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Considering the Grothendieck group modulo numerical equivalence, we obtain the finitely generated lattice G 0 (R) for a Noetherian local ring R. Let C CM (R) be the cone in G 0 (R) R spanned by cycles of maximal Cohen-Macaulay R-modules. We shall define the fundamental class µ R of R in G 0 (R) 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 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 µ R is in C CM (R) in the case where R is FFRT or F-rational.

Let R = k[[X 1 , . . . , X n+1 ]] be a formal power series ring over a perfect field k of prime characteristic p > 0, and let m = (X 1 , . . . , X n+1 ) be the maximal ideal of R. Suppose 0 = f ∈ m. In this paper, we introduce a function ξ f (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 ξ f (x) on the interval's endpoints. The F-signature of the pair, denoted by s(R, f t ), was defined in [3]. Our function ξ f (x) is integrable, and the integral 1 t ξ f (x)dx is just s(R, f t ) for any t ∈ [0, 1]. 1991 Mathematics Subject Classification. 13F25.