Let \(X\) be a noetherian, integral, separated, regular scheme. (We say a scheme is regular if all of its local rings are regular local rings.) Recall the definition of the Grothendieck group \(K(X)\) from (II, Ex. 6.10).
We define similarly another group \(K_1(X)\) using locally free sheaves: it is the quotient of the free abelian group generated by all locally free (coherent) sheaves, by the subgroup generated by all expressions of the form \(\mce-\mce'-\mce''\), whenever \(0 \to \mce' \to \mce \to \mce'' \to 0\) is a short exact sequence of locally free sheaves.
Clearly there is a natural group homomorphism \(\varepsilon: K_1(X) \to K(X)\). Show that \(\varepsilon\) is an isomorphism as follows.
Given a coherent sheaf \(\mcf\), use (Ex. 6.8) to show that it has a locally free resolution \(\mce_\bullet \to \mcf \to 0\). Then use (6.11A) and (Ex. 6.5) to show that it has a finite locally free resolution \[ 0 \to \mce_n \to \cdots \to \mce_1 \to \mce_0 \to \mcf \to 0. \]
For each \(\mcf\), choose a finite locally free resolution \(\mce_\bullet \to \mcf \to 0\), and let \(\delta(\mcf)=\sum_i (-1)^i \gamma(\mce_i)\) in \(K_1(X)\). Show that \(\delta(\mcf)\) is independent of the resolution chosen, that it defines a homomorphism of \(K(X)\) to \(K_1(X)\), and finally, that it is an inverse to \(\varepsilon\).
Solution
Write \([F]\) for the class of a coherent sheaf in \(K(X)\) and \([E]_v\) for the class of a locally free coherent sheaf in the group \(K_1(X)\) defined in the statement. The map \(\varepsilon\) sends \([E]_v\) to \([E]\). For a bounded complex of locally free coherent sheaves, put \(\chi_v(E_\bullet)=\sum_i(-1)^i[E_i]_v\). Complexes in this proof are homologically indexed.
1. Every coherent sheaf has a finite locally free resolution. More precisely, every resolution by locally free coherent sheaves has a locally free kernel in every sufficiently large degree.
Regular local rings are factorial [@Har10a, Remark II.6.11.1A], so P-AGH368KLEIMAN applies to \(X\). Successively resolving coherent kernels gives \[ K_0=F,\qquad 0\longrightarrow K_{j+1}\longrightarrow E_j\longrightarrow K_j\longrightarrow0, \] with all \(E_j\) locally free of finite rank and all \(K_j\) coherent.
At every point \(x\), the module \(F_x\) has finite projective dimension over the regular local ring \(\OO_{X,x}\) [@Har10a, Proposition III.6.11A]. Dimension shifting, as in P-AGH365HOMDIM, makes \((K_j)_x\) finite free for all sufficiently large \(j\). For each \(j\), the set \(W_j\) where \(K_j\) is locally free is open: a basis at a stalk extends to a basis on a neighborhood because its coherent presentation has kernel and cokernel vanishing near that point. These opens are increasing. Indeed, if \((K_j)_x\) is free, the stalk sequence splits, and \((K_{j+1})_x\) is a finite projective module over a local ring, hence free. They cover \(X\) by the local projective-dimension bound.
Quasi-compactness gives a finite subcover, and the increasing property gives one \(N\) with \(W_N=X\). Then \(K_N\) is locally free and truncation yields \[ 0\longrightarrow K_N\longrightarrow E_{N-1}\longrightarrow\cdots \longrightarrow E_0\longrightarrow F\longrightarrow0. \] The same argument applies to any such resolution. It uses the finite local dimension at each point, not a uniform bound on \(\dim X\). This proves (a).
2. A bounded exact complex \(V_\bullet\) of locally free coherent sheaves has \(\chi_v(V_\bullet)=0\).
Shift the indices so that the nonzero terms lie in degrees \(0,\ldots,t\). Put \(Z_0=V_0\) and \(Z_i=\ker(V_i\to V_{i-1})\) for \(i\ge1\). Exactness gives short exact sequences \[ 0\longrightarrow Z_i\longrightarrow V_i\longrightarrow Z_{i-1}\longrightarrow0 \quad(1\le i\le t),\qquad Z_t=0. \] Starting with \(Z_0\), every \(Z_i\) is locally free: locally the surjection onto the locally free \(Z_{i-1}\) splits, so its kernel is locally a direct summand of a finite free module. The defining relations of \(K_1(X)\) give \([V_i]_v=[Z_i]_v+[Z_{i-1}]_v\). Alternating these identities cancels every \(Z_i\), proving the assertion.
3. Any two finite locally free resolutions \(E_\bullet\to F\) and \(D_\bullet\to F\) admit a finite locally free resolution \(R_\bullet\to F\) with chain maps to both inducing \(\id_F\).
Set \(Z_0^R=Z_0^E=Z_0^D=F\). Suppose \(R_i\) and the comparison maps have been constructed through degree \(j-1\), exactly resolving \(F\) up to that degree. Let \(Z_j^R\) be the kernel at \(R_{j-1}\), and define \(Z_j^E,Z_j^D\) in the same way using the augmented target resolutions. The comparison maps already constructed induce maps \(Z_j^R\to Z_j^E\) and \(Z_j^R\to Z_j^D\). For \(j=0\) these are both \(\id_F\). Form the coherent pullback \[ T_j=Z_j^R\times_{Z_j^E\oplus Z_j^D}(E_j\oplus D_j). \] The map \(T_j\to Z_j^R\) is surjective because both target resolutions are exact. Choose a locally free coherent sheaf \(R_j\) surjecting onto \(T_j\), using P-AGH368KLEIMAN. Its projection to \(Z_j^R\) defines the next resolution differential, while its projections to \(E_j,D_j\) extend the chain maps. This constructs a locally free resolution \(R_\bullet\to F\) and comparison maps in all degrees.
By step 1, choose \(N\) larger than the lengths of both target resolutions such that \(Z_N^R\) is locally free. Truncate \(R\) by placing \(Z_N^R\) in degree \(N\) and zero in higher degrees. The maps to the targets still define chain maps: the image of \(Z_N^R\) in either target’s degree \(N-1\) lies in its kernel there, which is the image of its zero degree-\(N\) term. Thus those images are zero. The resulting finite resolution has the required comparisons.
4. The value \(\chi_v(E_\bullet)\) for a finite locally free resolution of \(F\) is independent of the chosen resolution.
Take \(R\) and its comparison \(\alpha:R_\bullet\to E_\bullet\) from step 3, omitting the augmentations when taking their homology and mapping cone. Both complexes have homology only in degree zero, where \(\alpha\) induces \(\id_F\); their negative-degree terms are zero. The mapping cone is therefore a bounded exact complex, by the long exact homology sequence of a mapping cone. Its degree-\(i\) term is \(E_i\oplus R_{i-1}\), so step 2 gives \[ 0=\chi_v(\operatorname{Cone}\alpha)=\chi_v(E_\bullet)-\chi_v(R_\bullet). \] Applying the same argument to the comparison with \(D_\bullet\) gives \(\chi_v(D_\bullet)=\chi_v(R_\bullet)\). Thus the two values agree, as required. Define \(\delta(F)\) to be this common value.
5. For every short exact sequence \(0\to F'\to F\to F''\to0\) of coherent sheaves, one has \(\delta(F)=\delta(F')+\delta(F'')\).
First construct a degreewise exact triple of locally free resolutions. For any short exact sequence \(0\to A\xrightarrow{a}B\xrightarrow{b}C\to0\) of coherent sheaves, choose epimorphisms \(u:W\twoheadrightarrow A\) and \(v:V\twoheadrightarrow B\) with \(W,V\) locally free of finite rank. The split exact sequence \[ 0\longrightarrow W\longrightarrow W\oplus V\longrightarrow V\longrightarrow0 \] maps to the given sequence by \(u\), \((w,v')\mapsto a(u(w))+v(v')\), and \(b\circ v\). All three vertical maps are epimorphisms, and their kernels form a short exact sequence by the snake lemma. Apply this construction repeatedly to the kernel sequences, starting from the given sequence of \(F',F,F''\). It produces locally free resolutions \(E'_\bullet,E_\bullet,E''_\bullet\) and a degreewise split exact sequence of these complexes, with their augmentations to \(F',F,F''\).
Step 1 allows one common degree \(N\) in which the three resolution kernels are locally free. Their kernel sequence is still short exact. Truncation in that degree therefore gives a short exact sequence of finite locally free resolutions; at the terminal degree it need not split globally, but its terms are locally free. The defining relations of \(K_1(X)\) give \([E_i]_v=[E'_i]_v+[E''_i]_v\) in every degree. Alternating and using step 4 proves the desired additivity. Hence \(\delta\) descends to a homomorphism \(K(X)\to K_1(X)\).
6. The homomorphism \(\delta\) is inverse to \(\varepsilon\).
For a locally free coherent sheaf \(E\), its length-zero resolution gives \(\delta(E)=[E]_v\). Thus \(\delta\varepsilon\) fixes every generator of \(K_1(X)\). For any coherent \(F\), break a finite locally free resolution into its coherent kernel short exact sequences. The defining relations of the coherent Grothendieck group then give \[ [F]=\sum_i(-1)^i[E_i]=\varepsilon\delta([F]). \] Thus \(\varepsilon\delta\) fixes every generator of \(K(X)\) as well. Consequently \[ \boxed{K_1(X)\xrightarrow[\varepsilon]{\cong}K(X), \qquad \varepsilon^{-1}([F])=\sum_i(-1)^i[E_i]_v.} \]
7. Q.E.D.
Step 1 proves (a). Steps <1>2–<1>4 prove resolution independence, step 5 proves additivity, and step 6 proves the inverse identities, completing (b).