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This is a dissertation in commutative algebra. The author introduces a new numerical invariant for local rings called symmetric signature. Given a local ring, the symmetric signature is defined by looking at the maximal free splitting of reflexive symmetric powers of the top-dimensional syzygy module of the residue field. The main motivation for this work is given by the F-signature, an important numerical invariant for local rings of positive characteristic which has been largely studied in the last decade. The dissertation contains a computation of the symmetric signature for two-dimensional ADE singularities and for cones over elliptic curves. In both cases, the values obtained coincide with the F-signature of such rings in positive characteristic. The thesis presents also a self-contained exposition of the Auslander correspondence, which is one of the main methods employed.

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Alessio Caminata received his Ph.D. in Mathematics in 2016 from the University of Osnabrück. After one year as a postdoc at the University of Neuchâtel, he moved to the University of Barcelona with a Marie Sklodowska-Curie individual fellowship.His research interests are in commutative algebra, algebraic geometry, and post-quantum cryptography.

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