The symmetric signature

TL;DR

This paper introduces symmetric signature invariants for local rings, computed for specific singularities, revealing their connection to F-signature in positive characteristic.

Contribution

It defines and computes symmetric signature invariants for certain singularities, linking them to existing invariants like F-signature.

Findings

Symmetric signatures for ADE singularities are 1/|G|.

For cones over elliptic curves, the differential symmetric signature is 0.

Values match the F-signature in positive characteristic.

Abstract

We define two related invariants for a $d$-dimensional local ring $(R,\mathfrak{m},k)$ called syzygy and differential symmetric signature by looking at the maximal free splitting of reflexive symmetric powers of two modules: the top dimensional syzygy module $\mathrm{Syz}^d_R(k)$ of the residue field and the module of K\"ahler differentials $\Omega_{R/k}$ of $R$ over $k$. We compute these invariants for two-dimensional ADE singularities obtaining $1/|G|$, where $|G|$ is the order of the acting group, and for cones over elliptic curves obtaining $0$ for the differential symmetric signature. These values coincide with the F-signature of such rings in positive characteristic.