Exterior powers and determinant of an algebraic bundle

PDF p. 8: Determinant in Proposition 3.2; PDF p. 17: Exterior powers in Corollary 6.10

Let $X$ be a scheme, $E$ a locally free ${\mathcal{O}}_X$-module of constant finite rank $r\ge 0$, and $i\ge 0$ an integer. Here locally free of rank $r$ means that $X$ has an open cover on which $E$ is isomorphic to ${\mathcal{O}}_X^r$; such a sheaf is also called a rank-$r$ algebraic vector bundle.

For an open $U\subseteq X$, form the degree-$i$ part of the exterior algebra $T_{{\mathcal{O}}_X(U)}(E(U))\mathrm{/}(v\otimes v:v\in E(U))$, where $T$ is the tensor algebra and the denominator is the two-sided ideal generated by the displayed tensors. Restriction maps make these modules a presheaf. Define ${\mathrm{\Lambda }}^{i}E$ to be its sheafification. In particular, ${\mathrm{\Lambda }}^{0}E={\mathcal{O}}_X$ canonically. Define the determinant line bundle by $\det E={\mathrm{\Lambda }}^{r}E$; a line bundle, or invertible sheaf, is a locally free module of rank one. Its isomorphism class is an element of $\mathrm{Pic}(X)$, the group of isomorphism classes of line bundles under tensor product.

The sheaf ${\mathrm{\Lambda }}^{i}E$ is locally free of rank $\left(\genfrac{}{}{0px}{}{r}{i}\right)$, with the convention $\left(\genfrac{}{}{0px}{}{r}{i}\right)=0$ for $i>r$; hence it is zero for $i>r$. For every scheme morphism $f:Y\to X$, there is a canonical isomorphism $f^{\ast }({\mathrm{\Lambda }}^{i}E)\cong {\mathrm{\Lambda }}^i(f^{\ast }E)$, natural in $E$ and compatible with composition of morphisms. In particular $f^{\ast }(\det E)\cong \det (f^{\ast }E)$. Here $f^{\ast }$ is the usual pullback of modules, locally given by extension of scalars.

Locally free sheaves of finite rank

Exterior Algebra Of A Finite Free Module

Picard group of a scheme

1.1 Restriction and any module homomorphism preserve tensors of the form $v\otimes v$, so they induce maps on the graded quotients, preserving identities and composition. Thus the displayed degree-$i$ modules form a presheaf, whose sheafification exists by [F3]; the resulting construction is functorial in $E$. On a trivializing open $U$, [F2] identifies the presheaf on every open subset of $U$ with the finite free sheaf having the increasing wedges as its basis. These identifications commute with restrictions, so sheafification gives that same free sheaf on $U$. [F1, F2, F3]

2.1 There are $\left(\genfrac{}{}{0px}{}{r}{i}\right)$ increasing $i$-tuples, none when $i>r$, and exactly one when $i=0$ or $i=r$. Consequently the ranks and zero assertion follow locally and hence on $X$; the empty wedge identifies ${\mathrm{\Lambda }}^{0}E$ with ${\mathcal{O}}_X$. The top exterior power is locally free of rank one, including when $r=0$, and therefore defines the stated Picard class. [F2, F4, step 1.1]

3.1 On a trivializing open $U$, pullback sends the basis of $E$ to a basis of $f^{\ast }E$ on $f^{-1}U$. Send the pullback of each increasing wedge to the wedge of the pulled-back basis vectors. This is an isomorphism by [F2]. A change of basis acts on the exterior bases by the corresponding minors: expanding wedges multilinearly gives these coefficients, including over rings of arbitrary characteristic. Pulling back the coefficients gives exactly the minors of the pulled-back change-of-basis matrix. Thus the local isomorphisms agree on overlaps and glue. The same wedge formula shows naturality and compatibility with composition; taking $i=r$ gives the determinant assertion. [given, F1, F2, step 1.1, algebra] ∎