# Tight closure, Frobenius maps and Frobenius splittings

### Abstract

### Planned Impact

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

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

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

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