Let \(X\) be a noetherian scheme. Define \(K(X)\) to be the quotient of the free abelian group generated by all coherent sheaves on \(X\), by the subgroup generated by all expressions \(\mcf - \mcf' - \mcf''\) whenever there is an exact sequence \(0 \to \mcf' \to \mcf \to \mcf'' \to 0\) of coherent sheaves on \(X\). If \(\mcf\) is a coherent sheaf, denote by \(\gamma(\mcf)\) its image in \(K(X)\).
If \(X = \AA^1_k\), then \(K(X) \cong \ZZ\).
If \(X\) is any integral scheme and \(\mcf\) a coherent sheaf, define the rank of \(\mcf\) to be \(\dim_K \mcf_\xi\), where \(\xi\) is the generic point of \(X\) and \(K = \OO_\xi\) is the function field of \(X\). Show that the rank function defines a surjective homomorphism \(\rank: K(X) \to \ZZ\).
If \(Y\) is a closed subscheme of \(X\), there is an exact sequence \[ K(Y) \to K(X) \to K(X - Y) \to 0, \] where the first map is extension by zero and the second is restriction.
For further information about \(K(X)\), and its applications to the generalized Riemann–Roch theorem, see Borel–Serre, Manin, and Hartshorne Appendix A.
Hint
For exactness in the middle of part (c), show that if \(\mcf\) is a coherent sheaf on \(X\) whose support is contained in \(Y\), then there is a finite filtration \(\mcf = \mcf_0 \supseteq \mcf_1 \supseteq \cdots \supseteq \mcf_n = 0\) such that each \(\mcf_i/\mcf_{i+1}\) is an \(\OO_Y\)-module. To show surjectivity on the right, use (Ex. 5.15).
Solution
Write \(\gamma_X(\mathcal F)\) for the class of a coherent sheaf in \(K(X)\). All schemes in the problem remain noetherian; the letter \(K\) in part (b) denotes the function field, not the group \(K(X)\).
1. On the affine line, rank gives \(\boxed{K(\AA_k^1)\cong\ZZ}\), with inverse \(n\mapsto n\gamma_{\AA^1}(\OO_{\AA^1})\).
Coherent sheaves on \(\Spec k[t]\) correspond exactly to finite \(k[t]\)-modules, and this correspondence is exact [@Har10a, Proposition II.5.4]. The structure theorem over the PID \(R=k[t]\) writes every such module as \[ M\cong R^r\oplus\bigoplus_{i=1}^sR/(f_i),\qquad f_i\ne0. \] The exact sequence \[ 0\longrightarrow R\xrightarrow{\cdot f_i}R\longrightarrow R/(f_i)\longrightarrow0 \] shows that each torsion summand has zero Grothendieck class. Thus \(\gamma(\widetilde M)=r\gamma(\OO_{\AA^1})\), so this one class generates the group. Tensoring with \(k(t)\) is exact and sends it to a one-dimensional vector space. Consequently \([M]\mapsto\dim_{k(t)}(M\otimes_Rk(t))\) is an additive inverse to the asserted map from \(\ZZ\); in particular, the generator has infinite order.
2. For any integral noetherian \(X\), rank induces a surjective homomorphism \(K(X)\to\ZZ\).
Taking the stalk at the generic point \(\xi\) is exact. For a coherent sheaf this stalk is finite-dimensional over the field \(\OO_{X,\xi}\). The dimension formula for a short exact sequence of finite-dimensional vector spaces therefore shows that rank respects every defining relation of \(K(X)\). Since \(\rank\OO_X=1\), every integer is the rank of an element \(n\gamma_X(\OO_X)\). This proves part (b) and also splits the rank homomorphism.
3. In part (c), let \(i:Y\hookrightarrow X\), \(j:U=X\setminus Y\hookrightarrow X\), and let \(\mathcal I\) be the ideal sheaf of \(Y\).If \(\mathcal F\) is coherent with support contained in \(Y\), then \(\mathcal I^N\mathcal F=0\) for some integer \(N\ge1\).
2.Choose generators \(a_1,\ldots,a_s\) of \(I\) and positive integers \(e_i\) with \(a_i^{e_i}M=0\).For \(N=1+\sum_i(e_i-1)\), every monomial of total degree \(N\) in these generators is divisible by some \(a_i^{e_i}\).Hence \(I^NM=0\).When \(I=0\), take \(N=1\).Taking the maximum of these exponents over a finite affine cover proves the global assertion.
The finite filtration \[ \mathcal F\supseteq\mathcal I\mathcal F\supseteq\cdots\supseteq\mathcal I^N\mathcal F=0 \] has coherent quotients annihilated by \(\mathcal I\). Each quotient is therefore \(i_*\mathcal G\) for a coherent \(\OO_Y\)-module \(\mathcal G\): on affines this is the identification of finite \(A\)-modules annihilated by \(I\) with finite \(A/I\)-modules, and these identifications glue.
4. Let \(H\subseteq K(X)\) be the subgroup generated by classes of coherent sheaves supported on \(Y\).Then \(\im(i_*:K(Y)\to K(X))=H\), and restriction induces a surjection\[ \overline{j^*}:K(X)/H\longrightarrow K(U). \]
Direct image under a closed immersion is exact and preserves coherence, so it induces \(i_*\) on Grothendieck groups. Every \(i_*\mathcal G\) is supported on \(Y\). Conversely, the filtration in step 3 expresses the class of every supported coherent sheaf as a sum of such direct-image classes. This proves \(\im i_*=H\).
5.Its class is therefore in the image of \(\overline{j^*}\), proving surjectivity.
5. Two coherent sheaves \(\mathcal F_1,\mathcal F_2\) on \(X\) whose restrictions to \(U\) are isomorphic have the same class in \(K(X)/H\).
Choose an isomorphism \(\alpha:\mathcal F_1|_U\to\mathcal F_2|_U\). Its graph is a coherent subsheaf of \((\mathcal F_1\oplus\mathcal F_2)|_U\). The subsheaf-extension assertion of P-AGH2515EXTCOH, step 5, extends it to a coherent subsheaf \[ \mathcal G\subseteq\mathcal F_1\oplus\mathcal F_2. \] Both projections \(p_i:\mathcal G\to\mathcal F_i\) are isomorphisms on \(U\). Their kernels and cokernels are coherent and supported on \(Y\). Factoring \(p_i\) through its image and using the two short exact sequences gives \[ \gamma_X(\mathcal G)-\gamma_X(\mathcal F_i) =\gamma_X(\ker p_i)-\gamma_X(\operatorname{coker}p_i)\in H. \] Thus both sheaves have the same class as \(\mathcal G\) modulo \(H\).
6. Extending coherent sheaves from \(U\) defines a homomorphism \(s:K(U)\to K(X)/H\) inverse to \(\overline{j^*}\).
For a coherent sheaf \(\mathcal E\) on \(U\), choose a coherent extension \(\mathcal F\) on \(X\) and assign the class \(\gamma_X(\mathcal F)+H\). Step 5 proves independence of the extension and of the chosen identification on \(U\).
5.The coherent quotient \(\mathcal F''=\mathcal F/\mathcal F'\) restricts to \(\mathcal E''\).The exact sequence on \(X\) then gives\[ \gamma_X(\mathcal F)+H =(\gamma_X(\mathcal F')+H)+(\gamma_X(\mathcal F'')+H). \]Thus the assignment respects every defining relation and induces \(s\).
Restricting an extension returns the original sheaf, so \(\overline{j^*}\circ s=\id\). Conversely, for a coherent sheaf \(\mathcal F\) on \(X\), its own class is one permitted extension of \(\mathcal F|_U\). Hence \(s\circ\overline{j^*}=\id\) on the generators of \(K(X)/H\), and therefore on the whole group.
7. Q.E.D.
Step 1 proves part (a), and step 2 proves part (b). Steps <1>4–<1>6 identify \(K(U)\) with \(K(X)/\im i_*\), proving the entire exact sequence in part (c).