The symmetric signature

TL;DR

This thesis introduces differential and syzygy symmetric signatures as invariants for local rings, linking them to F-signature in positive characteristic and exploring their computation for specific singularities.

Contribution

It defines and studies two new invariants, differential and syzygy symmetric signatures, and computes them for various classes of singularities, connecting them to existing invariants.

Findings

Symmetric signatures coincide with F-signature in positive characteristic.

Explicit computations for ADE and cyclic singularities.

Analysis of cones over elliptic curves.

Abstract

This is the author's Ph.D. thesis. We introduce two related invariants for local (and standard graded) rings called differential and syzygy symmetric signature. These are defined by looking at the maximal free splitting of the module of K\"ahler differentials and of the the top-dimensional syzygy module of the residue field respectively. We study and compute them for different classes of rings: two-dimensional ADE singularities, two-dimensional cyclic singularities, and cones over plane elliptic curves (for the differential symmetric signature). The values obtained coincide with the F-signature of such rings in positive characteristic. The thesis contains also a short survey on the Auslander correspondence for quotient singularities.