## II.5: Sheaves of Modules

### II.5.1.  #to-work

Let $\left(X, \OO_X\right)$ be a ringed space, and let $\mathcal{E}$ be a locally free $\OO_X$-module of finite rank. We define the dual of $\mathcal{E}$, denoted $\mce\dual$, to be the sheaf $\sheafhom_{\OO_X}(\mce, \OO_X)$.

a. Show that $(\mce\dual)\dual \cong \mathcal{E}$.

b. For any $\OO_X$-module $\mathcal{F}$,
\[
\sheafhom_{\OO_X}(\mce \tensor\mcf, \mcg) \cong \sheafhom_{\OO_X}(\mcf, \sheafhom_{\OO_X}(\mce,\mcg) )
.\]

d. (Projection Formula). If $f:\left(X, \OO_X\right) \rightarrow\left(Y, \OO_Y\right)$ is a morphism of ringed spaces, if $\mathcal{F}$ is an $\OO_X$-module, and if $\mathcal{E}$ is a locally free $\OO_Y$-module of finite rank, then there is a natural isomorphism 
\[
f_*\left(\mathcal{F} \otimes_{\OO_X} f^* \mathcal{E}\right) \cong f_*(\mathcal{F}) \otimes_{\OO_Y} \mathcal{E}
.\]

### II.5.2.  #to-work

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

a. To give an $\OO_X$-module is equivalent to giving an $R$-module $M$, a $K$-vector space $L$, and a homomorphism $\rho: M \otimes_R K \rightarrow L$.

b. That $\OO_X$-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 $\OO_X$-modules $\mathcal{F}$, there is a natural isomorphism
$$
\operatorname{Hom}_A(M, \Gamma(X, \mathcal{F})) \cong \operatorname{Hom}_{\OO_X}( \tilde{M}, \mathcal{F} )
$$


### II.5.4.  #to-work

Show that a sheaf of $\OO_X$-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.

a. 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 $k$.

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

c. 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).

a. 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 $\supp m = V(\Ann m )$, where $\Ann m$ is the annihilator of $m=\{a \in A \mid a m=0\}$.

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

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

d. For any ideal $\mathfrak{a} \subseteq A$, we define a submodule $\Gamma_{\mfa}(M)$ of $M$ by 
\[
\Gamma_{\mfa }(M)= \left\{m \in M \mid \mathfrak{a}^n m=0 \text{ for some } n > 0 \right\}
\]
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 $\mathcal{F}=\tilde{M}$.[^hint.2.5.qqqqqqq.1.d]

[^hint.2.5.qqqqqqq.1.d]: 
Hint: Use (Ex. 1.20) and (5.8) to show a priori that $\mathcal{H}_Z^0(\mathcal{F})$ is quasi-coherent. Then show that $\Gamma_{\mathrm{a}}(M) \cong \Gamma_Z(\mathcal{F})$.

e. Let $X$ be a noetherian scheme, and let $Z$ be a closed subset. If $\mathcal{F}$ is a quasicoherent (respectively, coherent) $\OO_X$-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.

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

b. $\mathcal{F}$ is locally free if and only if its stalks $\mathcal{F}_x$ are free $\OO_x$-modules for all $x \in X$.

c. $\mathcal{F}$ is invertible (i.e., locally free of rank 1) if and only if there is a coherent sheaf $\mathcal{G}$ such that $\mathcal{F} \otimes \mathcal{G} \cong \OO_X$.[^rmk.2.5.7.c]

[^rmk.2.5.7.c]: 
This justifies the terminology invertible: it means that $\mathcal{F}$ is an invertible element of the monoid of coherent sheaves under the operation $\otimes$.

### 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
\[
\varphi(x)=\operatorname{dim}_{k(x)} \mathcal{F}_x \otimes_{\OO_x} k(x),
\]
where $k(x)=\OO_x / m_x$ is the residue field at the point $x$. Use Nakayama's lemma to prove the following results.

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

b. If $\mathcal{F}$ is locally free, and $X$ is connected, then $\varphi$ is a constant function.

c. 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.

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

b. 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.[^hint.2.5.9.b]

c. 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 $\OO_X$-modules.


[^hint.2.5.9.b]: 
Hint: Use the methods of the proof of (5.19).

### 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).

a. For any homogeneous ideal $I \subseteq S$, we define the saturation $\bar{I}$ of $I$ to be $\left\{s \in S \mid\right.$ 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 $I=\bar{I}$. Show that $\bar{I}$ is a homogeneous ideal of $S$.

b. 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.

c. 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 $Y$.

d. 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** $S \fiberprod{A} T$ 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 $\OO(1)$ on $\operatorname{Proj}\left(S \times{ }_A T\right)$ is isomorphic to the sheaf $p_1^*\left(\OO_X(1)\right) \otimes p_2^*\left(\OO_Y(1)\right)$ 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 $\Proj(S\fiberprod A T) \injects \PP^N_A$, 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

a. 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.[^hint.2.5.12.a]

b. 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 $Z$.

[^hint.2.5.12.a]: 
Hint: Use a Segre embedding.

### 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 $\OO(1)$ on Proj $S^{(d)}$ corresponds via this isomorphism to $\OO_X(d)$.

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 $\PP_A^r$. 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.

a. Let $S$ be the homogeneous coordinate ring of $X$, and let $S^{\prime}=\bigoplus_{n \geqslant 0} \Gamma\left(X, \OO_X(n)\right)$. Show that $S$ is a domain, and that $S^{\prime}$ is its integral closure.[^hint.2.5.8.a]

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

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

d. 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 $\Gamma\left(\mathbf{P}^r, \OO_{\mathbf{P}^r}(n)\right) \rightarrow \Gamma\left(X, \OO_X(n)\right)$ is surjective.

[^hint.2.5.8.a]: 
Hint: First show that $X$ is integral. Then regard $S^{\prime}$ as the global sections of the sheaf of rings $\mathcal{S}=\bigoplus_{n \geq 0} \OO_X(n)$ on $X$, and show that $\mathcal{S}$ is a sheaf of integrally closed domains.

### 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 $\left.\mathcal{F}^{\prime}\right|_U \cong \mathcal{F}$.

a. 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 $\mathcal{F}_\alpha(U)$.

b. 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 $\left.\mathcal{F}^{\prime}\right|_U \cong \mathcal{F}$.[^2.5.15.b.hint]

c. 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 $\left.\mathcal{F}^{\prime}\right|_U \cong \mathcal{F}$.[^2.5.8.c.hint]

d. 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.[^hint.2.5.8.d]

e. As an extra corollary, show that on a noetherian scheme, any quasi-coherent sheaf $\mathcal{F}$ is the union of its coherent subsheaves.[^hint.2.5.8.e]

[^hint.2.5.8.e]: 
Hint: If $s$ is a section of $\mathcal{F}$ over an open set $U$, apply (d) to the subsheaf of $\left.\mathcal{F}\right|_U$ generated by $s$.

[^hint.2.5.8.d]: 
Hint: Cover $X$ with open affines, and extend over one of them at a time.

[^2.5.8.c.hint]: 
Hint: Use the same method, but replace $i_* \mathcal{F}$ by $\rho^{-1}\left(i_* \mathcal{F}\right)$, where $\rho$ is the natural map $\mathcal{G} \rightarrow i_*\left(\left.\mathcal{G}\right|_U\right)$.

[^2.5.15.b.hint]: 
Hint: Let $i: U \rightarrow X$ be the inclusion map. Show that $i_* \mathcal{F}$ is quasi-coherent, then use (a).

### 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 $\left(X, \OO_X\right)$ be a ringed space, and let $\mathcal{F}$ be a sheaf of $\OO_X$-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 $\OO_X(U)$-module. The results are $\OO_x$-algebras, and their components in each degree are $\OO_x$-modules.

a. 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.

b. 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 $\mathcal{F} \cong \mathcal{F}^{\top} \otimes \bigwedge^2 \mathcal{F}$.

c. 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 $S^r(\mathcal{F})$,
\[
S^r(\mathcal{F})=F^0 \supseteq F^1 \supseteq \ldots \supseteq F^r \supseteq F^{r+1}=0
\]
with quotients
\[
F^p / F^{p+1} \cong S^p\left(\mathcal{F}^{\prime}\right) \otimes S^{r-p}\left(\mathcal{F}^{\prime \prime}\right)
\]
for each $p$.

d. 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

e. Let $f: X \rightarrow Y$ be a morphism of ringed spaces, and let $\mathcal{F}$ be an $\OO_Y$-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 $i$.

a. 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[^hint2.5.17.a]

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

c. Let $Y$ be a scheme, and let $\mathcal{A}$ be a quasi-coherent sheaf of $\OO_Y$-algebras (i.e., a sheaf of rings which is at the same time a quasi-coherent sheaf of $\OO_Y$-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 $\spec \mathcal{A}$.[^hint.2.4.14.c]

d. If $\mathcal{A}$ is a quasi-coherent $\OO_Y$-algebra, then $f: X=$ Spec $\mathcal{A} \rightarrow Y$ is an affine morphism, and $\mathcal{A} \cong f_* \OO_X$. Conversely, if $f: X \rightarrow Y$ is an affine morphism, then $\mathcal{A}=f_* \OO_X$ is a quasi-coherent sheaf of $\OO_Y$-algebras, and $X \cong \operatorname{Spec} \mathcal{A}$.

e. Let $f: X \rightarrow Y$ be an affine morphism, and let $\mathcal{A}=f_* \OO_X$. Show that $f_*$ induces an equivalence of categories from the category of quasi-coherent $\OO_X$-modules to the category of quasi-coherent $\mathcal{A}$-modules (i.e., quasi-coherent $\OO_Y$-modules having a structure of $\mathcal{A}$-module).[^2.5.14.e]

[^2.5.14.e]: 
Hint: For any quasi-coherent $\mathcal{A}$-module $\mathcal{M}$, construct a quasi-coherent $\OO_X$-module $\tilde{\mathcal{M}}$, and show that the functors $f_*$ and $\sim$ are inverse to each other.

[^hint.2.4.14.c]: 
Hint: Construct $X$ by glueing together the schemes $\operatorname{Spec} \mathcal{A}(V)$, for $V$ open affine in $Y$.

[^hint2.5.17.a]: 
Hint: Reduce to the case $Y$ affine, and use (Ex. 2.17).

### 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** 
\[
g:\left(X, f,\left\{U_i\right\},\left\{\psi_i\right\}\right) \rightarrow\left(X^{\prime}, f^{\prime},\left\{U_i^{\prime}\right\},\left\{\psi_i^{\prime}\right\}\right)
\]
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 $X$.

a. 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 $\OO(U)\left[x_1, \ldots, x_n\right]$. 

    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 $\mathbf{V}(\mathcal{E})$.

b. 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 $\OO_Y$-module, which makes it a locally free $\OO_Y$-module of rank $n$.[^hint.2.5.18.b]

c. 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 $\operatorname{Hom}\left(\left.\mathcal{E}\right|_V, \OO_V\right)$. So $s$ determines an $\OO_V$-algebra homomorphism $S\left(\left.\mathcal{E}\right|_V\right) \rightarrow \OO_V$. 

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

d. 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.

[^hint.2.5.18.b]: 
Hint: It is enough to define the module structure locally, so we can assume $Y=\operatorname{Spec} A$ is affine, and $X=\mathbf{A}_Y^n$. Then a section $s: Y \rightarrow X$ comes from an $A$-algebra homomorphism $\theta: A\left[x_1, \ldots, x_n\right] \rightarrow$ $A$, which in turn determines an ordered $n$-tuple $\left\langle\theta\left(x_1\right), \ldots, \theta\left(x_n\right)\right\rangle$ of elements of $A$. Use this correspondence between sections $s$ and ordered $n$-tuples of elements of $A$ to define the module structure.]