Euler class of a regular bundle functional

For local sections $v_1,\dots ,v_i$ of $E$, define contraction by $d(v_1\wedge \cdots \wedge v_i)={\sum }_{j=1}^i(-1{)}^{j-1}s(v_j)v_1\wedge \cdots \widehat{v_j}\cdots \wedge v_i$. This formula is alternating and compatible with restriction, so it defines a global map ${\mathrm{\Lambda }}^{i}E\to {\mathrm{\Lambda }}^{i-1}E$. In a local basis it is the differential in [F2]; hence $d^2=0$ locally and therefore globally. The quotient map ${\mathcal{O}}_X\to {\mathcal{O}}_X\mathrm{/}I={\mathcal{O}}_Z$ is an augmentation because its composite with $d:E\to {\mathcal{O}}_X$ is zero.

At $x\in Z$, choose a basis of the free stalk $E_x$. The stalk of the complex is the Koszul complex of the corresponding values of $s$; the regularity hypothesis and [F2] give homology only in degree zero, where it is ${\mathcal{O}}_{X,x}\mathrm{/}{I}_x$. At $x\mathrm{\notin }Z$, some value of $s_x$ is a unit. If $v\in E_x$ satisfies $s_x(v)=1$, exterior multiplication by $v$ is a contracting homotopy: the contraction formula gives $d(v\wedge w)+v\wedge d(w)=w$. The quotient stalk is also zero there. Thus the augmentation induces an isomorphism on every cohomology stalk and is a quasi-isomorphism. For $r=0$ this says ${\mathcal{O}}_X\to {\mathcal{O}}_X$ is an isomorphism.

By [F1], $K(s)$ is bounded with finite locally free terms, so it is perfect. Step 2.1 shows it is acyclic away from $Z$; by [F3] it is an object of ${\mathrm{Perf}}_Z(X)$. Its object class in the degree-zero K-group is therefore well-defined, and this is the stated $e_Z(s)$. The construction uses the underlying closed set for support; it does not identify supported K-theory with coherent-sheaf G-theory.

In the ambient perfect category, filter $K(s)$ by its subcomplexes consisting of the terms in degrees at least a fixed integer. Each successive quotient is a single term ${\mathrm{\Lambda }}^{i}E$ placed in degree $-i$, and the short exact sequence of complexes is degreewise split. Its associated cofiber sequence gives the corresponding additive relation in [F4]. A shift changes the sign of an object class: the cofiber sequence $A\to 0\to A[1]$ gives $[A[1]]=-[A]$. Iterating the finite filtration yields $[K(s)]={\sum }_{i=0}^r(-1{)}^i[{\mathrm{\Lambda }}^{i}E]$. Naturality of the object-class map sends $e_Z(s)$ to $[K(s)]$ under inclusion, proving the displayed formula.

A section $t$ gives the dual map $t^{\vee }:E^{\vee }\to {\mathcal{O}}_X$ by evaluation, and its values in a dual basis are the coordinates of $t$ in the original basis. The dual of a finite locally free module is finite locally free of the same rank. Applying steps 1.1 through 4.1 to $t^{\vee }$ under the stated regularity hypothesis gives the section formula with $E^{\vee }$.