Applications of Frobenius maps

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 this project is to produce an algorithm to compute a crucial component involved in the tight closure operation, namely parameter-test-ideals and test-ideals. The approach taken by this project is to study this 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 tackling a relatively simple instance of this problem and the project will attempt the generalize those results.The study of injective hulls of the residue field of the ring yielded new insights into a certain widely studied set numerical invariants of algebraic sets, namely their jumping coefficients. This resulted in a proof that these invariants for surfaces defined by one condition form a discrete set of rational numbers. This project will attempt to generalize this result for other surfaces and it will try to produce an algorithm for computing these numbers.