31.29 Weil divisors on normal schemes
First we discuss properties of reflexive modules.
Lemma 31.29.1. Let X be an integral locally Noetherian normal scheme. For \mathcal{F} and \mathcal{G} coherent reflexive \mathcal{O}_ X-modules the map
(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{O}_ X) \otimes _{\mathcal{O}_ X} \mathcal{G})^{**} \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})
is an isomorphism. The rule \mathcal{F}, \mathcal{G} \mapsto (\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G})^{**} defines an abelian group law on the set of isomorphism classes of rank 1 coherent reflexive \mathcal{O}_ X-modules.
Proof.
Although not strictly necessary, we recommend reading Remark 31.12.9 before proceeding with the proof. Choose an open subscheme j : U \to X such that every irreducible component of X \setminus U has codimension \geq 2 in X and such that j^*\mathcal{F} and j^*\mathcal{G} are finite locally free, see Lemma 31.12.13. The map
\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(j^*\mathcal{F}, \mathcal{O}_ U) \otimes _{\mathcal{O}_ U} j^*\mathcal{G} \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(j^*\mathcal{F}, j^*\mathcal{G})
is an isomorphism, because we may check it locally and it is clear when the modules are finite free. Observe that j^* applied to the displayed arrow of the lemma gives the arrow we've just shown is an isomorphism (small detail omitted). Since j^* defines an equivalence between coherent reflexive modules on U and coherent reflexive modules on X (by Lemma 31.12.12 and Serre's criterion Properties, Lemma 28.12.5), we conclude that the arrow of the lemma is an isomorphism too. If \mathcal{F} has rank 1, then j^*\mathcal{F} is an invertible \mathcal{O}_ U-module and the reflexive module \mathcal{F}^\vee = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{F}, \mathcal{O}_ X) restricts to its inverse. It follows in the same manner as before that (\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{F}^\vee )^{**} = \mathcal{O}_ X. In this way we see that we have inverses for the group law given in the statement of the lemma.
\square
Lemma 31.29.2. Let X be an integral locally Noetherian normal scheme. The group of rank 1 coherent reflexive \mathcal{O}_ X-modules is isomorphic to the Weil divisor class group \text{Cl}(X) of X.
Proof.
Let \mathcal{F} be a rank 1 coherent reflexive \mathcal{O}_ X-module. Choose an open U \subset X such that every irreducible component of X \setminus U has codimension \geq 2 in X and such that \mathcal{F}|_ U is invertible, see Lemma 31.12.13. Observe that \text{Cl}(U) = \text{Cl}(X) as the Weil divisor class group of X only depends on its field of rational functions and the points of codimension 1 and their local rings. Thus we can define the Weil divisor class of \mathcal{F} to be the Weil divisor class of \mathcal{F}|_ U in \text{Cl}(U). We omit the verification that this is independent of the choice of U.
Denote \text{Cl}'(X) the set of isomorphism classes of rank 1 coherent reflexive \mathcal{O}_ X-modules. The construction above gives a group homorphism
\text{Cl}'(X) \longrightarrow \text{Cl}(X)
because for any pair \mathcal{F}, \mathcal{G} of elements of \text{Cl}'(X) we can choose a U which works for both and the assignment (31.27.5.1) sending an invertible module to its Weil divisor class is a homorphism. If \mathcal{F} is in the kernel of this map, then we find that \mathcal{F}|_ U is trivial (Lemma 31.27.6) and hence \mathcal{F} is trivial too by Lemma 31.12.12 and Serre's criterion Properties, Lemma 28.12.5. To finish the proof it suffices to check the map is surjective.
Let D = \sum n_ Z Z be a Weil divisor on X. We claim that there is an open U \subset X such that every irreducible component of X \setminus U has codimension \geq 2 in X and such that Z|_ U is an effective Cartier divisor for n_ Z \not= 0. To prove the claim we may assume X is affine. Then we may assume D = n_1 Z_1 + \ldots + n_ r Z_ r is a finite sum with Z_1, \ldots , Z_ r pairwise distinct. After throwing out Z_ i \cap Z_ j for i \not= j we may assume Z_1, \ldots , Z_ r are pairwise disjoint. This reduces us to the case of a single prime divisor Z on X. As X is (R_1) by Properties, Lemma 28.12.5 the local ring \mathcal{O}_{X, \xi } at the generic point \xi of Z is a discrete valuation ring. Let f \in \mathcal{O}_{X, \xi } be a uniformizer. Let V \subset X be an open neighbourhood of \xi such that f is the image of an element f \in \mathcal{O}_ X(V). After shrinking V we may assume that Z \cap V = V(f) scheme theoretically, since this is true in the local ring at \xi . In this case taking
U = X \setminus (Z \setminus V) = (X \setminus Z) \cup V
gives the desired open, thereby proving the claim.
In order to show that the divisor class of D is in the image, we may write D = \sum _{n_ Z < 0} n_ Z Z - \sum _{n_ Z > 0} (-n_ Z) Z. By additivity of the map constructed above, we may and do assume n_ Z \leq 0 for all prime divisors Z (this step may be avoided if the reader so desires). Let U \subset X be as in the claim above. If U is quasi-compact, then we write D|_ U = -n_1 Z_1 - \ldots - n_ r Z_ r for pairwise distinct prime divisors Z_ i and n_ i > 0 and we consider the invertible \mathcal{O}_ U-module
\mathcal{L} = \mathcal{I}_1^{n_1} \ldots \mathcal{I}_ r^{n_ r} \subset \mathcal{O}_ U
where \mathcal{I}_ i is the ideal sheaf of Z_ i. This is invertible by our choice of U and Lemma 31.13.7. Also \text{div}_\mathcal {L}(1) = D|_ U. Since \mathcal{L} = \mathcal{F}|_ U for some rank 1 coherent reflexive \mathcal{O}_ X-module \mathcal{F} by Lemma 31.12.12 we find that D is in the image of our map.
If U is not quasi-compact, then we define \mathcal{L} \subset \mathcal{O}_ U locally by the displayed formula above. The reader shows that the construction glues and finishes the proof exactly as before. Details omitted.
\square
Lemma 31.29.3. Let X be an integral locally Noetherian normal scheme. Let \mathcal{F} be a rank 1 coherent reflexive \mathcal{O}_ X-module. Let s \in \Gamma (X, \mathcal{F}). Let
U = \{ x \in X \mid s : \mathcal{O}_{X, x} \to \mathcal{F}_ x \text{ is an isomorphism}\}
Then j : U \to X is an open subscheme of X and
j_*\mathcal{O}_ U = \mathop{\mathrm{colim}}\nolimits (\mathcal{O}_ X \xrightarrow {s} \mathcal{F} \xrightarrow {s} \mathcal{F}^{[2]} \xrightarrow {s} \mathcal{F}^{[3]} \xrightarrow {s} \ldots )
where \mathcal{F}^{[1]} = \mathcal{F} and inductively \mathcal{F}^{[n + 1]} = (\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{F}^{[n]})^{**}.
Proof.
The set U is open by Modules, Lemmas 17.9.4 and 17.12.6. Observe that j is quasi-compact by Properties, Lemma 28.5.3. To prove the final statement it suffices to show for every quasi-compact open W \subset X there is an isomorphism
\mathop{\mathrm{colim}}\nolimits \Gamma (W, \mathcal{F}^{[n]}) \longrightarrow \Gamma (U \cap W, \mathcal{O}_ U)
of \mathcal{O}_ X(W)-modules compatible with restriction maps. We will omit the verification of compatibilities. After replacing X by W and rewriting the above in terms of homs, we see that it suffices to construct an isomorphism
\mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{O}_ X, \mathcal{F}^{[n]}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{O}_ U, \mathcal{O}_ U)
Choose an open V \subset X such that every irreducible component of X \setminus V has codimension \geq 2 in X and such that \mathcal{F}|_ V is invertible, see Lemma 31.12.13. Then restriction defines an equivalence of categories between rank 1 coherent reflexive modules on X and V and between rank 1 coherent reflexive modules on U and V \cap U. See Lemma 31.12.12 and Serre's criterion Properties, Lemma 28.12.5. Thus it suffices to construct an isomorphism
\mathop{\mathrm{colim}}\nolimits \Gamma (V, (\mathcal{F}|_ V)^{\otimes n}) \longrightarrow \Gamma (V \cap U, \mathcal{O}_ U)
Since \mathcal{F}|_ V is invertible and since U \cap V is equal to the set of points where s|_ V generates this invertible module, this is a special case of Properties, Lemma 28.17.2 (there is an explicit formula for the map as well).
\square
Lemma 31.29.4. Assumptions and notation as in Lemma 31.29.3. If s is nonzero, then every irreducible component of X \setminus U has codimension 1 in X.
Proof.
Let \xi \in X be a generic point of an irreducible component Z of X \setminus U. After replacing X by an open neighbourhood of \xi we may assume that Z = X \setminus U is irreducible. Since s : \mathcal{O}_ U \to \mathcal{F}|_ U is an isomorphism, if the codimension of Z in X is \geq 2, then s : \mathcal{O}_ X \to \mathcal{F} is an isomorphism by Lemma 31.12.12 and Serre's criterion Properties, Lemma 28.12.5. This would mean that Z = \emptyset , a contradiction.
\square
Lemma 31.29.6. Assumptions and notation as in Lemma 31.29.3. The following are equivalent
the inclusion morphism j : U \to X is affine, and
for every x \in X \setminus U there is an n > 0 such that s^ n \in \mathfrak m_ x \mathcal{F}^{[n]}_ x.
Proof.
Assume (1). Then for x \in X \setminus U the inverse image U_ x of U under the canonical morphism f_ x : \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}) \to X is affine and does not contain x. Thus \mathfrak m_ x \Gamma (U_ x, \mathcal{O}_{U_ x}) is the unit ideal. In particular, we see that we can write
with f_ i \in \mathfrak m_ x and g_ i \in \Gamma (U_ x, \mathcal{O}_{U_ x}). By Lemma 31.29.3 we have \Gamma (U_ x, \mathcal{O}_{U_ x}) = \mathop{\mathrm{colim}}\nolimits \mathcal{F}^{[n]}_ x with transition maps given by multiplication by s. Hence for some n > 0 we have
for some t_ i = s^ ng_ i \in \mathcal{F}^{[n]}_ x. Thus (2) holds.
Conversely, assume that (2) holds. To prove j is affine is local on X, see Morphisms, Lemma 29.11.3. Thus we may and do assume that X is affine. Our goal is to show that U is affine. By Cohomology of Schemes, Lemma 30.17.8 it suffices to show that H^ p(U, \mathcal{O}_ U) = 0 for p > 0. Since H^ p(U, \mathcal{O}_ U) = H^0(X, R^ pj_*\mathcal{O}_ U) (Cohomology of Schemes, Lemma 30.4.6) and since R^ pj_*\mathcal{O}_ U is quasi-coherent (Cohomology of Schemes, Lemma 30.4.5) it is enough to show the stalk (R^ pj_*\mathcal{O}_ U)_ x at a point x \in X is zero. Consider the base change diagram
\xymatrix{ U_ x \ar[d]_{j_ x} \ar[r] & U \ar[d]^ j \\ \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}) \ar[r] & X }
By Cohomology of Schemes, Lemma 30.5.2 we have (R^ pj_*\mathcal{O}_ U)_ x = R^ pj_{x, *}\mathcal{O}_{U_ x}. Hence we may assume X is local with closed point x and we have to show U is affine (because this is equivalent to the desired vanishing by the reference given above). In particular d = \dim (X) is finite (Algebra, Proposition 10.60.9). If x \in U, then U = X and the result is clear. If d = 0 and x \not\in U, then U = \emptyset and the result is clear. Now assume d > 0 and x \not\in U. Since j_*\mathcal{O}_ U = \mathop{\mathrm{colim}}\nolimits \mathcal{F}^{[n]} our assumption means that we can write
for some n > 0, f_ i \in \mathfrak m_ x, and g_ i \in \mathcal{O}(U). By induction on d we know that D(f_ i) \cap U is affine for all i: going through the whole argument just given with X replaced by D(f_ i) we end up with Noetherian local rings whose dimension is strictly smaller than d. Hence U is affine by Properties, Lemma 28.27.3 as desired.
\square
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