## View abstract

### Session S07 - Differential operators in algebraic geometry and commutative algebra

Friday, July 16, 14:00 ~ 14:30 UTC-3

## Differential powers in mixed characteristic

### Eloísa Grifo

#### University of Nebraska – Lincoln, United States of America   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloake5c3d5d7f30f38b26c1307f891427bc3').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addye5c3d5d7f30f38b26c1307f891427bc3 = 'gr&#105;f&#111;' + '&#64;'; addye5c3d5d7f30f38b26c1307f891427bc3 = addye5c3d5d7f30f38b26c1307f891427bc3 + '&#117;nl' + '&#46;' + '&#101;d&#117;'; var addy_texte5c3d5d7f30f38b26c1307f891427bc3 = 'gr&#105;f&#111;' + '&#64;' + '&#117;nl' + '&#46;' + '&#101;d&#117;';document.getElementById('cloake5c3d5d7f30f38b26c1307f891427bc3').innerHTML += '<a ' + path + '\'' + prefix + ':' + addye5c3d5d7f30f38b26c1307f891427bc3 + '\'>'+addy_texte5c3d5d7f30f38b26c1307f891427bc3+'<\/a>';

The differential operators version of the Zariski--Nagata theorem says that over a polynomial ring over a perfect field, the differential powers of a radical ideal coincide with its symbolic powers. In mixed characteristic, differential powers are larger than symbolic powers, but there is a version of Zariski--Nagata that works when we mix in p-derivations.

In singular rings, differential powers can still be used as tools to prove results about symbolic powers, even though the two notions no longer coincide. We will discuss a uniform Chevalley theorem for direct summands of polynomial rings in mixed characteristic by introducing a new type of differential powers, which do not require the existence of a p-derivation on the direct summand.

Joint work with Alessandro De Stefani (University of Genova, Italy) and Jack Jeffries (University of Nebraska--Lincoln, United States of America).