Alessio Caminata: Counting free summands in Frobenius and symmetric powers

To study singularities in positive characteristic, one can look at the asymptotic splitting behaviour of the local ring corresponding to the singularity viewed as a module over itself via powers of the Frobenius homomorphism. This leads to the definition of F-signature, which has received a lot of attention in recent years.

Inspired by this approach, we introduce the differential symmetric signature of a local (or standard graded) ring, which is defined by looking at the asymptotic splitting behavior of the reflexive symmetric powers of the module of Kähler differentials of the ring.

We study this new invariant and compare it with the F-signature in several examples, such as quotient singularities, cones over elliptic curves, and normal hypersurfaces with an isolated singularity.

The talk is based on a joint work with H. Brenner.