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.
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The zero sheaf and the structure sheaf \({\mathscr O}_X\) are \({\mathscr O}_X\)-modules in a natural way.
There is an obvious notion of \({\mathscr O}_X\)-module presheaf. (In the definition of \({\mathscr O}_X\)-module, replace sheaf by presheaf.) Let \({\mathscr F}\) be a presheaf of \({\mathscr O}_X\)-modules. The multiplication by scalars is a bilinear map \({\mathscr O}_X\times {\mathscr F}\to {\mathscr F}\) of presheaves of abelian groups with certain further properties. Since sheafification is a functor and is compatible with finite products, we obtain a natural \({\mathscr O}_X\)-module structure on the sheafification of \({\mathscr F}\). This allows us to carry over most constructions we know for modules over rings to the setting of \({\mathscr O}_X\)-modules: First define a presheaf version, then, if necessary, pass to the sheafification.
Given an \({\mathscr O}_X\)-module \({\mathscr F}\), an \({\mathscr O}_X\)-submodule of \({\mathscr F}\) is an \({\mathscr O}_X\)-module \({\mathscr G}\) such that for every \(U \subseteq X\) open, \({\mathscr G}(U) \subseteq {\mathscr F}(U)\) is an \({\mathscr O}_X(U)\)-submodule. The quotient \({\mathscr F}/{\mathscr G}\) then is defined as the sheafification of the presheaf \(U\mapsto {\mathscr F}(U)/{\mathscr G}(U)\). It is an \({\mathscr O}_X\)-module in a natural way by (2).
We can construct direct sums \(\bigoplus \), direct products \(\prod \) (each with arbitrary index sets), tensor products of \({\mathscr O}_X\)-modules and limits and colimits of \({\mathscr O}_X\)-modules, by doing the corresponding presheaf construction and then sheafifying (if necessary; for products, hence also for direct sums, and limits, the corresponding presheaf is already a sheaf).
Similarly we can construct kernels, cokernels and image sheaves of homomorphism of \({\mathscr O}_X\)-modules. These are compatible with passing to the stalks, and we obtain a notion of exact sequence of \({\mathscr O}_X\)-modules (see below).
Restriction to open subsets: For \({\mathscr F}\) an \({\mathscr O}_X\)-modules and \(U \subseteq X\) open, the restriction \({\mathscr F}_{X|U}\) is naturally an \({\mathscr O}_{X|U}\)-module.
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”). For an \({\mathscr O}_X\)-module \({\mathscr F}\) we define the dual \({\mathscr F}^\vee = \mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits _{{\mathscr O}_X}({\mathscr F}, {\mathscr O}_X)\).
Tensor product \({\mathscr F}\otimes _{{\mathscr O}_X}{\mathscr G}\), the sheafification of the presheaf \(U\mapsto {\mathscr F}(U)\otimes _{{\mathscr O}_X(U)}{\mathscr G}(U)\). This is compatible with passing to stalks, i.e., we have
\[ ({\mathscr F}\otimes _{{\mathscr O}_X}{\mathscr G})_x = {\mathscr F}_{X, x}\otimes _{{\mathscr O}_{X, x}}{\mathscr G}_x, \]because tensor product commutes with colimits.
Note that taking products and the \(\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits \) sheaf is not compatible with passing to the stalks in general. But compare Proposition 2.19/Problem 7.)
A homomorphism \({\mathscr F}\to {\mathscr G}\) of \({\mathscr O}_X\)-modules is called injective, if for every \(x\in X\), the induced map \({\mathscr F}_x\to {\mathscr G}_x\) is injective.
A homomorphism \({\mathscr F}\to {\mathscr G}\) of \({\mathscr O}_X\)-modules is called surjective, if for every \(x\in X\), the induced map \({\mathscr F}_x\to {\mathscr G}_x\) is bijective.
A sequence \({\mathscr F}'\to {\mathscr F}\to {\mathscr F}''\) of homomorphisms of \({\mathscr O}_X\)-modules is exact at \({\mathscr F}\) if for all \(x\in X\) the sequence \({\mathscr F}'_x\to {\mathscr F}_x\to {\mathscr F}''_x\) is exact.
We have seen in Algebraic Geometry 1 that a homomorphism is an isomorphism if and only if it is injective and surjective; and also that a homomorphism \({\mathscr F}\to {\mathscr G}\) is injective if and only if for all open subsets \(U \subseteq X\), the map \({\mathscr F}(U)\to {\mathscr G}(U)\) is injective, but that the analogous equivalence for surjective does not hold (requiring that all \({\mathscr F}(U)\to {\mathscr G}(U)\) are surjective implies that \({\mathscr F}\to {\mathscr G}\) is surjective, but is much stronger).
Since taking kernels and images is compatible with passing to stalks (i.e., the stalks of the kernel are the kernels of the homomorphisms induced on stalks, etc.) a sequence \({\mathscr F}'\xrightarrow {f} {\mathscr F}\xrightarrow {g} {\mathscr F}''\) is exact at \({\mathscr F}\) if and only if \(\ker (g)=\mathop{\rm im}(f)\).
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The category of \({\mathscr O}_X\)-modules is an abelian category.
An important class of examples of \({\mathscr O}_X\)-modules are locally free \({\mathscr O}_X\)-modules.
Let \({\mathscr F}\) be an \({\mathscr O}_X\)-module on a ringed space \(X\). Call \({\mathscr F}\)
free, if it is isomorphic to \({\mathscr O}_X^{(I)}:=\bigoplus _{i\in I}{\mathscr O}_X\) 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 \(j\).
For \(X\ne \emptyset \) 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). It is determined by \({\mathscr F}\). 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). (In fact, there is a natural homomorphism \({\mathscr L}\otimes _{{\mathscr O}_X}{\mathscr L}^\vee \to {\mathscr O}_X\) (this holds for any \({\mathscr O}_X\)-module), and we can check locally on \(X\) that it is an isomorphism. But then we may assume that \({\mathscr L}={\mathscr O}_X\), and the assertion is clear.) Hence \(\otimes \) induces the structure of a commutative group on the set of isomorphism classes of invertible sheaves in \(X\) (with neutral element the isomorphism class of the structure sheaf, also called the trivial line bundle). The resulting group is called the Picard group of \(X\) and denoted by \(\operatorname{Pic}(X)\).
Let \(f\colon X\to Y\) be a morphism of ringed spaces, \({\mathscr F}\) an \({\mathscr O}_Y\)-module.
We define
We obtain functors \(f_*\), \(f^*\) between the categories of \({\mathscr O}_X\)-modules and \({\mathscr O}_Y\)-modules.
For \(x\in X\), we have \({(f^*{\mathscr F})}_x \cong {\mathscr F}_{f(x)} \otimes _{{\mathscr O}_{Y, f(x)}}{\mathscr O}_{X, x}\). Since \(f^{-1}\) is exact, this shows that \(f^*\) is right exact. Recall also that \(f_*\) is left exact (on abelian sheaves, hence on \({\mathscr O}_X\)-modules). It is also immediate to check that for morphisms \(X\xrightarrow {f}Y\xrightarrow {g} Z\) of ringed spaces, there is an equality \((g\circ f)_* = g_*\circ f_*\) of functors. With a bit of work one can check that likewise, there is an isomorphism \((g\circ f)^* \cong f^*\circ g^*\) of functors. This also follows formally from the adjunction between \(f^*\) and \(f_*\).
We know already that for homomorphisms of abelian sheaves we have \(\operatorname{Hom}(f^{-1}{\mathscr G}, {\mathscr F}) \cong \operatorname{Hom}({\mathscr G}, f_*{\mathscr F})\). This isomorphism restricts to an isomorphism \(\operatorname{Hom}_{f^{-1}{\mathscr O}_Y}(f^{-1}{\mathscr G}, {\mathscr F})\cong \operatorname{Hom}_{{\mathscr O}_Y}({\mathscr G}, f_*{\mathscr F})\). On the other hand, similarly as for modules of rings, there are isomorphisms
functorial in \({\mathscr F}\) and \({\mathscr G}\).
Quasi-coherent \({\mathscr O}_X\)-modules
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Proving that the presheaf on the basis of principal opens is in fact a sheaf works in exactly the same way as for the structure sheaf of \(\operatorname{Spec}(A)\). Similarly, the same proof as for the structure sheaf shows that the stalk of \(\widetilde{M}\) at a point \(\mathfrak p\in \operatorname{Spec}A\) is the localization \(M_{\mathfrak p} = \left\{ \frac{m}{s};\ m\in M,\ s\in A\setminus {\mathfrak p}\right\} \) (where fractions \(\frac{m}{s}\) are to be understood as equivalence classes in the usual way). We may also identify \(M_{{\mathfrak p}} = M\otimes _A A_{{\mathfrak p}}\).
Taking \(M=A\), we obtain \(\widetilde{A} = {\mathscr O}_{\operatorname{Spec}(A)}\).
It is clear that \(\Gamma (\operatorname{Spec}(A), \widetilde{\varphi }) = \varphi \). On the other hand, starting with \(\Phi \) and defining \(\varphi :=\Gamma (\operatorname{Spec}(A), \Phi )\), for any \(f\in A\) we consider the diagram
![\begin{tikzcd}
M\ar[r, "\varphi"]\ar[d] & N\ar[d]\\
M_f\ar[r] & N_f.
\end{tikzcd}](images/img-0004.svg)
This is commutative with lower horizontal homomorphism \(\varphi _f\) by functoriality of localization, but also with lower horizontal homomorphism \(\Phi (D(f))\), since \(\Phi \) is a morphism of sheaves. But the upper horizontal homomorphism determines the lower one uniquely, so it follows that \(\Phi (D(f)) = \varphi _f\), as desired.
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. More precisely:
Let \(M'\to M\to M''\) be a sequence of \(A\)-modules. The sequence is exact at \(M\) if and only if the sequence \(\widetilde{M'}\to \widetilde{M}\to \widetilde{M''}\) is exact at \(\widetilde{M}\).
Let \(f\colon M\to N\) be a homomorphism of \(A\)-modules, and let \(\widetilde{f}\) be the associated homomorphism of \({\mathscr O}_X\)-modules. Then \(\ker (f)^\sim \cong \ker (\widetilde{f})\), \(\mathop{\rm im}(f)^\sim \cong \mathop{\rm im}(\widetilde{f})\), \({\rm coker}(f)^\sim \cong {\rm coker}(\widetilde{f})\). In particular, \(f\) is injective (or surjective, respectively) if and only \(\widetilde{f}\) is injective (surjective).
The functor \(\widetilde{\cdot }\) commutes with taking direct sums and colimits.
For Part (1) use that the exactness of \(M'\to M\to M''\) is equivalent to the exactness of \(M'_{\mathfrak p}\to M_{\mathfrak p}\to M''_{\mathfrak p}\) for all \({\mathfrak p}\in \operatorname{Spec}(A)\) (in fact, localization is an exact functor; and to go back one reduces to the statement that \(M=0\) if and only if all the localizations \(M_{\mathfrak p}= 0\), \({\mathfrak p}\in \operatorname{Spec}(A)\).
Part (2) follows from (1). Part (3) follows from the compatibility of direct sums and colimits with localizations.
Recall our notation \({\mathscr F}^{(I)} := \bigoplus _{i\in I}{\mathscr F}\).
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 \(X\) by finitely many principal open subsets \(D(g_i)\) such that \({\mathscr F}_{|D(g_i)}\) is the cokernel of a homomorphism of free modules and hence (Proposition 2.12, Proposition 2.11) 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 \({\mathscr F}\), and that localization is exact and commutes with finite products, to conclude that (iv) holds for \(U\) itself.
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
\[ \Gamma (U, {\mathscr F}\otimes {\mathscr G}) = \Gamma (U, {\mathscr F}) \otimes \Gamma (U, {\mathscr G}). \]
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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.
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 7.
The point here is that both properties can be checked locally on \(X\). For affine schemes we have already shown this statement in Algebraic Geometry 1 (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 \(\mathfrak a\subseteq A\)). However, with the new tools we can give a direct proof, and as a corollary again get the statement for affine schemes.
Let \({\mathscr I}\) be a quasi-coherent ideal sheaf. Then \(Z:=\operatorname{Supp}({\mathscr O}_X/{\mathscr I})\) is a closed subset of \(X\) and we need to check that the locally ringed space \((Z, i^{-1}({\mathscr O}_X/{\mathscr I}))\) is a scheme (where \(i\colon Z\to X\) denotes the inclusion map). This can be done locally on \(X\), so we may assume that \(X = \operatorname{Spec}(A)\) is affine, and hence that \({\mathscr I}\) is of the form \(\widetilde{{\mathfrak a}}\) for some ideal \({\mathfrak a}\subseteq A\). But then we may identify \((Z, i^{-1}({\mathscr O}_X/{\mathscr I}))\) with \(\operatorname{Spec}(A/{\mathfrak a})\) which obviously is a scheme.
Conversely, let \(i\colon Z\to X\) be a closed immersion with corresponding ideal sheaf \({\mathscr I}= \operatorname{Ker}({\mathscr O}_X\to i_*{\mathscr O}_Z)\), and let us show that \({\mathscr I}\) is quasi-coherent. Again we may work locally on \(X\), i.e., we may assume that \(X=\operatorname{Spec}(A)\) is an affine scheme. Moreover, we may also assume that \(Z\) is affine. In fact, since \(Z\) is a scheme, we find an open cover \(X=\bigcup U_i\) of \(X\) such that all \(Z\cap U_i\) are affine. At this point we do not know whether the \(U_i\) are affine but if \(D(f) \subseteq U_I \subseteq X\) is a principal open in \(X\), then \(Z\cap D(f)\) is a principal open in \(Z\cap U_i\), so also affine. Thus passing to a refinement of the original cover, if necessary, we find an affine open cover \(X = \bigcup U_i\) such that all \(Z\cap U_i\) are affine.
So now consider the situation that \(X\) and \(Z\) are both affine. But then \(i\colon Z\to X\) is a morphism of affine schemes, and since \({\mathscr O}_Z\) is a quasi-coherent \({\mathscr O}_Z\)-module, the direct image \(i_*{\mathscr O}_Z\) is a quasi-coherent \({\mathscr O}_X\)-module by Proposition 2.17. So \({\mathscr I}= \ker ({\mathscr O}_X\to i_*{\mathscr O}_Z)\) is quasi-coherent, as well.
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.
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\(\widetilde{M}\) 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. 2.19. 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 \({\mathscr O}_X\)-modules 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.)