Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0579.14013
Fulton, William; Gillet, Henri
Riemann-Roch for general algebraic varieties. (English)
[J] Bull. Soc. Math. Fr. 111, 287-300 (1983). ISSN 0037-9484

In this well-written article, the authors continue the program of removing hypotheses from the Riemann-Roch theorems. The starting point is the Grothendieck-Riemann-Roch theorem [{\it A. Borel} and {\it J.-P. Serre}, Bull. Soc. Math. Fr. 86(1958), 97-136 (1959; Zbl 0091.330)] proved for proper morphisms between smooth, quasiprojective varieties. Then {\it P. Baum}, {\it W. Fulton} and {\it R. D. MacPherson} [Publ. Math., Inst. Hautes Études Sci. 45, 101-145 (1975; Zbl 0332.14003)] eliminated the hypothesis that the schemes be smooth varieties, thus generalizing the Grothendieck-Riemann-Roch theorem to the category of quasiprojective schemes (over an arbitrary base field) and proper morphisms; the main point there was to consider homology, $A\sb*X\sb Q$, and to construct the natural (Riemann-Roch) transformation $\tau\sb X: K\sb 0X\to A\sb*X\sb Q$ such that $\tau\sb X({\cal O}\sb X)=Td(X)$, the Todd class of X. In this article, the authors eliminate the assumption of quasiprojectivity, proving a Riemann-Roch theorem in the category of algebraic schemes (over an arbitrary base field) and proper morphisms. Their method also works to generalize the topological K-theory Riemann- Roch theorem of {\it P. Baum}, {\it W. Fulton} and {\it R. D. MacPherson} [Acta Math. 143, 155-192 (1979; Zbl 0474.14004)] to algebraic ${\bbfC}$- schemes. The basic ingredients of the proof are the Riemann-Roch theorem for quasiprojective schemes, Chow's lemma allowing one to approximate any algebraic scheme X by a quasiprojective "envelope" X'$\to X$, and an exact sequence from algebraic K-theory (involving only $K\sb 0$ and $K\sb 1)$, which relates $\tau\sb{X'}$ to $\tau\sb X$. They also include a proof of an implicit assumption made by {\it H. Gillet} [Algebraic K- theory, Proc. Conf. Evanston 1980, Lect. Notes Math. 854, 141-167 (1981; Zbl 0478.14011)], concerning the equality of the Riemann-Roch transformations $\tau\sp S\sb X$ and $\tau\sp T\sb X$ from $K\sb 0X$ to $A\sp*X\sb Q$ for a given scheme X, regarded as a quasiprojective S- scheme or as a quasiprojective T-scheme.

MSC 2000:: *14C40 Riemann-Roch theorems
14C35 Appl. of methods of algebraic K-theory

Keywords: Grothendieck-Riemann-Roch theorem; algebraic schemes; algebraic ${\bbfC}$-schemes

Citations: Zbl 0091.330; Zbl 0332.14003; Zbl 0474.14004; Zbl 0478.14011

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]