Let \(X\) be a variety over an algebraically closed field \(k\), and let \(\OO_X\) be the sheaf of regular functions on \(X\).
Let \(Y\) be a closed subset of \(X\). For each open set \(U \subseteq X\), let \(\mci_Y(U)\) be the ideal in the ring \(\OO_X(U)\) consisting of those regular functions which vanish at all points of \(Y \intersect U\). Show that the presheaf \(U \mapsto \mci_Y(U)\) is a sheaf. It is called the sheaf of ideals \(\mci_Y\) of \(Y\), and it is a subsheaf of the sheaf of rings \(\OO_X\).
If \(Y\) is a subvariety, show that the quotient sheaf \(\OO_X / \mci_Y\) is isomorphic to \(i_* \OO_Y\), where \(i: Y \to X\) is the inclusion and \(\OO_Y\) is the sheaf of regular functions on \(Y\).
Now let \(X = \PP^1\) and let \(Y\) be the union of two distinct points \(P, Q \in X\). Then with \(\mcf = i_* \OO_P \oplus i_* \OO_Q\) there is an exact sequence of sheaves on \(X\) \[ 0 \to \mci_Y \to \OO_X \to \mcf \to 0. \] Show however that the induced map on global sections \(\Gamma(X, \OO_X) \to \Gamma(X, \mcf)\) is not surjective. This shows that the global section functor \(\Gamma(X, \wait)\) is not exact.
Again let \(X = \PP^1\) and let \(\OO\) be the sheaf of regular functions. Let \(\mck\) be the constant sheaf on \(X\) associated to the function field \(K\) of \(X\). Show that there is a natural injection \(\OO \to \mck\). Show that the quotient sheaf \(\mck / \OO\) is isomorphic to the direct sum of sheaves \(\bigoplus_{P \in X} i_P(I_P)\), where \(I_P\) is the group \(K/\OO_P\) and \(i_P(I_P)\) denotes the skyscraper sheaf given by \(I_P\) at the point \(P\).
Finally, show that in the case of (d) the sequence \[ 0 \to \Gamma(X, \OO) \to \Gamma(X, \mck) \to \Gamma(X, \mck/\OO) \to 0 \] is exact.
Part (e) is an analogue of the first Cousin problem in several complex variables; see Gunning and Rossi.
Solution
1. For a closed subset \(Y\subseteq X\), the assignment\[ \mathcal I_Y(U) = \{f\in\mathcal O_X(U):f|_{Y\cap U}=0\text{ pointwise}\} \]is a subpresheaf of rings of \(\mathcal O_X\).
If \(V\subseteq U\) and \[ f\in\mathcal I_Y(U), \] then for every \[ P\in Y\cap V \subseteq Y\cap U \] one has \[ (f|_V)(P)=f(P)=0. \] Thus \[ f|_V\in\mathcal I_Y(V). \] The set \(\mathcal I_Y(U)\) is clearly an ideal of \(\mathcal O_X(U)\), since sums of functions vanishing on \(Y\cap U\) vanish there and multiplying by an arbitrary regular function preserves vanishing.
2. The presheaf \(\mathcal I_Y\) is a sheaf, hence a sheaf of ideals in \(\mathcal O_X\).
Let \[ U=\bigcup_iU_i \] and suppose compatible sections \[ f_i\in\mathcal I_Y(U_i) \] are given. Since \(\mathcal O_X\) is a sheaf, the \(f_i\) glue uniquely to \[ f\in\mathcal O_X(U). \]
For any point \[ P\in Y\cap U, \] choose \(i\) with \(P\in U_i\). Then \[ f(P)=f_i(P)=0. \] Thus \[ f\in\mathcal I_Y(U). \] Uniqueness is inherited from \(\mathcal O_X\). Hence \(\mathcal I_Y\) is a sheaf.
3. If \(Y\) is a subvariety and\[ i:Y\hookrightarrow X \]is the inclusion, restriction of regular functions gives a sheaf morphism\[ \rho:\mathcal O_X\longrightarrow i_*\mathcal O_Y \]whose kernel is \(\mathcal I_Y\).
For every open \(U\subseteq X\), restriction gives \[ \rho_U:\mathcal O_X(U) \longrightarrow \mathcal O_Y(U\cap Y). \] These maps commute with further restrictions, so they define a sheaf morphism.
By definition, \[ f\in\ker\rho_U \] exactly when \(f\) vanishes as a regular function on \(U\cap Y\), equivalently when it vanishes at every point of \(U\cap Y\). This is precisely \[ f\in\mathcal I_Y(U). \] Thus \[ \ker\rho=\mathcal I_Y. \]
4. The morphism \(\rho\) is surjective as a morphism of sheaves.
It suffices to check surjectivity on stalks.
If \(P\notin Y\), then \[ (i_*\mathcal O_Y)_P=0, \] so surjectivity is automatic.
Let \(P\in Y\). Choose an affine open neighborhood \[ P\in V=\operatorname{Spec}A\subseteq X \] such that \[ Y\cap V=V(I)=\operatorname{Spec}(A/I) \] for the ideal defining the subvariety locally. The map on stalks is then \[ A_{\mathfrak p} \longrightarrow (A/I)_{\mathfrak p/I}, \] which is surjective. Hence \(\rho\) is surjective at every stalk.
5. Therefore\[ \boxed{ \mathcal O_X/\mathcal I_Y \cong i_*\mathcal O_Y. } \]
Steps 3 and 4 give an exact sequence \[ 0\longrightarrow\mathcal I_Y \longrightarrow\mathcal O_X \xrightarrow{\rho}i_*\mathcal O_Y \longrightarrow0. \] The sheaf first-isomorphism theorem gives the displayed quotient isomorphism.
6. Let now\[ X=\mathbb P^1_k, \qquad Y=\{P,Q\} \]for distinct \(k\)-points \(P,Q\), and let\[ \mathcal F=i_{P*}\mathcal O_P\oplus i_{Q*}\mathcal O_Q. \]Then\[ 0\longrightarrow\mathcal I_Y \longrightarrow\mathcal O_X \longrightarrow\mathcal F \longrightarrow0 \]is exact.
The reduced subvariety \(Y\) is the disjoint union of the two points \(P\) and \(Q\), so \[ \mathcal O_Y \cong \mathcal O_P\oplus\mathcal O_Q \] as a sheaf on \(Y\). Therefore \[ i_*\mathcal O_Y \cong i_{P*}\mathcal O_P\oplus i_{Q*}\mathcal O_Q =\mathcal F. \] Apply the exact sequence from 5.
7. The induced map on global sections\[ \Gamma(X,\mathcal O_X) \longrightarrow \Gamma(X,\mathcal F) \]is not surjective.
Every global regular function on the projective integral curve \(\mathbb P^1_k\) is constant, so \[ \Gamma(X,\mathcal O_X)=k. \] On the other hand, \[ \Gamma(X,\mathcal F) \cong k\oplus k. \] The restriction map sends \[ c\longmapsto(c,c). \] Its image is the diagonal subspace of \(k\oplus k\), which does not contain, for example, \[ (1,0). \] Hence the map is not surjective.
8. Thus the global-section functor is left exact but not exact.
Applying global sections to the short exact sequence of 6 gives \[ 0\to \Gamma(X,\mathcal I_Y) \to \Gamma(X,\mathcal O_X) \to \Gamma(X,\mathcal F), \] which is exact at the first two terms because \(\Gamma\) is left exact. Step 7 shows that the last map need not be surjective, so \(\Gamma\) is not right exact and hence not exact.
9. Continue with \(X=\mathbb P^1_k\).Let\[ K=k(X) \]be its function field and let \(\mathcal K\) be the constant sheaf associated to the abelian group \(K\).There is a natural injection\[ \boxed{\mathcal O_X\hookrightarrow\mathcal K.} \]
Every nonempty open subset of the irreducible variety \(X\) is irreducible, so the constant sheaf \(\mathcal K\) has \[ \mathcal K(U)=K \] for every nonempty open \(U\).
A regular function on a nonempty open \(U\) is, by definition, a rational function regular at every point of \(U\). Thus there is a natural injective map \[ \mathcal O_X(U)\hookrightarrow K=\mathcal K(U). \] These inclusions commute with restriction, giving the sheaf morphism. It is injective on every open set, hence a monomorphism.
10. For every point \(P\in X\), the stalk of the quotient sheaf is\[ \boxed{ (\mathcal K/\mathcal O_X)_P \cong K/\mathcal O_{X,P} =:I_P. } \]
Sheaf stalks are exact. Taking the stalk at \(P\) of \[ 0\longrightarrow\mathcal O_X \longrightarrow\mathcal K \longrightarrow\mathcal K/\mathcal O_X \longrightarrow0 \] gives \[ 0\longrightarrow\mathcal O_{X,P} \longrightarrow\mathcal K_P \longrightarrow (\mathcal K/\mathcal O_X)_P \longrightarrow0. \] Since \(X\) is irreducible, \[ \mathcal K_P=K, \] so the quotient stalk is \(K/\mathcal O_{X,P}\).
11. A section of \(\mathcal K/\mathcal O_X\) over an open set \(U\) has nonzero germ at only finitely many points of \(U\).
The support of any section of a sheaf is closed in \(U\). We show more directly that it is finite.
Every point of \(U\) has a neighborhood on which the section is represented by a section of \(\mathcal K\), hence by one rational function in \(K\). Since \(X=\mathbb P^1_k\) is noetherian, the open set \(U\) is quasicompact, so finitely many such neighborhoods cover \(U\).
A rational function on the nonsingular projective curve \(\mathbb P^1\) has only finitely many poles. On a neighborhood where the quotient section is represented by a rational function \(f\), its germ is zero at every point where \(f\) is regular. Thus the support on that neighborhood is contained in the finite pole set of \(f\).
Taking the finite union over a finite cover shows that the support of the original section is finite.
12. Sending a section to its germs defines a sheaf morphism\[ \Phi: \mathcal K/\mathcal O_X \longrightarrow \bigoplus_{P\in X}i_P(I_P). \]
By Hartshorne II.1.11, direct sums of sheaves on the noetherian space \(X\) are computed sectionwise as filtered direct limits of finite direct sums. Hence \[ \left(\bigoplus_{P\in X}i_P(I_P)\right)(U) \cong \bigoplus_{P\in U}I_P, \] the group of finite-support tuples.
Given \[ s\in(\mathcal K/\mathcal O_X)(U), \] define \[ \Phi_U(s)=(s_P)_{P\in U}. \] By 11 this tuple has finite support, so it belongs to the direct sum. Taking germs is compatible with restriction, so the maps \(\Phi_U\) form a sheaf morphism.
13. The morphism \(\Phi\) is an isomorphism:\[ \boxed{ \mathcal K/\mathcal O_X \cong \bigoplus_{P\in X}i_P(I_P). } \]
It suffices to check the map on stalks.
At a point \(Q\in X\), 10 gives \[ (\mathcal K/\mathcal O_X)_Q=I_Q. \] On the right, the stalk of the \(Q\)th skyscraper is \(I_Q\), while every skyscraper supported at a different closed point has zero stalk at \(Q\). Thus \[ \left(\bigoplus_{P\in X}i_P(I_P)\right)_Q \cong I_Q. \] Under these identifications, \(\Phi_Q\) is the identity map on \(I_Q\). Therefore \(\Phi\) is an isomorphism at every stalk, hence an isomorphism of sheaves.
14. Consequently,\[ \Gamma(X,\mathcal K/\mathcal O_X) \cong \bigoplus_{P\in X}K/\mathcal O_{X,P}. \]
13. As noted in 12, the direct sum is computed sectionwise on the noetherian space \(X\), so its global sections are the direct sum of the skyscraper groups.
15. Every finite collection of principal parts on \(\mathbb P^1\) is the collection of principal parts of a single rational function.
Choose the affine coordinate \[ x \] on \[ \mathbb A^1=\mathbb P^1\setminus\{\infty\}. \] Let a finite tuple \[ (\xi_P)_P \in \bigoplus_{P\in X}K/\mathcal O_{X,P} \] be given.
At a finite point \[ P=a\in k, \] the local parameter is \[ x-a. \] Every class \[ \xi_a\in K/\mathcal O_{X,a} \] has a unique representative consisting of its finite polar part \[ p_a(x) = \sum_{m=1}^{N_a} \frac{c_{a,m}}{(x-a)^m}. \]
At infinity, take the local parameter \[ t=1/x. \] Every class \[ \xi_\infty\in K/\mathcal O_{X,\infty} \] has a unique finite polar representative \[ p_\infty(t) = \sum_{m=1}^{N_\infty}d_mt^{-m} = \sum_{m=1}^{N_\infty}d_mx^m. \]
Now define \[ f(x) = p_\infty(1/x) + \sum_{a\in k}p_a(x), \] where only the finitely many \(a\) in the support of the given tuple occur.
For a fixed finite point \(a\), every summand \(p_b\) with \(b\ne a\) is regular at \(a\), and the polynomial \(p_\infty(1/x)\) is also regular there. Thus the principal part of \(f\) at \(a\) is exactly \(p_a\).
At infinity, every term \[ \frac{1}{(x-a)^m} \] tends to zero in the parameter \(t=1/x\) and is regular there, while the polynomial part has precisely the prescribed principal part \(p_\infty\). Hence \(f\) realizes every prescribed component \(\xi_P\).
16. The map on global sections\[ \Gamma(X,\mathcal K) \longrightarrow \Gamma(X,\mathcal K/\mathcal O_X) \]is surjective.
Since \(X\) is irreducible, \[ \Gamma(X,\mathcal K)=K. \] By 14, a section of the quotient is a finite collection of principal parts. Step 15 constructs a rational function whose image in every \[ K/\mathcal O_{X,P} \] is that prescribed principal part. Hence every global section of the quotient is in the image of \(K\).
17. Therefore\[ \boxed{ 0 \longrightarrow \Gamma(X,\mathcal O_X) \longrightarrow \Gamma(X,\mathcal K) \longrightarrow \Gamma(X,\mathcal K/\mathcal O_X) \longrightarrow0 } \]is exact.
Left exactness of global sections applied to \[ 0\to\mathcal O_X\to\mathcal K\to\mathcal K/\mathcal O_X\to0 \] gives exactness at the first two terms. Step 16 gives surjectivity of the final map.
Equivalently, the kernel consists of rational functions with no pole anywhere on \(\mathbb P^1\), hence the global regular functions, which are the constants \(k\).
18. Q.E.D.
Steps <1>1–<1>2 prove (a), <1>3–<1>5 prove (b), <1>6–<1>8 prove (c), <1>9–<1>13 prove (d), and <1>14–<1>17 prove (e).