The symmetric signature

Holger Brenner; Alessio Caminata

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arXiv · 1602.07184

The symmetric signature

Holger Brenner·Alessio Caminata

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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.

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Holger Brenner, Alessio Caminata. 2016-02-23. The symmetric signature. https://arxiv.org/abs/1602.07184

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