Common threads in the theories of Local Cohomology, D-modules and Tight Closure and their interactions

Lead Research Organisation: University of Sheffield
Department Name: Mathematics and Statistics

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.

During the course of development of the study of commutative rings of prime characteristic, various notions and techniques were introduced, e.g., a certain tight-closure operation of ideals, certain structures on ``large'' objects called local cohomology modules, and differential operators acting on these rings. The objects and their associated techniques have proved to be very successful in tackling algebraic and geometric problems, and the interactions between these concepts turned out to be especially fertile.

I propose to study these interactions further with the aid of a research assistant, and to apply the resulting techniques to the solution of several outstanding problems in my field.

Planned Impact

Good progress in any one of the objectives of this project would be regarded by many experts in my field as a significant mathematical advance. Moreover, the cross-fertilization of ideas across different areas in algebra has the potential to introduce completely new ways of thinking about these areas and novel techniques for the solution of problems.

In particular, any progress in the application of D-modules techniques to commutative algebra has the potential to make problems in this field amenable to the tools of D-module experts in the UK.

Work on this project will give an excellent opportunity for a young researcher to acquire a broad experience in several related areas of research.

It is very likely that the RA will be chosen from outside the UK, and by giving talks in seminars will present local staff and postgraduate students an opportunity to learn about research done in other research centres.

Publications

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Boix A (2014) An algorithm for producing F-pure ideals in Archiv der Mathematik

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Boix Alberto F. (2013) An algorithm for producing F-pure ideals in arXiv e-prints

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Bonacho Dos Anjos Henriques I (2016) Test, multiplier and invariant ideals in Advances in Mathematics

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Henriques (2014) Test, multiplier and invariant ideals in arXiv e-prints

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Katzman M (2014) Annihilators of Artinian modules compatible with a Frobenius map in Journal of Symbolic Computation

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Katzman M (2015) An extension of a theorem of Hartshorne in Proceedings of the American Mathematical Society

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KATZMAN M (2014) Rings of Frobenius operators in Mathematical Proceedings of the Cambridge Philosophical Society

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Katzman M (2017) Global parameter test ideals in Journal of Algebra

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Katzman M (2016) Global parameter test ideals

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Katzman M (2017) The Support of Local Cohomology Modules in International Mathematics Research Notices

 
Description The main outcomes of the funded research were:
1) The discovery of new algorithms for computing various invariants associated to prime characteristic methods,
2) Explicit descriptions of $F$-jumping coefficients in determinantal rings.
3) New insights into the properties of local cohomology modules in prime characteristic.
Exploitation Route In addition to the fundamental results yielded by our research, we are providing the commutative algebra community algorithmic methods for the calculation of interesting invariants.
Sectors Other

 
Description Collaboration with Prof. Wenliang Zhang 
Organisation University of Nebraska-Lincoln
Country United States 
Sector Academic/University 
PI Contribution Prof Zhang and I studiied various problems in various areas of this project. The PI visited Prof Zhang several times.
Collaborator Contribution Several papers were coauthored.
Impact Katzman M, Lyubeznik G, Zhang W. (2015). An extension of a theorem of Hartshorne. Proceedings of the American Mathematical Society, 144 (3), pp. 955-962 Katzman Mordechai, Lyubeznik Gennady, Zhang Wenliang. (2014). An extension of a theorem of Hartshorne. ArXiv e-prints, pp. arXiv:1408.0858 Katzman Mordechai, Zhang Wenliang. (2015). The support of local cohomology modules. ArXiv e-prints, pp. arXiv:1509.01519 Katzman M, Zhang W. (2014). Annihilators of Artinian modules compatible with a Frobenius map. Journal of Symbolic Computation, pp. 29-46 Katzman M, Zhang W. (2016) Global parameter test ideals, ArXiv e-prints, arXiv:1607.01947
Start Year 2012
 
Description Collaboration with University of Utah 
Organisation University of Utah
Department Department of Mathematics
Country United States 
Sector Academic/University 
PI Contribution Professors Schwede and Singh and myself looked at several problems related to the project. We coauthored a paper.
Collaborator Contribution We coauthored a paper.
Impact KATZMAN M, SCHWEDE K, SINGH A, ZHANG W. (2014). Rings of Frobenius operators. Mathematical Proceedings of the Cambridge Philosophical Society, 157 (01), pp. 151-167
Start Year 2013
 
Description Test ideals and multiplier ideals 
Organisation University of Genoa
Department Department of Mathematics
Country Italy 
Sector Academic/University 
PI Contribution Several reciprocal meetings yielded a colaboartion between the RA, I. Henriques and M. Varbaro (University of Genoa) which culminated in two papers.
Collaborator Contribution The contribution was in the form of collaboration in the writing of two articles, one of them already in print.
Impact Bonacho Dos Anjos Henriques I, Varbaro M. (2016). Test, multiplier and invariant ideals. Advances in Mathematics, pp. 704-732 Henriques Varbaro M. (2014). Test, multiplier and invariant ideals. ArXiv e-prints, pp. arXiv:1407.4324
Start Year 2013
 
Title Macaulay2 library for computations in prime characteristic 
Description The PI authored several functions in the PosChar Macaulay2 library which computes several objects studied in this project. 
Type Of Technology Software 
Year Produced 2013 
Open Source License? Yes  
Impact Many researchers in my field use these functions to generate examples and to test conjectures. 
URL http://www.math.uiuc.edu/Macaulay2/