Abstract
We introduce a measure of complexity for affine algebras and their finitely generated modules, in terms of the degrees of the polynomials used in their description. We then study how various cohomological operations and numerical invariants are uniformly bounded with respect to these complexities. We apply this to give first order characterisations of certain algebraic-geometric properties. This enables us to apply the Lefschetz Principle to transfer properties between various characteristics. As an application, we obtain the following version of the Zariski-Lipman Conjecture in positive characteristic: letR be the local ring of a pointP on a hypersurface over an algebraically closed fieldK such that the module ofK-invariant derivations onR is free, thenP is a non-singular point, provided the characteristic is larger than some bound only depending on the degree of the hypersurface.
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Schoutens, H. Bounds in cohomology. Isr. J. Math. 116, 125–169 (2000). https://doi.org/10.1007/BF02773216
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DOI: https://doi.org/10.1007/BF02773216