Riemann-Roch for 0-cycles on a singular variety

Abstract

Let $X$ be a quasi-projective scheme of dimension $d$ over an infinite field $k$, such that $X$ is reduced after removing all components of dimension $<d$, and let $X^{\ast }\subset X$ be a closed subset of dimension $<d$ such that $X\setminus X^{\ast }$ is regular of dimension $d$. Using a modification ${\text{CH}}^d(X,X^{\ast })$ of the Chow group of 0-cycles on a singular variety defined by C. Weibel and the author in 1985, we construct a Chern class map $c_d:K_0(X)\to {\text{CH}}^d(X,X^{\ast })$, a cycle class map from ${\text{CH}}^d(X,X^{\ast })$ onto a subgroup $F^d{K}_0(X)$ of $K_0(X)$, and prove a Riemann-Roch theorem computing the two compositions of these two maps.