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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Differential symmetric signature in high dimension


Authors: Holger Brenner and Alessio Caminata
Journal: Proc. Amer. Math. Soc. 147 (2019), 4147-4159
MSC (2010): Primary 13A50, 13D40, 13N05
DOI: https://doi.org/10.1090/proc/14458
Published electronically: June 27, 2019
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Abstract: We study the differential symmetric signature, an invariant of rings of finite type over a field, introduced in a previous work by the authors in an attempt to find a characteristic-free analogue of the F-signature. We compute the differential symmetric signature for invariant rings $ k[x_1,\dots ,x_n]^G$, where $ G$ is a finite small subgroup of GL$ (n,k)$, and for hypersurface rings $ k[x_1,\dots ,x_n]/(f)$ of dimension $ \geq 3$ with an isolated singularity. In the first case, we obtain the value $ 1/\vert G\vert$, which coincides with the F-signature and generalizes a previous result of the authors for the two-dimensional case. In the second case, following an argument by Bruns, we obtain the value 0, providing an example of a ring where differential symmetric signature and F-signature are different.


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Additional Information

Holger Brenner
Affiliation: Institut für Mathematik, Universität Osnabrück, Albrechtstrasse 28a, 49076 Osnabrück, Germany
Email: holger.brenner@uni-osnabrueck.de

Alessio Caminata
Affiliation: Institut de Matemàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
Email: alessio.caminata@unine.ch

DOI: https://doi.org/10.1090/proc/14458
Keywords: F-signature, symmetric signature, quotient singularities, K\"ahler differentials
Received by editor(s): November 28, 2017
Received by editor(s) in revised form: October 3, 2018
Published electronically: June 27, 2019
Additional Notes: The second author was supported by European Union’s Horizon 2020 research and innovation programme under grant agreement No. 701807.
Communicated by: Jerzy Weyman
Article copyright: © Copyright 2019 American Mathematical Society