Relative Mather discrepancy on arc spaces

Abstract

Given any generically étale morphism of varieties $f:X\to Y$, we define the relative Mather discrepancy function on the arc space $X_{\infty }$ of the domain and show that this function computes the dimension of the kernel of the differential map of the induced morphism on arc spaces $f_{\infty }:X_{\infty }\to Y_{\infty }$. We relate this result to the change-of-variable formula in motivic integration. We introduce the notion of $\hat{K}$-equivalence, which agrees with $K$-equivalence for smooth varieties, and prove that $\hat{K}$-equivalent varieties of arbitrary characteristic define the same class in the motivic ring.

v4: Final version, to appear in Michigan Math. J. 13 pages