite extension algebras (this is a theorem for rings containing a field, and a major open question in dimension four or higher, even if the regular ring is Z[x,y,z]). In particular, I am very interesed in developing an analogue for tight closure theory that could be applied to Noetherian rings that do not necessarily contain a field

Recently, I have been working with Tigran Ananyan on giving bounds on the projectdive dmension of an ideal generated by a specified number of forms of specified degrees that do not depend on the number of variables.  We have proved a conjecture of Michael Stillman that such bounds always exist.  It remains an open question to give best possible bounds: very few are known.