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 as opposed to a single vector bundle. It was originally introduced in Fulton's "intersection theory", as an algebraic counterpart of the similar construction in algebraic topology. The notion is used in particular in the Riemann–Roch-type theorem.
S. Bloch later generalized the notion in the context of arithmetic schemes (schemes over a Dedekind domain) for the purpose of giving #Bloch's conductor formula that computes the non-constancy of Euler characteristic of a degenerating family of algebraic varieties (in the mixed characteristic case).
Definitions.
Let "Y" be a pure-dimensional regular scheme of finite type over a field or discrete valuation ring and "X" a closed subscheme. Let formula_1 denote a complex of vector bundles on "Y"
that is exact on formula_3. The localized Chern class of this complex is a class in the bivariant Chow group of formula_4 defined as follows. Let formula_5 denote the tautological bundle of the Grassmann bundle formula_6 of rank formula_7 subbundles of formula_8. Let formula_9. Then the "i"-th localized Chern class formula_10 is defined by the formula:
where formula_12 is the projection and formula_13 is a cycle obtained from formula_14 by the so-called graph construction.
Example: localized Euler class.
Let formula_15 be as in #Definitions. If "S" is smooth over a field, then the localized Chern class coincides with the class
where, roughly, formula_17 is the section determined by the differential of "f" and (thus) formula_18 is the class of the singular locus of "f".