Isomorphism classes of vector bundles

PDF p. 2: Paragraph before Theorem A; PDF p. 11: Proof of Proposition 5.5, notation for the unit

Assume the Axiom of Choice and the suitable universes used for the supplied perfect K-theory construction. Let $X$ be a quasi-compact, quasi-separated scheme and $n$ a nonnegative integer. A vector bundle on $X$ is a finite locally free ${\mathcal{O}}_X$-module: each point has an open neighbourhood on which it is isomorphic to a finite direct sum of copies of ${\mathcal{O}}_X$. It has rank $n$ if every point has such a neighbourhood with exactly $n$ copies. An isomorphism of vector bundles is an invertible ${\mathcal{O}}_X$-linear morphism. Write ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_n(X)$ for the set of their isomorphism classes and $[E{]}_{\mathrm{i}\mathrm{s}\mathrm{o}}$ for the class of a rank-$n$ bundle $E$ in that set.

The supplied nonconnective perfect K-theory spectrum is $K(X)=\mathbb{K}(\mathrm{Perf}(X))$, and $K_0(X)={\pi }_{0}K(X)$. A bundle $E$, regarded as a complex concentrated in degree zero, is perfect. Write $[E]\in K_0(X)$ for its object class in this spectral K-zero group. The unit of the ring $K_0(X)$ is denoted $1=[{\mathcal{O}}_X]$. Here $[E]$ denotes a K-zero class, not an isomorphism class in ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_n(X)$.

If $X=\mathrm{Spec}A$ for a commutative unital ring $A$, the supplied affine comparison identifies this $K_0(X)$ with the vector-bundle Grothendieck group: the abelian group generated by bundle isomorphism classes subject to $[E]=[E^{\mathrm{\prime }}]+[E^{\mathrm{\prime }\mathrm{\prime }}]$ for every short exact sequence $0\to E^{\mathrm{\prime }}\to E\to E^{\mathrm{\prime }\mathrm{\prime }}\to 0$ of vector bundles. Under the comparison the two meanings of $[E]$ agree. This affine assertion makes no identification of perfect K-theory with coherent-sheaf G-theory.

1.1 The open subsets of $X$ form a set. For each indexed open cover drawn from that set, rank-$n$ bundles trivialized on the cover are described up to isomorphism by transition matrices in ${\mathrm{GL}}_n({\mathcal{O}}_X(U\cap V))$ satisfying the identity, inverse and cocycle conditions. These matrices form a set; changing trivializations acts by a set of invertible matrices, and its equivalence classes form a set. Taking the union over covers and identifying data defining isomorphic sheaves gives a set of isomorphism classes. Every rank-$n$ bundle occurs by its local triviality, and an isomorphism preserves local rank. Thus ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_n(X)$ is well-defined, including when $X$ is empty. [F1, F5, construct]

2.1 By local freeness, $E[0]$ is locally a bounded complex of finite locally free sheaves, so it is an object of $\mathrm{Perf}(X)$. The object-class map in [F3] assigns it a well-defined element of ${\pi }_{0}K(X)$, invariant under isomorphism. The unit statement in [F3] identifies the multiplicative unit with the class of ${\mathcal{O}}_X[0]$. [F2, F3, step 1.1]

3.1 If $X=\mathrm{Spec}A$ as stated, the hypotheses of [F4] hold; its comparison identifies the spectral and vector-bundle classes on every generator. This establishes the stated notation and affine comparison. No corresponding identification with coherent G-theory is used. [F4, step 2.1] ∎