Rank-three alternating exterior class

PDF p. 17: Proof of Corollary 6.10, definition of beta

Assume the Axiom of Choice and the suitable universes used in the vector-bundle classes definition. Let $X$ be a quasi-compact, quasi-separated scheme and let $E$ be a rank-three vector bundle on $X$: that is, a locally free ${\mathcal{O}}_X$-module locally isomorphic to ${\mathcal{O}}_X^3$. Write $K_0(X)={\pi }_0\mathbb{K}(\mathrm{Perf}(X))$ for the supplied degree-zero perfect K-theory group, and write $[V]$ for the class in it of a vector bundle $V$ viewed as a complex in degree zero.

The rank-three alternating exterior class of $E$ is $\beta (E)=[{\mathcal{O}}_X]-[E]+[{\mathrm{\Lambda }}^{2}E]-[\det E]\in K_0(X)\mathrm{.}$ Here ${\mathrm{\Lambda }}^{i}E$ is the $i$th exterior power and $\det E={\mathrm{\Lambda }}^{3}E$. Equivalently, in the functional convention, define ${\lambda }_{-1}(E):={\sum }_{i=0}^3(-1{)}^i[{\mathrm{\Lambda }}^{i}E]$; then $\beta (E)={\lambda }_{-1}(E)$. This notation applies the alternating exterior-power expression to the bundle $E$; no extension to arbitrary virtual classes is asserted.

Isomorphism classes of vector bundles

1.1 By [F2], ${\mathcal{O}}_X$, $E$, ${\mathrm{\Lambda }}^{2}E$ and $\det E$ are vector bundles. Their classes exist in the abelian group $K_0(X)$ by [F1], so their indicated sum and differences define an element of that group. An isomorphism $E\cong E^{\mathrm{\prime }}$ induces isomorphisms of the corresponding exterior powers by functoriality; consequently [F1] gives $\beta (E)=\beta (E^{\mathrm{\prime }})$. [given, F1, F2, F3]

2.1 Expanding the four terms in the defining sum for ${\lambda }_{-1}(E)$ and using ${\mathrm{\Lambda }}^{0}E\cong {\mathcal{O}}_X$ and ${\mathrm{\Lambda }}^{3}E=\det E$ gives precisely the displayed formula for $\beta (E)$. The higher exterior powers vanish, so the same expression is the full finite alternating exterior-power sum. [F2, step 1.1, algebra] ∎