Hilbert-Kunz multiplicity of quadrics decreases

Abstract

In this paper we prove that the Green ring of $\mathbb{Z}/p^e\mathbb{Z}$ is the $e$-fold tensor product of the Green ring of $\mathbb{Z}/p\mathbb{Z}$, and this isomorphism is given by $p$-adic expansions of integers. As an application of this isomorphism, we compute the Hilbert-Kunz function and Hilbert-Kunz multiplicity of Fermat quadrics. Then we use Gelfand transform of the Green ring to give an analytical expression of this Hilbert-Kunz multiplicity, and prove that it decreases with its characteristic, thus giving a positive answer to a conjecture of Yoshida.