EPSRC Reference: 
EP/I031405/1 
Title: 
Prime characteristic methods in commutative algebra 
Principal Investigator: 
Katzman, Dr M 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Pure Mathematics 
Organisation: 
University of Sheffield 
Scheme: 
Overseas Travel Grants (OTGS) 
Starts: 
06 August 2011 
Ends: 
05 August 2013 
Value (£): 
57,111

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


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Panel History: 

Summary on Grant Application Form 
Many theorems in Commutative Algebra can be proved by showing that:(1) if the theorem fails, one can find a counterexample 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 counterexample 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 parametertestideals and testideals. During the last few years I developed a new way to study these testideals via a duality which relates them to certain subobjects 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.I propose to expand my research of commutative rings of prime characteristic by continuing my collaboration with fellow researchers in my field who work in the US. These include Prof. Gennady Lyubeznik (University of Minnesota),Prof. Karl Schwede (Penn State University) and participants of the special programme in Commutative Algebra organized by the Mathematical Sciences Research Institute in California, which I would like to attend.

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Organisation Website: 
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