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 pure-dimensional regular scheme of finite type over a discrete valuation ring and X a closed subscheme. Let $E_{\bullet }$ denote a complex of vector bundles on Y

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

that is exact on $Y-X$. The localized Chern class of this complex is a class in the bivariant Chow group of $X\subset Y$ defined as follows. Let $\xi _{i}$ denote the tautological bundle of the Grassmann bundle $G_{i}$ of rank $\operatorname {rk} E_{i}$ subbundles of $E_{i}\otimes E_{i-1}$. Let $\xi =\prod (-1)^{i}\operatorname {pr} _{i}^{*}(\xi _{i})$. 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 $\eta :G_{n}\times _{Y}\dots \times _{Y}G_{m}\to X$ is the projection and $\gamma$ is a cycle on

Example: localized Euler class

Let $f:X\to S$ be as in #Definitions. If S is smooth over a field, then the localized Chern class coincides with the class

$(-1)^{\dim X}\mathbf {Z} (s_{f})$

where, roughly, $s_{f}$ is the section determined by the differential of f and (thus) $\mathbf {Z} (s_{f})$ is the class of the singular locus of f. Precisely, since the differential $df$

Bloch's conductor formula

See also: Grothendieck–Ogg–Shafarevich formula.

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.