Tight closure, Frobenius maps and Frobenius splittings

Lead Research Organisation: University of Sheffield
Department Name: Pure Mathematics

Abstract

Many theorems in Commutative Algebra can be proved by showing that:(1) if the theorem fails, one can find a counter-example in a ring of prime characteristic p (i.e., a ring which contains the ring of integers modulo a prime number p), and(2) no such counter-example exists in characteristic p.Step (2) above is often much easier to prove than in characteristic zero because of the existence of the Frobenius function f(r) which raises r to the pth power. This functon is an endomorphism of the rings, i.e., it has the property that f(r+s)=f(r)+f(s), and surprisingly, gives a good handle on many problems in characteristic p.A formal method to exploit the existence of these Frobenius function is the theory of Tight Closure which was first developed about 20 years ago to tackle old problems in the field. Since its inception it has been very successful in giving short and elegant solutions to hard old questions. Tight Closure also found surprising applications in other fields, especially in Algebraic Geometry.The essence of this theory is an operation which takes an ideal in a ring of commutative ring of characteristic p and produces another larger ideal with useful properties. This operation is very difficult to grasp, even in seemingly simple examples, and one of the aims of my recent research has been to produce an algorithm to compute a crucial component involved in the tight closure operation, namely parameter-test-ideals and test-ideals. During the last few years I developed a new way to study these test-ideals via a duality which relates them to certain sub-objects of certain large and complicated objects, namely injective hulls of the residue field of the ring. This approach has been very successful in exploring other problems as well.

Planned Impact

A successful outcome of this project has the potential to (1) raise the visibility and prestige of the British mathematical research, (2) incorporate new growing trends into British mathematics, (3) foster and strengthen international collaboration.

Publications


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Katzman M (2012) An algorithm for computing compatibly Frobenius split subvarieties in Journal of Symbolic Computation
Katzman M (2014) Castelnuovo-Mumford regularity and the discreteness of $F$-jumping coefficients in graded rings in Transactions of the American Mathematical Society
Katzman M (2011) An upper bound on the number of $F$-jumping coefficients of a principal ideal in Proceedings of the American Mathematical Society
Katzman M (2012) Two interesting examples of ?-modules in characteristic p >0 in Bulletin of the London Mathematical Society
 
Description he funded research yielded insights into the properties of commutative rings defined over fields of prime characteristic. It also discovered ways to compute certain important objects associated with these rings.
Exploitation Route This research has been cited by experts in my field and has provided the foundation for subsequent research.
Sectors Other
 
Description My research produced insights into rings of prime characteristic: (1) The paper "An upper bound on the number of $F$-jumping coefficients of a principal ideal" showed that certain invariants associated with local rings of prime characteristic cannot be too abundant. (2) The paper "Two interesting examples of D-modules in characteristic p>0" showed that the effort to find a sensible notion of holonomicity for D-modules of prime characteristic is doomed, and thus new notions replacing this should be sought. (3) The paper "Castelnuovo-mumford regularity and the discreteness of $f$-jumping coefficients in graded rings" uncovered a surprising connection between the growth of the Castelnuovo-Mumford refularity of Frobenius powers of ideals and the properties of F-jumping coefficients. (4) In "An algorithm for computing compatibly Frobenius split subvarieties" we produced the first ever algorithm for finding subvarieties which are preserved by a given Frobenius splitting, and thus opening a new area of research into algoorithmic prime characteristic methods.
First Year Of Impact 2010
Sector Other
Impact Types Cultural