Recall from (II, Ex. 6.10) the definition of the Grothendieck group \(K(X)\) of a noetherian scheme \(X\).
Let \(X\) be a projective scheme over a field \(k\), and let \(\mco_X(1)\) be a very ample invertible sheaf on \(X\). Show that there is a unique additive homomorphism \[ P: K(X) \to \QQ[z] \] such that for each coherent sheaf \(\mcf\) on \(X\), \(P(\gamma(\mcf))\) is the Hilbert polynomial of \(\mcf\) (Ex. 5.2).
Now let \(X=\PP_k^r\), where \(r\ge0\). For each \(i=0,1,\ldots,r\), let \(L_i\) be a linear space of dimension \(i\) in \(X\). Show that:
\(K(X)\) is the free abelian group generated by \(\ts{\gamma(\mco_{L_i})\st i=0,\ldots,r}\);
- the map \(P:K(X)\to\QQ[z]\) is injective.
Hint
Show that (b)(i) implies (b)(ii), and then prove them simultaneously by induction on \(r\), using (II, Ex. 6.10c).
Solution
Use the Grothendieck group of coherent sheaves defined in P-AGH2610KGROUP, and write \(\gamma_X(F)\) for the class of \(F\). A sheaf on a closed linear subspace is regarded on projective space by closed direct image.
1. Hilbert polynomials define the unique homomorphism in (a).
For any short exact sequence \(0\to F'\to F\to F''\to0\), twisting by the invertible sheaf \(\OO_X(n)\) is exact for every integer \(n\). Euler additivity P-AGH351EULERCHAR and the polynomial identity P-AGH352HILBPOLY give \[ P_F(n)=P_{F'}(n)+P_{F''}(n)\qquad(n\in\ZZ). \] The polynomials are therefore equal as polynomials: \(P_F=P_{F'}+P_{F''}\). The map from the free abelian group on coherent sheaves sending \(F\) to \(P_F\) annihilates every defining relation of \(K(X)\). It descends to the required additive homomorphism \(P\). The classes of coherent sheaves generate \(K(X)\), so its specified values determine it uniquely.
2. Every finite module over \(R=k[x_1,\ldots,x_r]\) has a finite resolution by finite free \(R\)-modules.
Take a finite presentation \(R^b\to R^a\to M\to0\) and set \(S=k[x_0,x_1,\ldots,x_r]\) with its standard grading. In each column of the presentation matrix, homogenize the entries to one common degree by inserting powers of \(x_0\). This gives a homogeneous map \(\bigoplus_j S(-d_j)\to S^a\) with finite graded cokernel \(N\) satisfying \[ N/(x_0-1)N\cong M. \] The equality follows by setting \(x_0=1\) in the presentation and using right exactness of tensor product.
The graded Hilbert syzygy argument gives a finite graded free resolution of \(N\) as follows. Construct a minimal graded free resolution \(F_\bullet\to N\) by successively taking homogeneous generators lifting a basis modulo \(\mathfrak m=(x_0,\ldots,x_r)\). The graded Nakayama argument shows that these generators generate: any nonzero quotient would have a lowest nonzero degree that could not lie in its own \(\mathfrak m\)-multiple. Noetherianness makes every kernel finite, so every \(F_i\) is a finite sum of shifts of \(S\). Minimality makes every differential matrix have entries in \(\mathfrak m\), and hence \(F_\bullet\otimes_S k\) has zero differentials.
The variables form an \(S\)-regular sequence, so their Koszul complex is a free resolution of \(k\) of length \(r+1\), as in D-DEFREGSQ [@Har10a, Proposition III.7.10A]. Computing \(\operatorname{Tor}^S_i(N,k)\) with this resolution gives zero for \(i>r+1\). Computing it with \(F_\bullet\) gives \(F_i\otimes_S k\), since the differentials there are zero. The equality of the two computations follows by tensoring the two free resolutions and taking homology in either order. Thus \(F_i\otimes_S k=0\) for \(i>r+1\), which forces the finite graded free module \(F_i\) itself to be zero. The resolution terminates.
Multiplication by \(x_0-1\) is injective on every graded module, in particular on \(N\): if a nonzero element has lowest nonzero homogeneous component \(v_d\), the degree-\(d\) component of \((x_0-1)v\) is \(-v_d\ne0\). The two-term free resolution of \(S/(x_0-1)=R\) therefore gives \(\operatorname{Tor}^S_i(N,R)=0\) for all \(i>0\). Tensoring the finite graded free resolution of \(N\) with \(R\) is consequently exact in positive degrees and has degree-zero homology \(M\). Its terms are finite free \(R\)-modules, proving the claim for arbitrary finite modules, not only graded ones.
3. Rank identifies \(K(\AA_k^r)\) with \(\ZZ\), generated by \(\gamma(\OO_{\AA^r})\).
2.Breaking it into short exact sequences gives\[ \gamma(\widetilde M)=\sum_i(-1)^i\operatorname{rank}(E_i)\,\gamma(\OO_{\AA^r}). \]Tensoring the resolution with the function field of \(R\) shows that the integer coefficient is the rank of \(M\).Rank is additive and sends \(\gamma(\OO_{\AA^r})\) to \(1\), by P-AGH2610KGROUP, part (b).Thus this generator has infinite order, and the rank map and \(n\mapsto n\gamma(\OO_{\AA^r})\) are inverse isomorphisms.This includes \(r=0\), for which \(R=k\).
4. For any linear space \(L_i\cong\PP_k^i\subseteq\PP_k^r\),\[ P\bigl(\gamma(\OO_{L_i})\bigr)=\binom{z+i}{i}. \]These polynomials are linearly independent over \(\QQ\).
The restriction of \(\OO_{\PP^r}(1)\) to \(L_i\) is its standard \(\OO(1)\). For sufficiently large \(n\), the Euler characteristic is the dimension of its global sections, which is \(\binom{n+i}{i}\) by the degree-\(n\) monomial basis [@Har10a, Theorems III.5.1 and III.5.2]. Closed direct image preserves cohomology [@Har10a, Lemma III.2.10], so the same computation is the Hilbert polynomial of \(\OO_{L_i}\) regarded on \(\PP^r\). Uniqueness of the polynomial proves the formula. Its degree is \(i\) with leading coefficient \(1/i!\). In a nonzero linear relation, the term with largest index having a nonzero coefficient would have strictly larger degree than all the others and could not cancel. Thus the polynomials are independent. In particular, once these classes generate \(K(\PP^r)\), this independence proves both their freeness and injectivity of \(P\), as indicated by the hint.
5. For every \(r\), a linear flag supplies a free basis of \(K(\PP_k^r)\), and \(P\) is injective.
Induct on \(r\). For \(r=0\), step 3 gives \(K(\PP_k^0)=K(\Spec k)=\ZZ\), generated by its structure sheaf; its polynomial is \(1\), so \(P\) is injective. For \(r>0\), take a coordinate hyperplane \(H\cong\PP_k^{r-1}\) and a flag of linear spaces \(F_0\subset\cdots\subset F_{r-1}=H\), with \(\dim F_i=i\). Set \(F_r=\PP_k^r\). The coherent localization sequence of P-AGH2610KGROUP, part (c), is \[ K(H)\longrightarrow K(\PP_k^r)\longrightarrow K(\AA_k^r)\longrightarrow0. \] By induction, the first group is generated by the classes of \(\OO_{F_i}\) for \(i<r\). By step 3, the last is generated by \(\OO_{\AA^r}\), and this generator lifts to \(\OO_{\PP^r}=\OO_{F_r}\). Exactness therefore shows that \(\gamma(\OO_{F_0}),\ldots,\gamma(\OO_{F_r})\) generate the middle group. Step 4 makes them linearly independent and makes \(P\) injective. This completes the simultaneous induction without assuming that the left map in the localization sequence is injective in advance.
6. The same basis assertion holds for the arbitrary linear spaces specified in (b), and\[ \boxed{K(\PP_k^r)=\bigoplus_{i=0}^r\ZZ\,\gamma(\OO_{L_i}), \qquad P\text{ is injective}.} \]
4.Injectivity from step 5 gives \(\gamma(\OO_{L_i})=\gamma(\OO_{F_i})\) in \(K(\PP^r)\).Thus the specified classes are exactly the basis classes already obtained, regardless of the relative positions of the \(L_i\).Both assertions in (b) follow.
7. Q.E.D.
Step 1 proves (a), and steps <1>2–<1>6 establish the free basis and injectivity required in (b).