Karl Schwede (Utah)

PDL C-38

**Title**: Singularities in mixed characteristic via alterations

**Abstract**: Multiplier ideals and test ideals are ways to measure singularities in characteristic zero and p > 0 respectively. In characteristic zero, multiplier ideals are computed by a sufficiently large blowup by comparing the canonical module of the base and the resolution. In characteristic p > 0, test ideals were originally defined via Frobenius, but under moderate hypotheses, can be computed via a sufficiently large alteration again via canonical modules. In mixed characteristic (for example over the p-adic integers) we show that the various mixed analogs of multiplier/test ideals can be computed via a single sufficiently large alteration, at least when one builds in a small perturbation term. Besides unifying the three pictures, this has various applications. This is joint work with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Kevin Tucker, Joe Waldron and Jakub Witaszek.