The symmetric signature

ABSTRACT

We define two related invariants for a d-dimensional local ring (R,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 ${Syz}_R^d\left(k\right)$ of the residue field and the module of Kähler differentials Ω_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.

KEYWORDS:

• ADE singularities
• F-signature
• free resolutions
• Kähler differentials

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13A35
• 13A50

Acknowledgements

We would like to thank Winfried Bruns, Ragnar–Olaf Buchweitz, Helena Fischbacher–Weitz, Lukas Katthän, and Yusuke Nakajima for their interest and many valuable comments. We thank the referee for showing us how to simplify the proof of Theorem 4.15 and pointing out Remark 4.6.