# Tight closure, Frobenius maps and Frobenius splittings

### Abstract

### 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.

## People |
## ORCID iD |

Mordechai Katzman (Principal Investigator) |

### Publications

*Two interesting examples of ?-modules in characteristic p >0*in Bulletin of the London Mathematical Society

*An algorithm for computing compatibly Frobenius split subvarieties*in Journal of Symbolic Computation

*An upper bound on the number of $F$-jumping coefficients of a principal ideal*in Proceedings of the American Mathematical Society

*Castelnuovo-Mumford regularity and the discreteness of $F$-jumping coefficients in graded rings*in Transactions of the American 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 |