arXiv subjects
Publications and source records attributed to Kosuke Ohta.
Let R be a formal power series ring over a perfect field k of prime characteristic p, and let m be the maximal ideal of R. Suppose f is a non-zero element in m. In this paper, we introduce a function xi (x) associated with a hypersurface defined on the closed interval [0, 1] in R. The Hilbert-Kunz function and the F-signature of a hypersurface appear as the values of our function xi (x) on the interval s endpoints.
Considering the Grothendieck group modulo numerical equivalence, we obtain the finitely generated lattice $\overline{G_0(R)}$ for a Noetherian local ring $R$. Let $C_{CM}(R)$ be the cone in $\overline{G_0(R)}_{\Bbb R}$ spanned by cycles of maximal Cohen-Macaulay $R$-modules. We shall define the fundamental class $\overline{μ_R}$ of $R$ in $\overline{G_0(R)}_{\Bbb R}$, which is the limit of the Frobenius direct images (divided by their rank) $[{}^e R]/p^{de}$ in the case ${ch}(R) = p > 0$. The homological conjectures are deeply related to the problems whether $\overline{μ_R}$ is in the Cohen-Macaulay cone $C_{CM}(R)$ or the strictly nef cone $SN(R)$ defined below. In this paper, we shall prove that $\overline{μ_R}$ is in $C_{CM}(R)$ in the case where $R$ is FFRT or F-rational.