Faculty Detail
Mel Hochster
Chair &J E McLaughlin Distinguished University Pro
3078 East Hall
Phone: (734) 9361310
Email:
hochster@umich.edu

Affiliation(s)

Algebra/Algebraic Geometry

Algebra/Algebraic Geometry

About
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 CohenMacaulay rings and modules (a finitely generated graded algebra over a field is CohenMacaulay 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 CohenMacaulay.
In recent years 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 CohenMacaulay 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 BrianÃ§onSkoda theorem on integral closures of ideals, and the result that regular rings are direct summands of their modulefinite 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.