The symmetric signature

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 ${\mathrm{Syz}}_R^d(k)$ of the residue field and the module of Kähler 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.

arXiv commentShortened the proofs of Proposition 2.8 and Theorem 3.15; modified Lemma 3.11; added Remark 3.6, Lemma 4.10, and Lemma 4.11; minor typos fixed; improved exposition; updated references