2_5x Sheaves of Modules

II.5: Sheaves of Modules

II.5.1. #to_work

Let \(\left(X, {\mathcal{O}}_X\right)\) be a ringed space, and let \(\mathcal{E}\) be a locally free \({\mathcal{O}}_X\)-module of finite rank. We define the dual of \(\mathcal{E}\), denoted \({\mathcal{E}} {}^{ \vee }\), to be the sheaf \(\mathop{\mathcal{H}\! \mathit{om}}_{{\mathcal{O}}_X}({\mathcal{E}}, {\mathcal{O}}_X)\).

• Show that \(({\mathcal{E}} {}^{ \vee }) {}^{ \vee }\cong \mathcal{E}\).

• For any \({\mathcal{O}}_X\)-module \(\mathcal{F}\), \begin{align*} \mathop{\mathcal{H}\! \mathit{om}}_{{\mathcal{O}}_X}({\mathcal{E}}\otimes{\mathcal{F}}, {\mathcal{G}}) \cong \mathop{\mathcal{H}\! \mathit{om}}_{{\mathcal{O}}_X}({\mathcal{F}}, \mathop{\mathcal{H}\! \mathit{om}}_{{\mathcal{O}}_X}({\mathcal{E}},{\mathcal{G}}) ) .\end{align*}

• (Projection Formula). If \(f:\left(X, {\mathcal{O}}_X\right) \rightarrow\left(Y, {\mathcal{O}}_Y\right)\) is a morphism of ringed spaces, if \(\mathcal{F}\) is an \({\mathcal{O}}_X\)-module, and if \(\mathcal{E}\) is a locally free \({\mathcal{O}}_Y\)-module of finite rank, then there is a natural isomorphism \begin{align*} f_*\left(\mathcal{F} \otimes_{{\mathcal{O}}_X} f^* \mathcal{E}\right) \cong f_*(\mathcal{F}) \otimes_{{\mathcal{O}}_Y} \mathcal{E} .\end{align*}

II.5.2. #to_work

Let \(R\) be a discrete valuation ring with quotient field \(K\), and let \(X=\operatorname{Spec} R\).

• To give an \({\mathcal{O}}_X\)-module is equivalent to giving an \(R\)-module \(M\), a \(K\)-vector space \(L\), and a homomorphism \(\rho: M \otimes_R K \rightarrow L\).

• That \({\mathcal{O}}_X\)-module is quasi-coherent if and only if \(\rho\) is an isomorphism.

II.5.3. #to_work

Let \(X=\operatorname{Spec} A\) be an affine scheme. Show that the functors \(\sim\) and \(\Gamma\) are adjoint, in the following sense: for any \(A\)-module \(M\), and for any sheaf of \({\mathcal{O}}_X\)-modules \(\mathcal{F}\), there is a natural isomorphism \begin{align*} \operatorname{Hom}_A(M, \Gamma(X, \mathcal{F})) \cong \operatorname{Hom}_{{\mathcal{O}}_X}( \tilde{M}, \mathcal{F} ) \end{align*}

II.5.4. #to_work

Show that a sheaf of \({\mathcal{O}}_X\)-modules \(\mathcal{F}\) on a scheme \(X\) is quasi-coherent if and only if every point of \(X\) has a neighborhood \(U\), such that \(\left.\mathcal{F}\right|_U\) is isomorphic to a cokernel of a morphism of free sheaves on \(U\). If \(X\) is noetherian, then \(\mathcal{F}\) is coherent if and only if it is locally a cokernel of a morphism of free sheaves of finite rank. (These properties were originally the definition of quasi-coherent and coherent sheaves.)

II.5.5. #to_work

Let \(f: X \rightarrow Y\) be a morphism of schemes.

• Show by example that if \(\mathcal{F}\) is coherent on \(X\), then \(f_* \mathcal{F}\) need not be coherent on \(Y\), even if \(X\) and \(Y\) are varieties over a field \(k\).

• Show that a closed immersion is a finite morphism (\(\S 3\)).

• If \(f\) is a finite morphism of noetherian schemes, and if \(\mathcal{F}\) is coherent on \(X\), then \(f_* \mathcal{F}\) is coherent on \(Y\).

II.5.6. Support. #to_work

Recall the notions of support of a section of a sheaf, support of a sheaf, and subsheaf with supports from (Ex. 1.14) and (Ex. 1.20).

• Let \(A\) be a ring, let \(M\) be an \(A\)-module, let \(X=\operatorname{Spec} A\), and let \(\mathcal{F}=\tilde{M}\). For any \(m \in M=\Gamma(X, \mathcal{F})\), show that \(\mathop{\mathrm{supp}}m = V(\operatorname{Ann}m )\), where \(\operatorname{Ann}m\) is the annihilator of \(m=\{a \in A \mathrel{\Big|}a m=0\}\).

• Now suppose that \(A\) is noetherian, and \(M\) finitely generated. Show that \(\mathop{\mathrm{supp}}\mathcal{F}=V( \operatorname{Ann}M)\).

• The support of a coherent sheaf on a noetherian scheme is closed.

• For any ideal \(\mathfrak{a} \subseteq A\), we define a submodule \(\Gamma_{{\mathfrak{a}}}(M)\) of \(M\) by \begin{align*} \Gamma_{{\mathfrak{a}}}(M)= \left\{m \in M \mathrel{\Big|}\mathfrak{a}^n m=0 \text{ for some } n > 0 \right\} \end{align*} Assume that \(A\) is noetherian, and \(M\) any \(A\)-module. Show that \(\Gamma_{\mathfrak{a}}(M)^{\sim} \cong \mathcal{H}_Z^0(\mathcal{F})\), where \(Z=V(\mathfrak{a})\) and \(\mathcal{F}=\tilde{M}\). ¹

• Let \(X\) be a noetherian scheme, and let \(Z\) be a closed subset. If \(\mathcal{F}\) is a quasicoherent (respectively, coherent) \({\mathcal{O}}_X\)-module, then \(\mathcal{H}_Z^0(\mathcal{F})\) is also quasicoherent (respectively, coherent).

II.5.7. #to_work

Let \(X\) be a noetherian scheme, and let \(\mathcal{F}\) be a coherent sheaf.

• If the stalk \(\mathcal{F}_x\) is a free \({\mathcal{O}}_x\)-module for some point \(x \in X\), then there is a neighborhood \(U\) of \(x\) such that \(\left.\mathcal{F}\right|_U\) is free.

• \(\mathcal{F}\) is locally free if and only if its stalks \(\mathcal{F}_x\) are free \({\mathcal{O}}_x\)-modules for all \(x \in X\).

• \(\mathcal{F}\) is invertible (i.e., locally free of rank 1) if and only if there is a coherent sheaf \(\mathcal{G}\) such that \(\mathcal{F} \otimes \mathcal{G} \cong {\mathcal{O}}_X\). ²

II.5.8. #to_work

Again let \(X\) be a noetherian scheme, and \(\mathcal{F}\) a coherent sheaf on \(X\). We will consider the function \begin{align*} \varphi(x)=\operatorname{dim}_{k(x)} \mathcal{F}_x \otimes_{{\mathcal{O}}_x} k(x), \end{align*} where \(k(x)={\mathcal{O}}_x / m_x\) is the residue field at the point \(x\). Use Nakayama’s lemma to prove the following results.

• The function \(\varphi\) is upper semi-continuous, i.e., for any \(n \in \mathbf{Z}\), the set \(\{x \in X \mathrel{\Big|}\varphi(x) \geqslant n\}\) is closed.

• If \(\mathcal{F}\) is locally free, and \(X\) is connected, then \(\varphi\) is a constant function.

• Conversely, if \(X\) is reduced, and \(\varphi\) is constant, then \(\mathcal{F}\) is locally free.

II.5.9. #to_work

Let \(S\) be a graded ring, generated by \(S_1\) as an \(S_0\)-algebra, let \(M\) be a graded \(S\) module, and let \(X=\) Proj S.

• Show that there is a natural homomorphism \(\alpha: M \rightarrow \Gamma_*(\tilde{M})\).

• Assume now that \(S_0=A\) is a finitely generated \(k\)-algebra for some field \(k\), that \(S_1\) is a finitely generated \(A\)-module, and that \(M\) is a finitely generated \(S\)-module. Show that the map \(\alpha\) is an isomorphism in all large enough degrees, i.e., there is a \(d_0 \in \mathbf{Z}\) such that for all \(d \geqslant d_0, \alpha_d: M_d \rightarrow \Gamma(X, \tilde{M}(d))\) is an isomorphism. ³

• With the same hypotheses, we define an equivalence relation \(\approx\) on graded \(S\)-modules by saying \(M \approx M^{\prime}\) if there is an integer \(d\) such that \(M_{\geqslant d} \cong M_{\geqslant d}^{\prime}\). Here \(M_{\geqslant d}=\bigoplus_{n \geqslant d} M_n\). We will say that a graded \(S\)-module \(M\) is quasifinitely generated if it is equivalent to a finitely generated module.

  Now show that the functors \(\sim\) and \(\Gamma_*\) induce an equivalence of categories between the category of quasi-finitely generated graded \(S\)-modules modulo the equivalence relation \(\approx\), and the category of coherent \({\mathcal{O}}_X\)-modules.

II.5.10. #to_work

Let \(A\) be a ring, let \(S=A\left[x_0, \ldots, x_r\right]\) and let \(X=\operatorname{Proj} S\). We have seen that a homogeneous ideal \(I\) in \(S\) defines a closed subscheme of \(X\) (Ex. 3.12), and that conversely every closed subscheme of \(X\) arises in this way (5.16).

• For any homogeneous ideal \(I \subseteq S\), we define the saturation \(\mkern 1.5mu\overline{\mkern-1.5muI\mkern-1.5mu}\mkern 1.5mu\) of \(I\) to be \(\left\{s \in S \mathrel{\Big|}\right.\) for each \(i=0, \ldots, r\), there is an \(n\) such that \(\left.x_i^n s \in I\right\}\). We say that \(I\) is saturated if \(I=\mkern 1.5mu\overline{\mkern-1.5muI\mkern-1.5mu}\mkern 1.5mu\). Show that \(\mkern 1.5mu\overline{\mkern-1.5muI\mkern-1.5mu}\mkern 1.5mu\) is a homogeneous ideal of \(S\).

• Two homogeneous ideals \(I_1\) and \(I_2\) of \(S\) define the same closed subscheme of \(X\) if and only if they have the same saturation.

• If \(Y\) is any closed subscheme of \(X\), then the ideal \(\Gamma_*\left(\mathcal{I}_Y\right)\) is saturated. Hence it is the largest homogeneous ideal defining the subscheme \(Y\).

• There is a 1-1 correspondence between saturated ideals of \(S\) and closed subschemes of \(X\).

II.5.11. #to_work

Let \(S\) and \(T\) be two graded rings with \(S_0=T_0=A\). We define the Cartesian product \(S \underset{\scriptscriptstyle {A} }{\times} T\) to be the graded ring \(\bigoplus_{d \geqslant 0} S_d \otimes_A T_d\). If \(X=\operatorname{Proj} S\) and \(Y=\operatorname{Proj} T\), show that \(\operatorname{Proj}\left(S \times{ }_A T\right) \cong X \times{ }_A Y\), and show that the sheaf \({\mathcal{O}}(1)\) on \(\operatorname{Proj}\left(S \times{ }_A T\right)\) is isomorphic to the sheaf \(p_1^*\left({\mathcal{O}}_X(1)\right) \otimes p_2^*\left({\mathcal{O}}_Y(1)\right)\) on \(X \times Y\).

The Cartesian product of rings is related to the Segre embedding of projective spaces (I, Ex. 2.14) in the following way. If \(x_0, \ldots, x_r\) is a set of generators for \(S_1\) over \(A\), corresponding to a projective embedding \(X \hookrightarrow \mathbf{P}_A^r\), and if \(y_0, \ldots, y_s\) is a set of generators for \(T_1\), corresponding to a projective embedding \(Y \hookrightarrow P_A^s\), then \(\left\{x_i \otimes y_j\right\}\) is a set of generators for \(\left(S \times{ }_A T\right)_1\), and hence defines a projective embedding \(\mathop{\mathrm{Proj}}(S \underset{\scriptscriptstyle {A} }{\times} T) \hookrightarrow{\mathbf{P}}^N_A\), with \(N=r s+r+s\). This is just the image of \(X \times Y \subseteq \mathbf{P}^r \times \mathbf{P}^s\) in its Segre embedding.

II.5.12. #to_work

• Let \(X\) be a scheme over a scheme \(Y\), and let \(\mathcal{L}, \mathcal{M}\) be two very ample invertible sheaves on \(X\). Show that \(\mathcal{L} \otimes \mathcal{M}\) is also very ample. ⁴

• Let \(f: X \rightarrow Y\) and \(g: Y \rightarrow Z\) be two morphisms of schemes. Let \(\mathcal{L}\) be a very ample invertible sheaf on \(X\) relative to \(Y\), and let \(\mathcal{M}\) be a very ample invertible sheaf on \(Y\) relative to \(Z\). Show that \(\mathcal{L} \otimes f^* \mathcal{M}\) is a very ample invertible sheaf on \(X\) relative to \(Z\).

II.5.13. #to_work

Let \(S\) be a graded ring, generated by \(S_1\) as an \(S_0\)-algebra. For any integer \(d>0\), let \(S^{(d)}\) be the graded ring \(\bigoplus_{n \geqslant 0} S_n^{(d)}\) where \(S_n^{(d)}=S_{n d}\). Let \(X=\) Proj \(S\). Show that Proj \(S^{(d)} \cong X\), and that the sheaf \({\mathcal{O}}(1)\) on Proj \(S^{(d)}\) corresponds via this isomorphism to \({\mathcal{O}}_X(d)\).

This construction is related to the \(d\)-uple embedding (I, Ex. 2.12) in the following way. If \(x_0, \ldots, x_r\) is a set of generators for \(S_1\), corresponding to an embedding \(X \hookrightarrow \mathbf{P}_A^r\), then the set of monomials of degree \(d\) in the \(x_i\) is a set of generators for \(S_1^{(d)}=S_d\). These define a projective embedding of Proj \(S^{(d)}\) which is none other than the image of \(X\) under the \(d\)-uple embedding of \(\mathbf{P}_A^r\).

II.5.14. #to_work

Let \(A\) be a ring, and let \(X\) be a closed subscheme of \({\mathbf{P}}_A^r\). We define the homogeneous coordinate ring \(S(X)\) of \(X\) for the given embedding to be \(A\left[x_0, \ldots, x_r\right] / I\), where \(I\) is the ideal \(\Gamma_*\left(\mathcal{I}_X\right)\) constructed in the proof of \((5.16)\). Of course if \(A\) is a field and \(X\) a variety, this coincides with the definition given in (I, §2)! Recall that a scheme \(X\) is normal if its local rings are integrally closed domains.

A closed subscheme \(X \subseteq \mathbf{P}_A^r\) is projectively normal for the given embedding, if its homogeneous coordinate ring \(S(X)\) is an integrally closed domain (cf. (I, Ex. 3.18)).

Now assume that \(k\) is an algebraically closed field, and that \(X\) is a connected, normal closed subscheme of \(\mathbf{P}_k^r\). Show that for some \(d>0\), the \(d\)-uple embedding of \(X\) is projectively normal, as follows.

• Let \(S\) be the homogeneous coordinate ring of \(X\), and let \(S^{\prime}=\bigoplus_{n \geqslant 0} \Gamma\left(X, {\mathcal{O}}_X(n)\right)\). Show that \(S\) is a domain, and that \(S^{\prime}\) is its integral closure. ⁵

• Use (Ex. 5.9) to show that \(S_d=S_d^{\prime}\) for all sufficiently large \(d\).

• Show that \(S^{(d)}\) is integrally closed for sufficiently large \(d\), and hence conclude that the \(d\)-uple embedding of \(X\) is projectively normal.

• As a corollary of (a), show that a closed subscheme \(X \subseteq \mathbf{P}_A^r\) is projectively normal if and only if it is normal, and for every \(n \geqslant 0\) the natural map \(\Gamma\left(\mathbf{P}^r, {\mathcal{O}}_{\mathbf{P}^r}(n)\right) \rightarrow \Gamma\left(X, {\mathcal{O}}_X(n)\right)\) is surjective.

II.5.15. Extension of Coherent Sheaves. #to_work

We will prove the following theorem in several steps: Let \(X\) be a noetherian scheme, let \(U\) be an open subset, and let \(\mathcal{F}\) be a coherent sheaf on \(U\). Then there is a coherent sheaf \(\mathcal{F}^{\prime}\) on \(X\) such that \(\left.\mathcal{F}^{\prime}\right|_U \cong \mathcal{F}\).

• On a noetherian affine scheme, every quasi-coherent sheaf is the union of its coherent subsheaves. We say a sheaf \(\mathcal{F}\) is the union of its subsheaves \(\mathcal{F}\) if for every open set \(U\), the group \(\mathcal{F}(U)\) is the union of the subgroups \(\mathcal{F}_\alpha(U)\).

• Let \(X\) be an affine noetherian scheme, \(U\) an open subset, and \(\mathcal{F}\) coherent on \(U\). Then there exists a coherent sheaf \(\mathcal{F}^{\prime}\) on \(X\) with \(\left.\mathcal{F}^{\prime}\right|_U \cong \mathcal{F}\). ⁶

• With \(X, U, \mathcal{F}\) as in (b), suppose furthermore we are given a quasi-coherent sheaf \(\mathcal{G}\) on \(X\) such that \(\left.\mathcal{F} \subseteq \mathcal{G}\right|_U\). Show that we can find \(\mathcal{F}^{\prime}\) a coherent subsheaf of \(\mathcal{G}\), with \(\left.\mathcal{F}^{\prime}\right|_U \cong \mathcal{F}\). ⁷

• Now let \(X\) be any noetherian scheme, \(U\) an open subset, \(\mathcal{F}\) a coherent sheaf on \(U\), and \(\mathcal{G}\) a quasi-coherent sheaf on \(X\) such that \(\left.\mathcal{F} \subseteq \mathcal{G}\right|_U\). Show that there is a coherent subsheaf \(\mathcal{F}^{\prime} \subseteq \mathcal{G}\) on \(X\) with \(\left.\mathcal{F}^{\prime}\right|_U \cong \mathcal{F}\). Taking \(\mathcal{G}=i_* \mathcal{F}\) proves the result announced at the beginning. ⁸

• As an extra corollary, show that on a noetherian scheme, any quasi-coherent sheaf \(\mathcal{F}\) is the union of its coherent subsheaves. ⁹

II.5.16. Tensor Operations on Sheaves. #to_work

First we recall the definitions of various tensor operations on a module. Let \(A\) be a ring, and let \(M\) be an \(A\)-module.

• Let \(T^n(M)\) be the tensor product \(M \otimes \ldots \otimes M\) of \(M\) with itself \(n\) times, for \(n \geqslant 1\). For \(n=0\) we put \(T^0(M)=A\). Then \(T(M)=\bigoplus_{n \geqslant 0} T^n(M)\) is a (noncommutative) \(A\)-algebra, which we call the tensor algebra of \(M\).

• We define the symmetric algebra \(S(M)=\bigoplus_{n \geqslant 0} S^n(M)\) of \(M\) to be the quotient of \(T(M)\) by the two-sided ideal generated by all expressions \(x \otimes y-y \otimes x\), for all \(x, y \in M\). Then \(S(M)\) is a commutative \(A\)-algebra. Its component \(S^n(M)\) in degree \(n\) is called the \(n\)th symmetric product of \(M\). We denote the image of \(x \otimes y\) in \(S(M)\) by \(x y\), for any \(x, y \in M\). As an example, note that if \(M\) is a free \(A\)-module of rank \(r\), then \(S(M) \cong\) \(A\left[x_1, \ldots, x_r\right]\)

• We define the exterior algebra \(\bigwedge(M)=\bigoplus_{n \geqslant 0} \bigwedge^n(M)\) of \(M\) to be the quotient of \(T(M)\) by the two-sided ideal generated by all expressions \(x \otimes x\) for \(x \in M\). Note that this ideal contains all expressions of the form \(x \otimes y+y \otimes x\), so that \(\bigwedge(M)\) is a skew commutative graded \(A\)-algebra. This means that if \(u \in\) \(\bigwedge^r(M)\) and \(v \in \bigwedge^s(M)\), then \(u \wedge v=(-1)^{r s} v \wedge u\) (here we denote by \(\wedge\) the multiplication in this algebra; so the image of \(x \otimes y\) in \(\bigwedge^2(M)\) is denoted by \(x \wedge y\) ). The \(n\)th component \(\bigwedge^n(M)\) is called the \(n\)th exterior power of \(M\).

Now let \(\left(X, {\mathcal{O}}_X\right)\) be a ringed space, and let \(\mathcal{F}\) be a sheaf of \({\mathcal{O}}_X\)-modules. We define the tensor algebra, symmetric algebra, and exterior algebra of \(\mathcal{F}\) by taking the sheaves associated to the presheaf, which to each open ’set \(U\) assigns the corresponding tensor operation applied to \(\mathcal{F}(U)\) as an \({\mathcal{O}}_X(U)\)-module. The results are \({\mathcal{O}}_x\)-algebras, and their components in each degree are \({\mathcal{O}}_x\)-modules.

• Suppose that \(\mathcal{F}\) is locally free of rank \(n\). Then \(T^r(\mathcal{F}), S^r(\mathcal{F})\), and \(\bigwedge^r(\mathcal{F})\) are also locally free, of ranks \(n^r,\left(\begin{array}{c}n+r-1 \\ n-1\end{array}\right)\), and \(\left(\begin{array}{c}r \\ r\end{array}\right)\) respectively.

• Again let \(\mathcal{F}\) be locally free of rank \(n\). Then the multiplication map \(\bigwedge^r \mathcal{F} \otimes\) \(\bigwedge^{n-r} \mathcal{F} \rightarrow \bigwedge^n \cdot \mathcal{F}\) is a perfect pairing for any \(r\), i.c., it induces an isomorphism of \(\bigwedge^r \mathcal{F}\) with \(\left(\bigwedge^{n-r} \mathcal{F}\right)^{\ulcorner} \otimes \bigwedge^n \mathcal{F}\). As a special case, note if \(\mathcal{F}\) has rank 2 , then \(\mathcal{F} \cong \mathcal{F}^{\top} \otimes \bigwedge^2 \mathcal{F}\).

• Let \(0 \rightarrow \mathcal{F}^{\prime} \rightarrow \mathcal{F} \rightarrow \mathcal{F}^{\prime \prime} \rightarrow 0\) be an exact sequence of locally free sheaves. Then for any \(r\) there is a finite filtration of \(S^r(\mathcal{F})\), \begin{align*} S^r(\mathcal{F})=F^0 \supseteq F^1 \supseteq \ldots \supseteq F^r \supseteq F^{r+1}=0 \end{align*} with quotients \begin{align*} F^p / F^{p+1} \cong S^p\left(\mathcal{F}^{\prime}\right) \otimes S^{r-p}\left(\mathcal{F}^{\prime \prime}\right) \end{align*} for each \(p\).

• Same statement as (c), with exterior powers instead of symmetric powers. In particular, if \(\mathcal{F}^{\prime}, \mathcal{F}, \mathcal{F}^{\prime \prime}\) have ranks \(n^{\prime}, n, n^{\prime \prime}\) respectively, there is an isomorphism

• Let \(f: X \rightarrow Y\) be a morphism of ringed spaces, and let \(\mathcal{F}\) be an \({\mathcal{O}}_Y\)-module. Then \(f^*\) commutes with all the tensor operations on \(\mathcal{F}\), i.e., \(f^*\left(S^n(\mathcal{F})\right)=\) \(S^n\left(f^* \mathcal{F}\right)\) etc.

II.5.17. Affine Morphisms. #to_work

A morphism \(f: X \rightarrow Y\) of schemes is affine if there is an open affine cover \(\left\{V_i\right\}\) of \(Y\) such that \(f^{-1}\left(V_i\right)\) is affine for each \(i\).

• Show that \(f: X \rightarrow Y\) is an affine morphism if and only if for every open affine \(V \subseteq Y, f^{-1}(V)\) is affine ¹⁰

• An affine morphism is quasi-compact and separated. Any finite morphism is affine.

• Let \(Y\) be a scheme, and let \(\mathcal{A}\) be a quasi-coherent sheaf of \({\mathcal{O}}_Y\)-algebras (i.e., a sheaf of rings which is at the same time a quasi-coherent sheaf of \({\mathcal{O}}_Y\)-modules). Show that there is a unique scheme \(X\), and a morphism \(f: X \rightarrow Y\), such that for every open affine \(V \subseteq Y, f^{-1}(V) \cong \operatorname{Spec} \mathcal{A}(V)\), and for every inclusion \(U \hookrightarrow V\) of open affines of \(Y\), the morphism \(f^{-1}(U) \hookrightarrow f^{-1}(V)\) corresponds to the restriction homomorphism \(\mathcal{A}(V) \rightarrow \mathcal{A}(U)\). The scheme \(X\) is called \(\operatorname{Spec}\mathcal{A}\). ¹¹

• If \(\mathcal{A}\) is a quasi-coherent \({\mathcal{O}}_Y\)-algebra, then \(f: X=\) Spec \(\mathcal{A} \rightarrow Y\) is an affine morphism, and \(\mathcal{A} \cong f_* {\mathcal{O}}_X\). Conversely, if \(f: X \rightarrow Y\) is an affine morphism, then \(\mathcal{A}=f_* {\mathcal{O}}_X\) is a quasi-coherent sheaf of \({\mathcal{O}}_Y\)-algebras, and \(X \cong \operatorname{Spec} \mathcal{A}\).

• Let \(f: X \rightarrow Y\) be an affine morphism, and let \(\mathcal{A}=f_* {\mathcal{O}}_X\). Show that \(f_*\) induces an equivalence of categories from the category of quasi-coherent \({\mathcal{O}}_X\)-modules to the category of quasi-coherent \(\mathcal{A}\)-modules (i.e., quasi-coherent \({\mathcal{O}}_Y\)-modules having a structure of \(\mathcal{A}\)-module). ¹²

II.5.18. Vector Bundles. #to_work

Let \(Y\) be a scheme. A (geometric) vector bundle of rank \(n\) over \(Y\) is a scheme \(X\) and a morphism \(f: X \rightarrow Y\), together with additional data consisting of an open covering \(\left\{U_i\right\}\) of \(Y\), and isomorphisms \(\psi_i: f^{-1}\left(U_i\right) \rightarrow \mathbf{A}_{U_i}^n\), such that for any \(i, j\), and for any open affine subset \(V=\operatorname{Spec} A \subseteq U_i \cap U_j\), the automorphism \(\psi=\psi_j \circ \psi_i^{-1}\) of \(\mathbf{A}_V^n=\operatorname{Spec} A\left[x_1, \ldots, x_n\right]\) is given by a linear automorphism \(\theta\) of \(A\left[x_1, \ldots, x_n\right]\), i.e., \(\theta(a)=a\) for any \(a \in A\), and \(\theta\left(x_i\right)=\) \(\sum a_{i j} x_j\) for suitable \(a_{i j} \in A\).

An isomorphism \begin{align*} g:\left(X, f,\left\{U_i\right\},\left\{\psi_i\right\}\right) \rightarrow\left(X^{\prime}, f^{\prime},\left\{U_i^{\prime}\right\},\left\{\psi_i^{\prime}\right\}\right) \end{align*} of one vector bundle of rank \(n\) to another one is an isomorphism \(g: X \rightarrow X^{\prime}\) of the underlying schemes, such that \(f=f^{\prime} \circ g\), and such that \(X, f\), together with the covering of \(Y\) consisting of all the \(U_i\) and \(U_i^{\prime}\), and the isomorphisms \(\psi_i\) and \(\psi_i^{\prime} \circ g\), is also a vector bundle structure on \(X\).

• Let \(\mathcal{E}\) be a locally free sheaf of rank \(n\) on a scheme \(Y\). Let \(S(\mathcal{E})\) be the symmetric algebra on \(\mathcal{E}\), and let \(X=\operatorname{Spec} S(\mathcal{E})\), with projection morphism \(f: X \rightarrow Y\). For each open affine subset \(U \subseteq Y\) for which \(\left.\mathcal{E}\right|_U\) is free, choose a basis of \(\mathcal{E}\), and let \(\psi: f^{-1}(U) \rightarrow \mathbf{A}_U^n\) be the isomorphism resulting from the identification of \(S(\mathcal{E}(U))\) with \({\mathcal{O}}(U)\left[x_1, \ldots, x_n\right]\).

  Then \((X, f,\{U\},\{\psi\})\) is a vector bundle of rank \(n\) over \(Y\), which (up to isomorphism) does not depend on the bases of \(\mathcal{E}_U\) chosen. We call it the geometric vector bundle associated to \(\mathcal{E}\), and denote it by \(\mathbf{V}(\mathcal{E})\).

• For any morphism \(f: X \rightarrow Y\), a section of \(f\) over an open set \(U \subseteq Y\) is a morphism \(s: U \rightarrow X\) such that \(f \circ s=\mathrm{id}_U\). It is clear how to restrict sections to smaller open sets, or how to glue them together, so we see that the presheaf \(U \mapsto\{\) set of sections of \(f\) over \(U\}\) is a sheaf of sets on \(Y\), which we denote by \(\mathcal{S}(X / Y)\).

  Show that if \(f: X \rightarrow Y\) is a vector bundle of rank \(n\), then the sheaf of sections \(\mathcal{S}(X / Y)\) has a natural structure of \({\mathcal{O}}_Y\)-module, which makes it a locally free \({\mathcal{O}}_Y\)-module of rank \(n\). ¹³

• Again let \(\mathcal{E}\) be a locally free sheaf of rank \(n\) on \(Y\), let \(X=\mathbf{V}(\mathcal{E})\), and let \(\mathcal{S}=\) \(\mathcal{S}(X / Y)\) be the sheaf of sections of \(X\) over \(Y\). Show that \(\mathcal{S} \cong \mathcal{E}^2\), as follows. Given a section \(s \in \Gamma\left(V, \mathcal{E}^{\curlyvee}\right)\) over any open set \(V\), we think of \(s\) as an element of \(\operatorname{Hom}\left(\left.\mathcal{E}\right|_V, {\mathcal{O}}_V\right)\). So \(s\) determines an \({\mathcal{O}}_V\)-algebra homomorphism \(S\left(\left.\mathcal{E}\right|_V\right) \rightarrow {\mathcal{O}}_V\).

  This determines a morphism of spectra \(V=\operatorname{Spec} {\mathcal{O}}_V \rightarrow \operatorname{Spec} S\left(\left.\mathcal{E}\right|_V\right)=\) \(f^{-1}(V)\), which is a section of \(X / Y\). Show that this construction gives an isomorphism of \(\mathcal{E}^2\) to \(\mathcal{S}\).

• Summing up, show that we have established a one-to-one correspondence between isomorphism classes of locally free sheaves of rank \(n\) on \(Y\), and isomorphism classes of vector bundles of rank \(n\) over \(Y\). Because of this, we sometimes use the words “locally free sheaf” and “vector bundle” interchangeably, if no confusion seems likely to result.