Vector-bundle K-theory equals coherent K-theory on regular quasi-projective schemes

Surjections from locally free sheaves. By [F2] there is an integer $\nu$ with $\mathcal{F}\otimes L^{\otimes \nu }$ globally generated. Since $X$ is quasi-compact (finite type over a field) and quasi-separated, finitely many global sections $s_0,\dots ,s_r$ generate $\mathcal{F}\otimes L^{\otimes \nu }$: the cokernel of the map ${⨁}_{i=0}^r{\mathcal{O}}_X\to \mathcal{F}\otimes L^{\otimes \nu }$ defined by the $s_i$ vanishes on a neighbourhood of each point for a suitable finite selection, and finitely many such neighbourhoods cover $X$. Untwisting by $L^{-\nu }$ gives a surjection $E_0:={⨁}_{i=0}^r{L}^{-\nu }\twoheadrightarrow \mathcal{F}$ from a finite locally free sheaf.

Bounded locally free resolutions. Define coherent subsheaves $K_0=\mathcal{F}$ and $K_{i+1}=\ker (E_i\to K_i)$ for the surjections produced by step 1.1, so that $0\to K_{i+1}\to E_i\to K_i\to 0$ is exact with $E_i$ finite locally free; all $K_i$ are coherent by [F1]. At a point $x$, the local ring $R={\mathcal{O}}_{X,x}$ is regular of dimension at most $n$ by [F1], so $R$ has global dimension at most $n$ by [F3]; hence the $n$-th syzygy of the stalk ${\mathcal{F}}_x$ is projective over $R$, and therefore free, because a finitely generated projective module over a local ring is free by [F3]. Since this holds at every point, $K_n$ is a coherent sheaf with free stalks, hence finite locally free by [F3]. Thus $\mathcal{F}$ admits the finite locally free resolution $0\to K_n\to E_{n-1}\to \cdots \to E_0\to \mathcal{F}\to 0$.

Independence of the resolution. Let $E_{\bullet },F_{\bullet }$ be bounded locally free resolutions of the same coherent sheaf $H$. Choose $N$ at least their lengths and $n$, padding by zero terms. Construct a resolution $G_{\bullet }$ with degreewise surjective maps to both. Start with $K_0(G)=K_0(E)=K_0(F)=H$. If $K_i(G)$ surjects onto $K_i(E),K_i(F)$, form the coherent sheaf $P_i=(E_i{\times }_{K_i(E)}{K}_i(G)){\times }_{K_i(G)}(F_i{\times }_{K_i(F)}{K}_i(G))$. Its projections onto $E_i,F_i,K_i(G)$ are surjective by local lifting through the given surjections. Cover $P_i$ by a finite locally free $G_i$ using step 1.1. Taking augmentation kernels gives surjections $K_{i+1}(G)\to K_{i+1}(E),K_{i+1}(F)$ by the kernel calculation in a diagram of short exact sequences. At degree $N$ use $G_N=K_N(G)$, locally free by the dimension bound in step 2.1, mapping onto the terminal syzygies $E_N,F_N$. The kernel complexes of $G_{\bullet }\to E_{\bullet },F_{\bullet }$ are bounded acyclic complexes of locally free sheaves: degreewise surjections between locally free sheaves split locally. Such a complex has zero alternating class, because starting at its lowest degree its successive cycle sheaves are locally free and the resulting short exact sequences telescope. The alternating classes of $E_{\bullet },F_{\bullet },G_{\bullet }$ therefore agree.

Additivity. Given $0\to H^{\mathrm{\prime }}\to H\to H^{\mathrm{\prime }\mathrm{\prime }}\to 0$, construct compatible resolutions term by term. Put $K_0^{\mathrm{\prime }}=H^{\mathrm{\prime }}$, $K_0=H$, $K_0^{\mathrm{\prime }\mathrm{\prime }}=H^{\mathrm{\prime }\mathrm{\prime }}$. Suppose $0\to K_i^{\mathrm{\prime }}\to K_i\to K_i^{\mathrm{\prime }\mathrm{\prime }}\to 0$ is exact. Choose locally free covers $C_i\to K_i^{\mathrm{\prime }\mathrm{\prime }}$ and $A_i\to K_i{\times }_{K_i^{\mathrm{\prime }\mathrm{\prime }}}{C}_i$ by step 1.1. The composite $A_i\to C_i$ surjects, so its kernel $B_i$ is locally free; the induced $B_i\to K_i^{\mathrm{\prime }}$ also surjects, by local lifting. Taking the kernels of the three augmentations yields $0\to K_{i+1}^{\mathrm{\prime }}\to K_{i+1}\to K_{i+1}^{\mathrm{\prime }\mathrm{\prime }}\to 0$. After $n$ stages all three syzygies are locally free by the dimension bound; take them as terminal terms. This produces a short exact sequence of bounded locally free resolutions of $H^{\mathrm{\prime }},H,H^{\mathrm{\prime }\mathrm{\prime }}$. Alternating classes add term by term, and independence in step 3.1 gives additivity for every choice of resolutions.

The comparison isomorphism. Both maps are well defined and additive: the natural map $K^0(X)\to K_0(X)$ sends the class of a finite locally free sheaf to its class in $K_0(X)$ and respects the exact-sequence relations of [F4], while the assignment sending the class of a coherent sheaf $H$ to the alternating class of any bounded locally free resolution of $H$ is well defined by step 3.1 and additive by step 4.1, hence descends to a homomorphism $K_0(X)\to K^0(X)$. The composite $K^0(X)\to K_0(X)\to K^0(X)$ is the identity because a finite locally free sheaf is its own length-zero resolution, and the composite $K_0(X)\to K^0(X)\to K_0(X)$ is the identity because the alternating class of a resolution of $H$ maps to $[H]$ in $K_0(X)$ by the telescoping exact-sequence relations. Therefore the natural map is an isomorphism, and in particular it applies to every smooth quasi-projective variety over a field, which is regular and quasi-projective by [F5].