Characteristic Classes

Abstract

In this section we consider a fixed commutative ring k. Let us recall that the inclusion i: Z → X of a closed subspace gives rise to two functors

$$ {\text{Sh(Z,k)}}\begin{array}{*{20}{c}} {\mathop{ \leftarrow }\limits^{{{{i}^{!}}}} } \\ {\mathop{ \to }\limits_{{{{i}_{*}}}} } \\ \end{array} {\text{Sh(X,k)}} $$

which are mutually adjoints

$${\text{Hom}({\text{i}_*}\text{E},\text{F})}={\text{Hom}(\text{F},{\text{i}^!}\text{F})}$$

(1.1)

The functor i^! is left exact and carries injectives into injectives. We shall study the derived functor R^Pi^!, p ∈ℤ.