Localized Chern class

In algebraic geometry, a localized Chern class is a variant of a Chern class, that is defined for a chain complex of vector bundles. It was originally introduced in (Fulton 1998, Example 18.1.3.). Bloch then generalized the notion in the context of arithmetic schemes (roughly, schemes over a Dedekind domain) for the purpose of defining a localized self-intersection class.

Definitions

Let Y be a regular scheme of finite type over a discrete valuation ring and X closed subscheme. A localized Chern class is a class in the bivariant Chow group of $X\subset Y$ defined as follows. Let $E_{\bullet }$ denote a complex of vector bundles on Y exact off X:

$0=E_{n-1}\to E_{n}\to \dots \to E_{m}\to E_{m-1}=0$.

Let $\xi _{i}$ denote the tautological bundle of the Grassmann bundle $G_{i}$ of subbundles of $E_{i}\otimes E_{i-1}$ of rank equal to $\operatorname {rk} E_{i}$. Let $c(\xi )=\prod c(\xi _{i})^{(-1)^{i}}$. Then the i-th localized Chern class $c_{i,X}^{Y}(E_{\bullet })$ is defined by the formula:

$c_{i,X}^{Y}(E_{\bullet })\cap \alpha =\eta _{*}(c_{i}(\xi )\cap \gamma )$

where $\eta :G_{i}\gamma$ is a cycle on <math

References

• S. Bloch, “Cycles on arithmetic schemes and Euler characteristics of curves,” Algebraic geometry, Bowdoin, 1985, 421–450, Proc. Symp. Pure Math. 46, Part 2, Amer. Math. Soc., Providence, RI, 1987.
• Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323, section B.7
• K. Kato and T. Saito, “On the conductor formula of Bloch,” Publ. Math. IHES 100 (2005), 5-151.