Vistoli, Angelo Equivariant Grothendieck groups and equivariant Chow groups. (English) Zbl 0784.14006 Classification of irregular varieties, minimal models and abelian varieties, Proc. Conf., Trento/Italy 1990, Lect. Notes Math. 1515, 112-133 (1992). This paper discusses an equivariant version of the Riemann-Roch theorem. Let \(k\) be a field and \(G\) an algebraic group over \(k\). Let \(X\) be a good \(G\)-scheme – that is, the action of \(G\) is proper and the stabilizer of any geometric point of \(X\) is finite and reduced. Then (proposition 2.1) there is a unique ring homomorphism \(ch:K_ 0 \bigl(X \| G \bigr) \to A \bigl( X \| G \bigr)\) compatible with pullbacks, and (2.2) a group homomorphism \(\tau_ X:K_ 0' \bigl( X \| G \bigr) \to\)CH\(\bigl( X \| G \bigr)\) which commutes with proper pushforwards and is onto. The homomorphisms \(ch\) and \(\tau\) satisfy several other properties which I will not list. The concepts mentioned above are all defined in this paper or in a previous paper of the author [Invent. Math. 97, No. 3, 613-670 (1989; Zbl 0694.14001)]. For example, \[ K_ 0'\bigl( X \| G \bigr)=\text{(Grothendieck group of } G\text{-equivariant coherent sheaves on} X) \otimes \mathbb{Q}, \] and CH\(\bigl( X \| G)\) is the Chow group with rational coefficients of the quotient stack \(X \| G\). The theorems mentioned above are not proved in this paper. Proofs will appear in a forthcoming paper of the author. Note that if the action of \(G\) is trivial then \(\tau_ X\) is an isomorphism [W. Fulton, “Intersection theory” (1984; Zbl 0541.14005); corollary 18.3.2]. If the \(G\)-action is not trivial then this need not be true. The kernel is described if \(X\) is smooth, and conjectured in more general cases. The paper concludes with a discussion of a totally split torus acting linearly on a finite dimensional vector space.For the entire collection see [Zbl 0744.00029]. Reviewer: L.G.Roberts (Kingston / Ontario) Cited in 1 ReviewCited in 4 Documents MSC: 14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry 14C05 Parametrization (Chow and Hilbert schemes) 18F30 Grothendieck groups (category-theoretic aspects) Keywords:equivariant Grothendieck groups; equivariant Chow groups; good scheme; Riemann-Roch theorem; Chow group; quotient stack; totally split torus Citations:Zbl 0694.14001; Zbl 0541.14005 PDFBibTeX XMLCite \textit{A. Vistoli}, Lect. Notes Math. 1515, 112--133 (1992; Zbl 0784.14006)