Green ring of Z/p^e Z equals e-fold tensor product of Green ring of Z/p Z by p-adic expansions, enabling explicit Hilbert-Kunz multiplicity for Fermat quadrics that decreases with characteristic and answers Yoshida's conjecture.
Irrational Hilbert-Kunz multiplicities
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abstract
We interpret Hilbert-Kunz theory of a graded ring of positive characteristic in terms of Frobenius asymptotic of cohomology of vector bundles on projective varieties. With this method we show that for almost all prime numbers there exist three-dimensional quartic hypersurface domains and modules of finite length with irrational Hilbert-Kunz multiplicity. From this we deduce that also the Hilbert-Kunz multiplicity of a local noetherian domain might be an irrational number.
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2026 1verdicts
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Hilbert-Kunz multiplicity of quadrics decreases
Green ring of Z/p^e Z equals e-fold tensor product of Green ring of Z/p Z by p-adic expansions, enabling explicit Hilbert-Kunz multiplicity for Fermat quadrics that decreases with characteristic and answers Yoshida's conjecture.