# 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.

(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.

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.

## People |
## ORCID iD |

Mordechai Katzman (Principal Investigator) |

### Publications

Boix Alberto F.
(2013)

*An algorithm for producing F-pure ideals*in arXiv e-prints
Katzman M
(2014)

*Annihilators of Artinian modules compatible with a Frobenius map*in Journal of Symbolic Computation
Boix A
(2014)

*An algorithm for producing F-pure ideals*in Archiv der Mathematik
KATZMAN M
(2014)

*Rings of Frobenius operators*in Mathematical Proceedings of the Cambridge Philosophical Society
Katzman Mordechai
(2014)

*An extension of a theorem of Hartshorne*in arXiv e-prints
Henriques
(2014)

*Test, multiplier and invariant ideals*in arXiv e-prints
Katzman M
(2015)

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

*The support of local cohomology modules*in arXiv e-prints
Bonacho Dos Anjos Henriques I
(2016)

*Test, multiplier and invariant ideals*in Advances in Mathematics
Katzman M
(2016)

*Global parameter test ideals*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/ |