2 \({\mathscr O}_X\)-modules
General references: [ GW1 ] Ch. 7, [ H ] II.5.
Definition and basic properties
Let \((X, {\mathscr O}_X)\) be a ringed space. An \({\mathscr O}_X\)-module is a sheaf \({\mathscr F}\) of abelian groups on \(X\) together with maps
giving each \({\mathscr F}(U)\) the structure of an \({\mathscr O}_X(U)\)-module, and which are compatible with the restriction maps for open subsets \(U'\subseteq U \subseteq X\).
An \({\mathscr O}_X\)-module homomorphism \({\mathscr F}\to {\mathscr G}\) between \({\mathscr O}_X\)-modules \({\mathscr F}\), \({\mathscr G}\) on \(X\) is a sheaf morphism \({\mathscr F}\to {\mathscr G}\) such that for all open subsets \(U\subseteq X\), the map \({\mathscr F}(U)\to {\mathscr G}(U)\) is a homomorphism of \({\mathscr O}_X(U)\)-modules. We denote the set of \({\mathscr O}_X\)-module homomorphisms from \({\mathscr F}\) to \({\mathscr G}\) by \(\operatorname{Hom}_{{\mathscr O}_X}({\mathscr F}, {\mathscr G})\); this is an \({\mathscr O}_X(X)\)-module (and in particular an abelian group).
We obtain the category (\({\mathscr O}_X\)-Mod) of \({\mathscr O}_X\)-modules.
\({\mathscr O}_X\),
submodules and quotients,
\(\oplus \), \(\prod \), \(-\otimes _{{\mathscr O}_X}-\), (filtered) colimits,
kernels, cokernels, image, exactness; these are compatible with passing to the stalks, and exactness can be checked on stalks,
restriction to open subsets: \({\mathscr F}_{X|U}\), \(U\subseteq X\) open,
The Hom sheaf \(\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits _{{\mathscr O}_X}({\mathscr F}, {\mathscr G})\), defined by \(U\mapsto \operatorname{Hom}_{{\mathscr O}_U}({\mathscr F}_{|U}, {\mathscr G}_{|U})\) (this is a sheaf, by “gluing of morphisms of sheaves”), duals: \({\mathscr F}^\vee = \mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits _{{\mathscr O}_X}({\mathscr F}, {\mathscr O}_X)\).
The principle for most of these constructions is the following: Use the corresponding construction for modules over a ring for sections on opens of \(X\), and then sheafify, if necessary. For products (and hence also for finite direct sums), kernels, and the Hom sheaf, the sheafification step is not required. For quotients, infinite direct sums, tensor products, colimits, cokernels and images, in general the presheaf obtained from the corresponding construction for modules is not a sheaf, so one has to sheafify.
The operations of taking kernels, cokernels, images, direct sums, tensor products, colimits are compatible with passing to stalks. (For products and the \(\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits \) sheaf, this does not hold in general. But compare Proposition 2.17/Problem 15.)
The category of \({\mathscr O}_X\)-modules is an abelian category.
Let \({\mathscr F}\) be an \({\mathscr O}_X\)-module on the ringed space \(X\). We call \({\mathscr F}\)
free, if it is isomorphic to \(\bigoplus _{i\in I}{\mathscr O}_X\) for some set \(I\),
locally free, if there exists an open covering \(X=\bigcup _j U_j\) of \(X\) such that \({\mathscr F}_{|U_j}\) is a free \({\mathscr O}_{U_j}\)-module for each \(j\).
The rank of a free \({\mathscr O}_X\)-module is the cardinality of \(I\) as above (we usually regard it in \(\mathbb {Z}\cup \{ \infty \} \), without making a distinction between infinite cardinals). The rank of a locally free \({\mathscr O}_X\)-module is a function \(X\to \mathbb {Z}\cup \{ \infty \} \) which is locally constant on \(X\) (i.e., on each connected component of \(X\), there is an integer giving the rank).
An invertible sheaf or line bundle on \(X\) is a locally free sheaf of rank \(1\).
For \({\mathscr L}\) invertible, there is a natural isomorphism \({\mathscr L}\otimes _{{\mathscr O}_X}{\mathscr L}^\vee \cong {\mathscr O}_X\) (whence the name), cf. Problem 14. Hence \(\otimes \) induces a group structure on the set of isomorphism classes of invertible sheaves in \(X\). The resulting group is called the Picard group of \(X\) and denoted by \(\operatorname{Pic}(X)\).
April 26,
2023
Let \(f\colon X\to Y\) be a morphism of ringed spaces, \({\mathscr F}\) an \({\mathscr O}_Y\)-module.
We define
For \(x\in X\), we have \({(f^*{\mathscr F})}_x \cong {\mathscr F}_{f(x)} \otimes _{{\mathscr O}_{Y, f(x)}}{\mathscr O}_{X, x}\).
We obtain functors \(f_*\), \(f^*\) between the categories of \({\mathscr O}_X\)-modules and \({\mathscr O}_Y\)-modules.
Quasi-coherent \({\mathscr O}_X\)-modules
In the situation of the definition, the stalk of \(\widetilde{M}\) at a point \(\mathfrak p\in \operatorname{Spec}A\) is the localization \(M_{\mathfrak p}\).
May 3,
2023
By applying the proposition to \(M=A\), we also see that for an \(A\)-module \(N\), \(\widetilde{N}\) is zero if and only if \(N\) is zero.
The construction \(M\mapsto \widetilde{M}\) is compatible with exactness, kernels, cokernels, images, direct sums, filtered inductive limits. (Cf. [ GW1 ] Prop. 7.14 for a more precise statement.)
For a morphism \(f\colon X\to Y\) of ringed spaces and a quasi-coherent \({\mathscr O}_Y\)-module \({\mathscr G}\), the pull-back \(f^*{\mathscr G}\) is a quasi-coherent \({\mathscr O}_X\)-module (since \(f^{-1}\) is exact and tensor product is a right exact functor). The direct image \(f_*\) preserves the property of quasi-coherence (only) under certain conditions.
Locally free \({\mathscr O}_X\)-modules are quasi-coherent.
Clearly, for a ring \(A\) and an \(A\)-module \(M\), \(\widetilde{M}\) is a quasi-coherent \({\mathscr O}_{\operatorname{Spec}A}\)-module. We will see below that the converse is true as well:
For a ringed space \(X\) and \(f\in \Gamma (X, {\mathscr O}_X)\), we write \(X_f := \{ x\in X;\ f_x \in {\mathscr O}_{X,x}^\times \} \), an open subset of \(X\). We obtain a homomorphism
for every \({\mathscr O}_X\)-module \({\mathscr F}\).
For every affine open \(\operatorname{Spec}A = U \subseteq X\), there exists an \(A\)-module \(M\) such that \({\mathscr F}_{|U} \cong \widetilde{M}\).
There exists a covering \(X = \bigcup _i U_i\) by affine open subschemes \(U_i = \operatorname{Spec}A_i\) and \(A_i\)-modules \(M_i\) such that \({\mathscr F}_{|U_i} \cong \widetilde{M_i}\) for all \(i\).
The \({\mathscr O}_X\)-module \({\mathscr F}\) is quasi-coherent.
For every affine open \(\operatorname{Spec}A = U \subseteq X\) and every \(f\in A\), the homomorphism \({\Gamma (U, {\mathscr F})}_f \to \Gamma (D(f), {\mathscr F})\) is an isomorphism.
Note that we can phrase (iv) equivalently as saying that the natural map \(\Gamma (U, {\mathscr F})^\sim \to {\mathscr F}_{|U}\) is an isomorphism.
The implications (iv) \(\Rightarrow \) (i) \(\Rightarrow \) (ii) \(\Rightarrow \) (iii) are relatively easy. To show (iii) \(\Rightarrow \) (iv), we may assume \(X=U=\operatorname{Spec}A\) and we can cover \(X\) be finitely many principal open subsets \(D(g_i)\) such that \({\mathscr F}_{|D(g_i)}\) is of the form \(\widetilde{M_i}\). In particular, (iv) holds for \({\mathscr F}_{|D(g_i)}\), and similarly for \({\mathscr F}_{D(g_ig_j)}\). Now use the sheaf property of \({\mathscr F}\) to conclude that (iv) holds for \(U\) itself.
May 8,
2023
The statements of the following corollary can be checked locally on \(X\), hence it is enough to show the corresponding claims for modules in the image of the \(\widetilde{\cdot }\) functor. For Part (3) use that tensor product is compatible with localization.
Kernels, cokernels, images of \({\mathscr O}_X\)-module homomorphisms between quasi-coherent \({\mathscr O}_X\)-modules are quasi-coherent.
Direct sums of quasi-coherent \({\mathscr O}_X\)-modules are quasi-coherent.
Let \({\mathscr F}\), \({\mathscr G}\) be quasi-coherent \({\mathscr O}_X\)-module. Then \({\mathscr F}\otimes _{{\mathscr O}_X}{\mathscr G}\) is quasi-coherent, and for every affine open \(U\subseteq X\) we have
\[ \Gamma (U, {\mathscr F}\otimes {\mathscr G}) = \Gamma (U, {\mathscr F}) \otimes \Gamma (U, {\mathscr G}). \]
Let \(N\) be an \(B\)-module, then \(f_*(\widetilde{N}) = \widetilde{N_{[A]}}\) where \(N_{[A]}\) is \(N\), considered as an \(A\)-module via \(\Gamma (f)\colon A\to B\).
Let \(M\) be an \(A\)-module, then \(f^*(\widetilde{M}) = \widetilde{M\otimes _{A}B}\).
The first part can easily be checked directly. For the second part, use that we already know that \(f^*\widetilde{M}\) is quasi-coherent, the Yoneda lemma and adjunction (or in other words, uniqueness of the left adjoint functor of \(f_*\)).
On an affine scheme, this coincides with the corresponding definitions in terms of modules (via \(M\mapsto \widetilde{M}\)). Note that every \({\mathscr O}_X\)-module of finite presentation is quasi-coherent. On a noetherian scheme, every quasi-coherent \({\mathscr O}_X\)-module of finite type is of finite presentation.
For all \(x \in X\) and for each \({\mathscr O}_X\)-module \({\mathscr G}\), the canonical homomorphism of \({\mathscr O}_{X,x}\)-modules
\[ {\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits _{{\mathscr O}_X}({\mathscr F},{\mathscr G})}_x \to \operatorname{Hom}_{{\mathscr O}_{X,x}}({\mathscr F}_x,{\mathscr G}_x) \]is bijective.
Let \({\mathscr F}\) and \({\mathscr G}\) be \({\mathscr O}_X\)-modules of finite presentation. Let \(x \in X\) be a point and let \(\theta \colon {\mathscr F}_x \overset {\sim }{\to }{\mathscr G}_x\) be an isomorphism of \({\mathscr O}_{X,x}\)-modules. Then there exists an open neighborhood \(U\) of \(x\) and an isomorphism \(u\colon {\mathscr F}{}_{\vert }{}_{U} \overset {\sim }{\to }{\mathscr G}{}_{\vert }{}_{U}\) of \({\mathscr O}_U\)-modules with \(u_x = \theta \).
Problem 15.
Problem 18.
May 10,
2023
The point here is that both properties can be checked locally on \(X\), and that for affine schemes we have already shown this statement (which amounts to saying that every closed subscheme of an affine scheme \(\operatorname{Spec}A\) has the form \(\operatorname{Spec}A/\mathfrak a\) for some ideal \(\mathfrak a\subseteq A\).
We hence obtain an inclusion-reversing bijection between the set of closed subschemes of a scheme \(X\) and the set of quasi-coherent ideal sheaves in \({\mathscr O}_X\), mapping
a quasi-coherent ideal sheaf \({\mathscr I}\) to \(Z:= (\operatorname{Supp}({\mathscr O}_X/{\mathscr I}), i^{-1}({\mathscr O}_X/{\mathscr I}))\), where \(i\colon \operatorname{Supp}({\mathscr O}_X/{\mathscr I}) \to X\) denotes the inclusion,
a closed subscheme \(Z\subseteq X\) to \(\operatorname{Ker}({\mathscr O}_X \to i_*{\mathscr O}_Z)\), where \(i\colon Z\to X\) denotes the inclusion morphism.
We denote the closed subscheme corresponding to a quasi-coherent ideal sheaf \({\mathscr I}\) by \(V({\mathscr I})\).
There is an obvious “commutative algebra way” of writing down, for an \(A\)-module \(M\), the condition that \(\widetilde{M}\) is locally free.
\(\widetilde{M}\) is a locally free \({\mathscr O}_{\operatorname{Spec}A}\)-module.
\(M\) is locally free, i.e., there exist \(f_1, \dots , f_n\in A\) generating the unit ideal such that for all \(i\), the \(A_{f_i}\)-module \(M_{f_i}\) is free.
For all \({\mathfrak p}\in \operatorname{Spec}A\), the \(A_{{\mathfrak p}}\)-module \(M_{{\mathfrak p}}\) is free.
The \(A\)-module \(M\) is flat.
We have the implications (i) \(\Leftrightarrow \) (ii) \(\Rightarrow \) (iii) \(\Rightarrow \) (iv).
If \(M\) is an \(A\)-module of finite presentation, then all the four properties are equivalent.
Part (1) is easy. Part (2) is more difficult. The implication (iii) \(\Rightarrow \) (ii), for finitely presented \(M\), follows from Prop. 2.17. See [ GW1 ] Prop. 7.40.
There are several ways to show (iv) \(\Rightarrow \) (iii). One can proceed in a fairly elementary fashion, using the “equational criterion of flatness”, see [ M2 ] Theorem 7.10, Theorem 7.12. Alternatively, using the Tor functor (the “left derived functor of the tensor product”), one can proceed as follows. We may assume that \(A\) is local with maximal ideal \(\mathfrak m\). Lift the elements of an \(A/\mathfrak m\)-basis of \(M/\mathfrak mM\) to \(M\). By the Lemma of Nakayama this induces a surjection \(A^n\to M\). Let \(K\) denote its kernel. We want to show that \(K=0\). Since \(M\) is of finite presentation, \(K\) is a finitely generated \(A\)-module ( [ M2 ] Theorem 2.6). Furthermore, the flatness of \(M\) implies that \(\mathop{\rm Tor}_1^A(A/\mathfrak m, M) = 0\). Therefore the short exact sequence
remains exact after tensoring \(-\otimes _A A/\mathfrak m\). Since by construction the homomorphism \(A^n\to M\) becomes an isomorphism after tensoring with the residue class field, this shows that \(K\otimes _A A/\mathfrak m\). Applying the Lemma of Nakayama again, we obtain that \(K=0\), as desired.
There is an obvious analogous theorem for \({\mathscr O}_X\)-module on a scheme \(X\), where we define
Let \(X\) be a scheme. An \({\mathscr O}_X\)-module \({\mathscr F}\) is called flat, if for all \(x\in X\) the stalk \({\mathscr F}_x\) is a flat \({\mathscr O}_{X,x}\)-module.
More generally, given an \({\mathscr O}_X\)-module \({\mathscr F}\) and a morphism \(f\colon X\to Y\) we say that \({\mathscr F}\) is \(f\)-flat or flat over \(Y\), if for all \(x\in X\) the stalk \({\mathscr F}_x\) is a flat \({\mathscr O}_{Y, f(x)}\)-module (via \(f^\sharp _x\colon {\mathscr O}_{Y, f(x)}\to {\mathscr O}_{X, x}\)).
If \(A\) is a domain, then every flat \(A\)-module \(M\) is torsion-free (i.e., multiplication by \(s\) is injective for all \(s\in A\setminus \{ 0\} )\). The converse holds only rarely; it does hold if \(A\) is a principal ideal domain and \(M\) is finitely generated.
Let \(A\) be a principal ideal domain. Then every finitely generated locally free (in the sense of condition (i\('\)) in the theorem) \(A\)-module is free. (Use the structure theorem for finitely generated modules over principal ideal domains.)
It is a difficult theorem (conjectured by Serre, proved independently by Quillen and Suslin) that every locally free sheaf of finite type on \(\mathbb {A}^n_k\), \(k\) a field, is free. The same statement holds even for \(k\) a discrete valuation ring.
It will not be relevant in the course, but in fact in the previous two items the hypothesis of finite type can be omitted. In fact, whenever \(R\) is a ring which is noetherian and such that \(\operatorname{Spec}R\) is connected, then every locally free \(R\)-module which is not finitely generated is free. One way to show this is to combine the paper [ Ba ] by H. Bass with the difficult theorem that the property of a module of being “projective” can be checked Zariski-locally on \(\operatorname{Spec}A\) ( [ Stacks ] 058B), which shows that all locally free \(R\)-modules, finitely generated or not, are projective. Maybe there is also a more direct way, without talking about projective modules?
On the other hand, even for an affine scheme \(X\), a locally free \({\mathscr O}_X\)-module is usually not a projective object in the category (\({\mathscr O}_X\)-Mod).
Let \(A\) be a noetherian unique factorization domain. Then every invertible sheaf on \(\operatorname{Spec}A\) is free.
See the answers to this question (mathoverflow.net/q/54356) for examples of non-free locally free modules over \(\operatorname{Spec}A\) for factorial (and even, in addition, regular) noetherian rings \(A\).
Let \(A\) be a domain, and let \(M\) be a locally free \(A\)-module of rank \(1\). Then \(M\) is isomorphic to a fractional ideal, i.e., to a finitely generated sub-\(A\)-module of \(K:=\operatorname{Frac}(A)\). (Cf. Problem 8 for a converse statement in the case that \(A\) is a Dedekind domain.)