Solutions to The Rising Sea (Vakil)

Proof:

Continuing to let \( \pi: V \to M \) denote our rank \( n \) vector bundle, over \( U_i \cap U_j \) our transition function makes the following diagram commute:

where the \( \phi_k \) denote the local trivializations. Letting \( s : M \to V \) be our global section and \( \vec{s_i} = \res{M, U_i}(s) \), recall that since \( \pi \circ \vec{s_i} = \textrm{id}_{U_i} \) (by definition of a section), \( \textrm{Im}\,\vec{s_i} \subset \pi^{-1}(U) \). By abuse of notation, we will continue to let \( \vec{s_i} \) denote \( \res{U_i}{ U_i \cap U_j} (\vec{s_i}) \) and \( \vec{s_j} \) denote \( \res{U_j}{U_i \cap U_j}(\vec{s_j}) \) (notice these must be equal by the sheaf axioms, which is already a hint in the right direction). In fact, we will botch notation a little bit worse since we will in fact be using \( \vec{s_i} \) to refer to the composition of our section with the trivialization \( \phi_i \) in the following diagram:

Thus, we think of \( \vec{s_i} \) as going along the outermost left wall in this case, and \( \vec{s_j} \) as going along the right wall. Most of the paths commute in the diagram by definition (though it is not explicitly stated in Vakil whether transition functions must commute with the projection onto the first component / dotted arrow, this can be easily shown) — thus, we must have that \( T_{ij} \vec{s_i} = \vec{s_j} \).

Proof:

On any open set \( U_i \), we have that the sheaf-hom is given by \(\mathscr{Hom}_{\OO_X} (\FF , \GG )(U_i) = \textrm{Hom}_{\OO_X}(\FF\vert_{U_i}, \GG\vert_{U_i})\). Therefore, we may suppose without loss of generality that \( \FF \) and \( \GG \) are free on \(X\). If we let \( \phi : \FF \to \OO_X^{\oplus m}\) and \( \psi : \GG \to \OO_X^{\oplus n} \) denote the trivializations, we get a natural identification of \( \mathscr{Hom}_{\OO_X} (\FF , \GG ) \) with \( \mathscr{Hom}_{\OO_X} (\OO_X^{\oplus m}, \OO_X^{\oplus n} ) \) — it is a somewhat standard fact (not proved in Chapter 1 it seems) that \( \textrm{Hom}(A, -) \) commutes with direct sums for finitely generated \(A\), and \( \textrm{Hom}(-, B) \) commutes with finite direct sums. In either case, we get that \( \mathscr{Hom}_{\OO_X} (\FF , \GG ) \) is isomorphic to \( \mathscr{Hom}_{\OO_X} (\OO_X , \OO_X )^{\oplus mn} \). But \( \mathscr{Hom}(\OO_X, \OO_X) \cong \OO_X \) by looking on affine open sets and considering \( \phi \mapsto \phi(1) \).

Proof:

The transition functions in the rank \( n \) case should not be much different from the rank 1 case: we have a diagram of the form

We know by the previous problem that \( \mathscr{Hom}(\EE, \OO_X) \) is locally free, since the isomorphism \( \phi_i : \EE\vert_{U_i} \cong (\OO\vert_{U_i})^{\oplus n} \) induces an isomorphism \( \mathscr{Hom}(\EE, \OO_X)(U_i) \cong \mathscr{Hom}(\OO^{\oplus n}, \OO) \cong \OO^{\oplus n} \). Then those very same transition functions \( T_{ij} \in \textrm{GL}_n( \OO(U_i \cap U_j) ) \) allow us to define transition functions \( \mathscr{Hom}(\EE, \OO)(U_i) \to \mathscr{Hom}(\EE, \OO)(U_j) \). In particular, for any \( f \in \mathscr{Hom}(\EE, \OO_X)(U_i \cap U_j) \), we precompose with \( \phi_i(U_i \cap U_j) \) to obtain a morphism \( \widetilde{\phi} : \OO^{\oplus n}(U_i \cap U_j) \to \OO(U_i \cap U_j) \). Then \( \widetilde{\phi} \circ T_{ij}^{-1} \) is precisely the desired function in terms of our other trivialisation on \( U_j \) (by commutivity of the diagram above).

To see the second claim that \( \EE \cong (\EE)^{\vee \vee} \), we simply use Exercise 2.4.E and reduce to the level of stalks. But then the problem reduces to the claim that \( M \cong M^{\vee \vee} \) for finite free modules, which is a slight generalization of the standard linear algebra fact (for example, see Exercise 10.5.11 in Dummit & Foote)

Proof:

Recall from Exercise 2.6.J that by \( \FF \otimes \GG \) we are referring to the sheafification of the presheaf \( U \mapsto \FF(U) \otimes_{\OO_X(U)} \GG(U) \). Now suppose that \( \{ U_i \} \) is an open cover with trivializations \( \phi_i : \FF\vert_{U_i} \to \OO_X^{\oplus m}\vert_{U_i} \) and \( \psi_i : \GG\vert_{U_i} \to \OO_X^{\oplus n}\vert_{U_i} \). Then we ge a diagram of the form

The fact that free modules commute with the tensor product is a result of Exercise 1.3.M, giving us the top ismorphism. By the universal properties of tensor product and sheafification, there is then a natural isomorphism \( (\FF \otimes \GG)\vert_{U_i} \to \OO_X^{\oplus mn}\vert_{U_i} \) (to be more precise, the universal properties give us the map, and then one must show that since \( \phi_i \) and \( \psi_i \) are isomorphisms the resulting \( \phi_i \otimes \psi_i \) is an isomorphism by possibly quoting freeness). Therefore, the sheaf \( \FF \otimes \GG \) is locally free.

To show that \( \FF \otimes \FF^{\vee} \cont \OO_X \), we work on the level of stalks — we know by Exercise 2.6.J(b) that the stalk of the tensor of sheaves is simply the tensor of the stalks (since colimits commute with the tensor product). But then this is equivalent to the statement that

for any free \( R \)-mod of rank 1. The isomorphism is clearly given by evaluation \( \textrm{ev}( m \otimes f ) = f(m) \), similar to the double dual.

Proof:

Following the hint, we know from §2.6 that exactness may be checked at the level of stalks. Since \( \FF \) is locally isomorphic to \( \OO_X^{\oplus n}\vert_{U_i} \) on some affine cover, each stalk \( \FF_p \cong \OO_{X,p}^{\oplus n} \). Since \( M \otimes_R R^{\oplus n} \cong M^{\oplus n} \) for any \( R \)-module \(M\), exactness of the sequence

implies exactness of

Proof:

Although not immediately apparent, the result will follow from the tensor-hom adjunction (c.f Exercise 1.5.D) by checking on the level of stalks. However, we must first prove another useful isomorphism:

Unfortunately we have not yet proved that natural map \( \mathscr{Hom}(\EE, \FF)_p \to \textrm{Hom}(\EE_p, \FF_p) \) is an isomorphism whenever \( \EE \) is coherent (i.e. locally finitely presented), so on the level of some affine set \( U \), let \( f \in \FF(U), \ell \in \mathscr{Hom}(\EE\vert_U, \OO\vert_U) \) and consider the map \( \phi_U \) given by sending \( f \otimes \ell \) to (the map)

for any \( V \subset U \). Since \( \EE\vert_U \cong \OO\vert_U^{\oplus n} \), we have that \( \FF \otimes \EE^\vee \cong (\FF \otimes \OO_X^\vee)^{\oplus n} \) and \( \mathscr{Hom}(\EE, \FF) \cong \mathscr{Hom}(\OO_X, \FF)^{\otimes n} \). Then reducing to \( n = 1 \), we need to show \( \FF \otimes \OO_X^\vee \cong \mathscr{Hom}( \OO_X, \FF) \). On the level of modules, this is equivalent to showing

Now any \( R \)-module morphism \( f: R \to N \), where \( N \) is an arbitrary \(R\)-mod, is uniquely determined by \( f(1) \in N \). So we get isomorphisms \( \textrm{Hom}_R (R, R) \cong \R \) (as modules) and \( \textrm{Hom}_R (R, M) \cong M \) proving the claim.

Proof:

Since \( M \otimes_R R \cong M \cong R \otimes_R M \), it should be clear that \( \OO_X\) is the identity under tensor product; moreover, Exercise 13.1.D tells us that the inverse of any locally free rank one \( \EE \) is simply \( \EE^\vee \). Though it is not true that \( A \otimes B = B \otimes A \), as we are considering objects up to isomorphism it will be true that the isomorphism classes of \( \EE \otimes \FF \) and \( \FF \otimes \EE \) are the same, making \( \Pic X \) abelian. Associativity follows from the usual associativity of the tensor product (together with some universal property stuff of sheafification making things unique up to unique isomorphism).

Proof:

A morphism of locally ringed spaces comes with the data of a morphism of sheaves \( \pi^\sharp : \OO_Y \to \OO_X \). If we let \( \{ U_i \} \) be an affine cover of \(Y\) with trivializations \(\phi_i : \GG\vert_{U_i} \xrightarrow{\sim} \OO_Y\vert_{U_i}^{\oplus n} \), then taking \( V_i = \pi^{-1}(U_i) \) we may define

Since the trivializations \( \phi_i \) agree on overlap, this gives a well-defined locally free sheaf (since \( \pi^\ast \OO_Y \vert_{U_i} = \OO_X \vert_{V_i}\)) on \(X\).

It is immediate that when our morphism \( \pi = \textrm{id}_X \) is the identity, the induced map of local rings is the identity is as well and thus the pullback map is the identity. Therefore, showing that \( \Pic \) gives a contravariant functor reduces to showing that it preserves composition. Letting \( \rho : Y \to Z \) be another morphism of ringed spaces and \( \GG \) now a locally free sheaf of rank \( n \) on \(Z\), we know on some sufficiently small \( U \subset Z \) that \( \GG\vert_U \cong \OO_Z\vert_U^{\oplus n} \). Taking \( V = \rho^{-1}(U) \) so that \( \rho^\sharp \OO_Z\vert_U = \OO_Y\vert_V \), the original construction above shows that this makes \( \rho^\ast \GG \) locally free on \( Y \) since it is the data of \( n \) copies of \( \rho^\ast \OO_Z\vert_U \) (applied to the image of the trivialization). But then taking \( W = \pi^{-1}(V) = (\rho \circ \pi)^{-1}(U) \) so that \( \(pi^\sharp \circ \rho^\sharp) \OO_Z\vert_U = (\rho \circ \pi)^\sharp \OO_Z\vert_U = \OO_X \), it becomes immediate that \( ( \rho \circ \pi )^\ast \GG = \pi^ast \rho^\ast \GG \) as desired.

Proof:

Following the hint, for any \(x \in X \) the data of the stalk \( x \in X \) is an equivalence class of \( (s_1, U) \) for some neighborhood \( U \ni x \) and \( s_1 \in \FF(U) \). Without loss of generality, we may restrict \( U \) such that \( \FF\vert_U \cong \OO_X\vert_U^{\oplus n} \) in which case we may abuse notation and write \( s_1 = (f_1, \dots, f_n) \) (it would really be the image of \(s_1\) under the trivialization). Then we say \( s(x) = 0 \) if \( x \in V(f_1) \cap \dots \cap V(f_n) \); in fact, we simply define \( V(s)\vert_U := V(f_1) \cap \dots \cap V(f_n) \) as a closed subscheme. Now suppose \( (s_1, U) \cong (s_2, V) \) such that there exists some dense open subset \( W \subset U \cap V \) with \( s_1\vert_W = s_2 \vert_W\). Again we may assume \( V \) is sufficiently small so that \( \FF\vert_V \cong \OO_X\vert_V^{\oplus n} \) and \( s_2 = (g_1, \dots, g_n) \). On \( W \) we clearly have that \( \cap_i V(f_i)\vert_W = \cap_i V(g_i)\vert_W \) — however, since the \( f_i \) and \(g_i \) are by definition continuous and \( W \subset U \cap V\) dense, we have that \( f_i = g_i \) on \(W\) implies \( f_i = g_i \) on all \( U \cap V \) so that \( V(s_1)\vert_{U \cap V} = V(s_2)\vert_{U \cap V} \).

Proof:

As before we restrict to the case that \( X = \spec A \) for \( A \) Noetherian, integrally closed domain — upon further restriction to another open affine, we may assume without loss of generality that our sheaf \( \FF \) is in fact free such that \( \FF \cong \OO_{\spec A}^{\oplus n} \) (globally, this simply tells us that \( \Gamma(\spec A, \FF) = A^{\oplus n} \)). Recall that the prime ideals of \( A^{\oplus n} \) are of the form \( A \oplus \dots \oplus \pp_j \oplus A \dots \oplus A \) such that all components except one must be the entire ring \(A\). By the original Hartogs Lemma in Theorem 11.3.11, we have

proving the claim.

Proof:

A rational section of an invertible sheaf \( \FF \) is some \( s \in \Gamma(X, \FF \otimes_{\OO_X} K(X)) \) — by reducing to some affine \( \spec A \) (where \(A\) is a Noetherian integrally closed domain), we may copy the proof of Exercise 12.5.H word-for-word (we do not even need the previous exercise, since invertible sheaves are rank 1 and thus our section still looks like \( f_1 / f_2 \in K(A)\)).

Proof:

1. First notice that since \( X \) is already affine, if \( \FF \) was quasicoherent then we would necessarily have \( M = \Gamma(X, \FF) = k(t) \), so that \( \FF\vert_{D(f)} \cong k(t)_f \) for any \( f \in k[t] \). However, this obviously can't be true sine we can find distinguished open sets \( D(f) \) with \( [(t)] \notin D(f) \). For example, \( (t) \notin D(t^2) \) by construction, so by definition of the skyscraper sheaf we have \( \FF(D(t^2)) = 0 \), whereas \( \widetilde{k(t)}(D(t^2)) = k(t)_{t^2} \cong k(t) \).

2. This is farily immediate since \( \{ \eta \} \) is dense implying every open set \( U \) must necessarily contain \( \eta \) and thus the skyscraper sheaf supported at the generic point is just the constant sheaf equal to \( k(t) \). Since \( \A^1_k \) is affine, we must have \( M = \Gamma(X, \FF) \) so it is easy to see that in this case \( \FF \cong \widetilde{k(t)} \). Since the localization of \( k(t)_f \cong k(t) \) for any \( f \in k[t] \), we have that the restriction morphisms are alyways the identity (which is also true by virtue of \( \FF \) being the constant sheaf which has identity restriction morphisms).

Proof:

On every distinguished open affine \( D(f) \), \( \widetilde{ k(t) }(D(t)) = k(t) \) but \( k(t) \not\cong \bigoplus_{i \in I} k[t] \); \( k(t) \) is a field while \( \bigoplus_{i \in I} k[t] \) has non-zero maximal ideals (in the infinite case, the axiom of choice must be used).

Proof:

Since all locally free sheaves \( \FF \) are in fact quasicoherent, we have that \( \FF \cong \widetilde{M} \) and on sufficiently small open sets \( D(f) \), we have \( \FF\vert_{D(f)} \cong \OO_{\spec A}\vert_{D(f)}^{\oplus n} \cong M_f \). That is to say that \( A_f^{\oplus n} \cong M_f \) on a sufficiently small cover of \( \spec A \) — since \( \spec A \) is quasicompact, we may take finitely many such \( f_i \), say \( f_1, \dots, f_r \). Then as \( M_{f_i} \) is finitely generated on all \( f_i \), we may use the idea from Exercise 3.5.B to show that \( M \) itself is finitely generated. By the classification of finitely-generated modules over a PID, \( M \) is necessarily projective (has no torsion) and thus free.

Proof:

Because Exercise 2.5.D shows us that since the \( D(f_i) \) cover \( \spec A \) and our isomorphisms satisfy the cocycle condition, the sheaf we obtain by gluing together is unique up to unique isomorphism. As \( \widetilde{M}(D(f_i)) \cong M_i \) by the commuting diagram shown in the text and the sheaf obtained from Exercise 2.5.D by gluing must also necessarily satisfy this property, the isomorphism immediately follows.

Proof:

This should just follow from the fact that since \( M_i \cong M_{f_i} \), \( M_j \cong M_{f_j} \), we have

Proof:

This follows from Exercise 2.4.A.

Proof:

This follows from Exercise 2.5.C

Proof:

This follows from Remark 2.5.3

Proof:

If \( \FF \) is quasicoherent, then by taking \( M = \Gamma(\spec A, \FF) \) we know that \( \Gamma(\spec A_f, \FF) \cong M_f \) showing that we necessarily have the desired isomorphism.

For the reverse direction, suppose that the map \( \alpha \) is an isomorphism. Again taking \( M \) to be the \( A \)-module \( \Gamma(\spec A, \FF)\), the supposition tells us that \( \FF(D(f)) \cong M_f \). By Theorem 2.5.1 in Vakil, this uniquely defines (up to unique isomorphism) a sheaf on all of \( \spec A\). As this necessarily agrees with \( \widetilde{M} \) on our distinguished base, we then have \( \FF\vert_{\spec A} \cong \widetilde{M}\) everywhere.

Proof:

By quasicompactness there is a finite cover \( \{ U_i \} = \{ \spec A_i \} \) of \(X\) and by quasiseparatedness each \( U_{ij} \) may be covered by finitely many \( U_{ijk} \). Following the hint, by the identity and gluability axioms the sequence

is exact. Since \(\otimes_{\Gamma (X, \OO_X )} \Gamma (X, \OO_X )_f\) is exact, we get an exact sequence

Since localization commutes with finite products by Exercise 1.3.F(a), we know that the first term is simply \( \oplus_i \Gamma(U_i, \FF)_f \) which is isomorphic to \( \oplus_i \Gamma( \spec (A_i)_{f_i}, \FF ) \) by the previous exercise. By a similar argument for the last term, we get an exact sequence

However, from the proof of the QCQS lemma 7.3.6 we know that the kernel of that last morphism is precisely \( \Gamma(X_f, \FF) \) so that \( \Gamma(X, \FF)_f \\cong \Gamma(X_f, \FF) \).

Proof:

We will make use of the previous two exercises here; let \( \spec B \subset Y \) and take \( g \in B \). By assumption \( \pi^{-1}(\spec B) \) is both quasiseparated and quasicompact. Letting \( \pi^\ast : \Gamma( \spec B, \OO_Y ) \to \Gamma( \pi^{-1}(\spec B), \OO_X ) \) denote the morphism of sheaves and \( f = \pi^\ast g \). Then by the previous exercise, we have an isomorphism

so that by Exercise 13.3.D we have that \( \pi_\ast \FF\) is quasicoherent.

Proof:

Although it was not shown in Chapter 1, localization commutes with intersections; therefore we may use the fact that the nilradical is the intersection of all prime ideals to show that \( \mathfrak{N}(A)_f \cong \mathfrak{N}(A_f) \). In a more constructive manner, suppose \( S \) is a multiplicatively closed subset of a ring \(A\) and let \( a/s \in S^{-1}\mathfrak{N}(A) \). Then \( a^n = 0 \) in \(A\), so that \( (a/s)^n = 0 \) in \( S^{-1} A \) or \( S^{-1}\mathfrak{N}(A) \hookrightarrow \mathfrak{N}(S^{-1}A) \). Conversely, if \( a/s = 0 \in \mathfrak{N}(S^{-1}A) \) then \( a^n/s^n = 0 \in S^{-1}A \) implying \( s^\prime \cdot a^n = 0 \) in \( A \). Assuming \( S \) has no zerodivisors (as otherwise \( S^{-1}A \) would be the zero ring), this implies that \( a^n = 0 \) as desired.

By Exercise 13.3.D, this implies that the sheaf of nilpotents (which could be defined as the ideal sheaf of the reduction morphism \( Y^{red} \to Y \) c.f. §8.3.10) is quasicoherent.

Proof:

Let \( \{U_i = \spec A_i\} \) be a finite cover of open affines of \(X\) (by quasicompactness) and assume that each \( U_i\) is sufficietly small so that \( \mathscr{L}\vert_{U_i} \cong \OO_X\vert_{U_i} \). Moreover, let \( \FF\vert_{U_i} = \widetilde{M_i} \) for some \( A_i \)-module \(M_i\). Let \( s_i \) be the image of \( s \) in \( A_i \) so that locally \( X_s \) is of the form \( D(s_i) \). Then for any section \( f \in \Gamma(X_s, \FF) \), by the exercises above we know that \( f \vert_{U_i} \in \Gamma(D(s_i), \FF) \cong (M_i)_{s_i} \) so that \( s_i^{n_i} f\vert_{U_i} \in M_i \). Taking \( n \) to be the maximum of all such \( n_i \) so that this works on each affine set, we know that since the \( s_i \) agree on overlaps and the \( f\vert_{U_i} \) glue by construction, we have some \( t_i \in M_i \) that are locally of the form \( s_i^{n_i} f\vert_{U_i} \) on each \( U_i \). As \(X\) is quasiseparated, we may use the same approach on finite covers of intersections to assume without loss of generality that the same choice of \( n \) works everywhere. Then the restriction of \( s^n f \in \Gamma(X, \FF \otimes \mathscr{L}^{\otimes n}) \) maps to \( f_\vert{U_i} \) on open affines by construction, proving the claim.

Proof:

Take \( X = \coprod_{i = 0}^\infty \spec \Z \) and on each copy of \( \spec \Z \), define \( \FF\vert_{\spec \Z} = \Q \). We may simply let \( \mathscr{L} \) be \( \OO_X \) in this case, and take \( f = (p, p, \dots) \in \Gamma(X, \FF ) \). Then \( (\prod_{i=0}^\infty \Q)_{f} \neq \prod_{i=0}^\infty \Q_p \), as only the right side contains \( (1, 1/p, 1/p^2, \dots) \).

Proof:

This is equivalent to the fact that \( 0 \to L \to M \to N \to 0 \) is exact if and only if \( 0 \to \widetilde{L} \to \widetilde{M} \to \widetilde{N} \to 0 \) is exact. To prove the reverse direction first, suppose \( 0 \to L \to M \to N \to 0 \) is exact. Since localization is exact, \( 0 \to L_\pp \to M_\pp \to N_\pp \to 0 \) is exact so our sequence of sheafifications is exact at stalks and thus exact.

Conversely, suppose \( \pp \in \spec A \) is arbitrary and consider

By assumption the second row is exact; however, by commutivity we know that since \( (\ker \beta(X) / \im \alpha(X))_\pp = 0 \) and \( \pp \) was arbitrary, \( \ker \beta(X) = \im \alpha(X) \).

Proof:

By the previous exercise, we know that both three-term sequences are exact (as they are exact on an affine cover) so that \( \FF \) must locally be isomorphic to the sheaffification of the cokernel of \( A_i^{\oplus I} \to A_i^{\oplus J} \).

For the reverse direction, suppose that \( \FF \) is quasicoherent so that on a sufficiently small affine cover we have \( \FF_{U_i} \cong \widetilde{M} \). Letting \( \{ m_i \}_{i \in I} \) be a set of generators, we obtain a surjective morphsim \( \phi : A^{\oplus I} \to M \) given by \( (a_i)_{i \in I} \mapsto \sum_{i \in I} a_i m_i \). Applying the same idea to \( \ker \phi \), one obtains an exact sequence

Applying the previous exercise proves the claim.

Proof:

1. Suppose \( \FF\vert_U \cong \widetilde{M} \) for some \(A\)-module \( M \). By Exercise 13.4.A we know that \( 0 \to \widetilde{A}^{\oplus a} \to \widetilde{M} \to \widetilde{A}^{\oplus b} \) is exact if and only if \( 0 \to A^{\oplus a} \to M \to A^{\oplus b} \to 0 \) is exact. Since the last term is obviously free, the splitting lemma tells us that this sequence splits. More precisely, since \(M\) must have free rank at least \(b\) (since the last map is surjective), if we take \( m_1, \dots, m_b, \dots, m_n \) to be generators of \( M \) then there is a map \( u : A^{\oplus b} \to M \) defined by sending \( e_i \mapsto m_i \) so that \( g \circ u = \textrm{id}_{A^{\oplus b}} \) (where \( g : M \to A^{\oplus b} \) is the last map in the sequence). The proof of the splitting lemma shows that \( M \cong A^{\oplus b} \oplus A^{\oplus a} \).

2. Let \( U = \spec A_1 \) and \( V = \spec A_2 \) be two affine opens sets in our cover; by part (a) above we may assume without loss of generality that \( \FF\vert_U \cong \widetilde{A_1}^{\oplus a + b} \) and \( \FF\vert_V \cong \widetilde{A_2}^{\oplus a + b} \). Then the transition function \( T_{12} \in \textrm{GL}_{a+b}(\OO( U \cap V )) \) must make the following diagram commute:

   

   That is to say it must commute with the transition functions \( \textrm{GL}_a ( \OO(U \cap V) ) \) and \( \textrm{GL}_b (\OO(U \cap V)) \). By freeness and the fact that our original sequence splits, this implies that the transition functions are upper block triangular matrices, with blocks corresponding to the respective transition functions.

Proof:

1. As suggested in the hint, at any point \( p \in X \) we know that for a sufficiently small neighborhood, exactness of our sequence in (13.5.1.1) is equivalent to exactness of

   $$ \begin{CD} 0 @>>> \widetilde{N} @>>> \widetilde{A^{\oplus m}} @>{\phi}>> \widetilde{A^{\oplus n}} @>>> 0 \end{CD} $$

   which itself is equivalent to exactness of the respective \(A\) modules by Exercise 13.4.A. By some choice of basis, \( \phi \) corresponds to an \( n \times m \) matrix; since it is necessarily surjective, the rank of \( M \) is \( n \) so that there exist \( n \) linearly independent columns. As the property of \( M \) being rank \( n \) is an open condition (given by the nonvanishing of \( n \times n\)-minors). Since this nonvanishing dependend on our choice of basis, our covers by these open subsets are in bijection with our \( n \) columns of \( M \) — therefore we may assume without loss of generality that we have restricted to a single chart \( X = \spec A \) such that the first \( n \) columns of \( M \) necessarily generate the image in \( A^{\oplus n} \). Further yet, after a change of coordinates we may assume that \( M \) is the identity on the first \( n \) columns and \( 0 \) afterwards. Since our sequence is exact, \( \FF^{\prime}(\spec A) = N \) is precisely the kernel of this map, which in this case is just the orthogonal complement of the copy of \( A^{\oplus n} \) (the image) sitting inside \( A^{\oplus m} \) which is precisely \( A^{\oplus (m - n)} \).

2. Considering the example referenced, on \( \A^1_k = \spec k[t] \) we have the exact sequence of \( k[t] \) modules

   $$ \begin{CD} 0 @>>> (t) @>>> k[t] @>>> k @>>> 0 \end{CD} $$

   However, \( k \) is certainly not a free \( k[t] \) module, so the corresponding cokernel in \( \textrm{QCoh}(X) \) certainly isn't as well.

Proof:

As with the previous exercise, we know that quasicoherent sheaves are essentially determined at the level of stalks. The universal property of sheafification as a functor tells us we only really need to concern ourselves with the tensor presheaf \( (\FF \otimes \GG)(U) = \FF(U) \otimes_{\OO_X(U)} \GG(U) \), as showing an isomorphism on the level of presheaves will simply factor through to the sheafification. Thus, we may assume without loss of generality \(X = \spec A \). Then by definition of the tensor presheaf, we necessarily have \( (\FF \otimes \GG)(X) = M \otimes_A N \); since the tensor product commutes with localization, we have \((M \otimes_A N)_f \cong M_f \otimes_{A_f} N_f\) for any \( f \in A \), so that as \( \FF \) and \( \GG \) are quasicoherent, it must be the case that \( \FF \otimes \GG \cong \widetilde{ M \otimes_A N } \) (say by checking this isomorphism on our distinguished affine base).

Proof:

Since \( \FF \) is quasicoherent, on a sufficiently small affine open set \( \spec A \) we have \( \FF\vert_{\spec A} \cong \widetilde{M} \) for some \( A \)-module \( M \). Then we simply define \( (T^n \FF)(\spec A) := T^n (M) \). Since the tensor product commutes with localization, this implies on any distinguished open subset \( D(f) \) that \( T^n \FF (D(f)) = (T^n (M))_f = T^n(M_f) \) so that by Important Exercise 13.3.D we obtain \( T^n \FF = \widetilde{ T^n(M) } \). An identical argument shows that \( \sym^n \FF \) and \( \bigwedge^n \FF \) are quasicoherent.

If \( \FF \) is rank \( r \), it is easy to see from the affine local description that the rank of \( T^n \FF \) is \( nr \). For those familiar with the multiset \( \left(\kern-.3em\left(\genfrac{}{}{0pt}{}{n}{r}\right)\kern-.3em\right) = { n + r - 1 \choose r }\), since repititions are allowed in \( \sym^n (V) \) we have that the dimension is precisely \( {n + r - 1 \choose r} \). Since repititions are not allowed in \( \bigwedge^n \FF \), the rank can be easily checked to be \( {n \choose r} \).

Proof:

Using the second approach mentioned in the hint (not typed out since it is the argument below), consider a small enough affine open set \( \spec A \) such that our exact sequence is of the form

Letting \( e_1, \dots, e_p \) be the standard basis for \( A^{\oplus p} \) and \( f_1, \dots, f_q \) be the standard basis for \( A^{\oplus q} \), we let \( e^\prime_1 \dots, e^\prime_p \) denote the images in \( A^{\oplus p + q} \) and \( f_1^\prime, \dots, f_q^\prime \) denote the lifts (since the last map is surjective). By exactness, we know that \( f_i^\prime \) is well-defined modulo any \( e_j^\prime \). Since

we define our filtration \( \GG_p\vert_{\spec A} \) as \( \GG_p := \bigoplus_{i= p}^r \sym^i \FF^\prime\vert_{\spec A} \otimes_{\OO\vert_{\spec A}} \FF^{\prime \prime}\vert_{\spec A} \). From this construction, it is clear that \( \GG_p / \GG_{p+1} = \sym^p \FF^\prime \vert_{\spec A} \otimes_{\OO_X \vert_{\spec A}} \FF^{\prime \prime }\vert_{\spec A}\).

Lastly we wish to show that the construction is independent of bases chosen. Let \( \overline{e}_1, \dots, \overline{e}_p \) be another choice of basis of \( A^{\oplus p} \), and \( T \) the change of basis matrix so that \( e_j = \sum_k T_{kj} \overline{e}_k \). Then if \( \overline{e}_{i_1} \otimes \dots \otimes \overline{e}_{i_r} \) is any new basis vector in \( \sym^r \FF^\prime \) (with \( 1 \leq i_1 \leq \dots \leq i_r \leq p \)), we clearly have by linearity of the tensor that

expands as a sum over our original basis \( e_{i_1} \otimes \dots \otimes e_{i_r} \) (for \( 1 \leq i_1 \leq \dots \leq i_r \leq p \) ). An identical argument may be used for the lifts \( f_j^\prime \)

Proof:

The idea is completely analogous to the previous proof; the main caveat is that when showing independence of basis, if \( \overline{e}_1, \dots, \overline{e}_p \) is another choice of basis of \( A^{\oplus p} \), then when expanding the sum

any terms \( e_{i_1} \otimes \dots \otimes e_{i_r} \) with repeated \( i_k \) will vanish by construction of the exterior algebra. Thus, our result will be in terms of \( e_{i_1} \otimes \dots \otimes e_{i_r} \) for \( 1 \leq i_1 < \dots < i_r \leq p \) which is the same as the original basis for \( \bigwedge^r A^{\oplus p} \).

To see that \( \det \FF = (\det \FF^{\prime} ) \otimes (\det \FF^{\prime \prime})\), consider our filtration

Now for any \( p \neq a \), notice that in

we must have either \( p > a \) or \( a + b - p > b \) so that the wedge product vanishes (as some basis vector must be repeated). Thus, \( \GG^0 = \GG^1 = \dots = \GG^a \) and \( \GG^p = 0 \) for \( p > a \) proving the claim.

Proof:

Restricting to a sufficiently small affine open subset, we again consider our \( \FF \) as locally of the form \( A^{\oplus n} \). Taking \( e_1, \dots, e_n \) as a basis, we define our map \(\bigwedge^r \FF \times \bigwedge^{n-r}\FF \to \bigwedge^{n} \FF\) by

By definition of the exterior algebra, if the \( \{ i_1, \dots, i_r \} \) and \( \{ j_1, \dots, j_{n-r} \} \) overlap then the map evaluates to \( 0 \). Otherwise, \{ i_1, \dots, i_r, j_1, \dots, j_{n-r} \} is some permutation \( \sigma \) of \( \{1, \dots, n \} \) by the pigeonhole principle, so we may reorder the image in such a way that \((e_{i_1} \otimes \dots \otimes e_{i_r}) , \ \ (e_{j_1}, \dots, e_{j_{n-r}}) \) is mapped to \( (-1)^{\textrm{sgn}\, \sigma} e_1 \otimes \dots \otimes e_n \). This clearly maps through to the tensor product since our map is literally defined using the tensor product (in fact, our map is simply the wedge product). By the tensor-hom adjunction, this induces an isomorphism \( \bigwedge^r \AA^{\oplus n} \to \textrm{Hom}( \bigwedge^{n-r} A^{\oplus n}, \bigwedge^{n} A^{\oplus n}) \). We will see in Exercise 13.7.B that the codomain is really \(( \bigwedge^{n-r} A^{\oplus n})^\vee \otimes \bigwedge^{n} A^{\oplus n} \)

Proof:

Let \( \phi_i : \FF_i \to \FF_{i+1} \) denote our maps and notice that by exacness of the original sequence, we get short exact sequences

By Exercise 13.5.F, we have \( \det \FF_i = \det ( \ker \phi_i ) \otimes \det (\ker \phi_{i+1}) \). Now the first map is injective by exacness, so \( \ker \phi_1 = 0 \). Moreover, we have a telescoping tensor so that the result must be \( \OO_X \) by Exercise 13.1.D

Proof:

Following the hint, we have a commuting diagram of the form

Now \( \gamma \) is obviously the identity, so by the snake lemma our exact sequence is

Thus, \( \ker \alpha \cong \ker \beta \) and \( \coker \alpha \cong \coker \beta \), so as \( \beta \) is a morphism between finite rank modules we see that \( \ker \alpha \) and \( \coker \alpha \) are finitely generated. Since \( K \) is finitely generated, this tells us \( \textrm{Im}\, \alpha \) is finitely generated so that by the first isomorphism theorem, \( \ker \phi \) must be finitely generated.

Proof:

The first two cases are immediate since localization is exact; e.g. if \( M \) is finitely presented, then there is an exact sequence

Since localization commutes with arbitrary direct sums

is exact. Now suppose \( M \) is coherent and we have an arbitrary map \(\phi_f : (A_f)^{\oplus p} \to M_f \). This lifts to a map \( \phi : A^{\oplus p} \to M \), so by assumption \( \ker \phi \) is finitely generated. Since \( \ker \phi \to (A_f)^{\oplus p} \to M_f \) vanishes by commutivity, there is a natural morphism \( \ker \phi \to \ker \phi_f \). Thus, we get a commuting diagram

Without even having to invoke the snake lemma, we know that \( \ker \phi_f \) must be the localization of \( ( \ker \phi)_f \) (again since localization is exact). Since \( \ker \phi \) is finitely generated \( A \)-module, \( \ker \phi_f \) is a finitely generated \(A_f\) module.

Proof:

To see that \( M \) is necessarily finitely generated, fix some \( m \in M \). Then in each \( M_{f_i} \), \( m/1 \) is the image of some \( \frac{a_1n_1}{f_i^{k_1}} + \dots +\frac{ a_rn_r}{f_i^{k_r}} \); multiplying by a sufficiently large power \( {N_i} \) of \( f_i \) gives us \( f_i^{N_i} m = a_1 n_1 + \dots + a_r n_r \) where the \( a_j \in A \) and \( n_j \in M \). Since the \( f_i \) generate the unit ideal, say \( b_1f_1 + \dots + b_n f_n = 1 \), we may exponentiate by \( N_1N_2\dots N_n \) and multiply by \( m \) to see that the collection of all such \( \{ n_i\} \) form a generating set for \( M \).

To see that finitely presented is a local condition, we know by the previous paragraph there must at the very least exist a finite surjection \( \phi : A^{\oplus n} \to M \). Consider \( \ker \phi \); the localization at each \( f_i \) is finitely generated by assumption and Exercise 13.6.A. However by the previous paragraph, being finitely generated is a local condition so that \( \ker \phi \) is itself finitely generated.

The idea for coherent \( M_{f_i} \) is the same; if \( \phi : A^{\oplus p} \to M \) is any morphism, the induced maps \( \phi_{f_i} : A_{f_i}^{\oplus p} \to M_{f_i} \) have finite kernels which are precisely the localizations of the kernels of \( \phi \) by exactness of localization. Again using the fact that finitely-generated is a local property we see that \( \ker \phi \) must be finitely generated.

Proof:

1. As indicated in the hint, we restrict to a sufficiently small affine open set \( U = \spec A \) so that \( \FF\vert_U \cong \widetilde{ M} \) for some finitely presented \( A \)-module \( M \) and \( \GG\vert_U \cong \widetilde{N} \). In particular we get an exact sequence of the form

   $$ \begin{CD} \OO^{\oplus q}\vert_U @>>> \OO^{\oplus p}\vert_U @>>> \FF\vert_U @>>> 0 \end{CD} $$

   Then similar to Exercise 1.6.G we may use the left-exactness of \( \textrm{Hom}_A(-, N) \) and the fact that \( \textrm{Hom}_A(A^{\oplus r}, N) \cong N^r \) to get an exact sequence

   $$ \begin{CD} 0 @>>> \textrm{Hom}( \FF\vert_U, \GG\vert_U ) @>>> \GG\vert_U^{\oplus p} @>>> \GG\vert_U^{\oplus q} \end{CD} $$

   Using the fact that quasi-coherent sheaves form an abelian category (they are closed under kernels and direct sums) so it follows that \( \textrm{Hom}( \FF\vert_U, \GG\vert_U ) \) is also quasicoherent.

2. The proof is identical to that above by Proposition 13.6.3 in the text.

3. This essentially follows from the affine-local construction and Exercise 1.6.F; to be precise we have on each sufficiently small open affine \( U = \spec A \) that \( \textrm{Hom}_A(M, -) \) is a left-exact functor. By Exercise 1.6.F(a) we know that this respects localization at each \( f \in A \), so by Exercise 13.3.D this gives us a left exact functor on the category of quasicoherent sheaves. The idea is similar for \( \mathscr{Hom}(-, \GG) \)

Proof:

Similar to the vector space case, we define a map \( \psi : \FF^\vee \otimes \GG \to \mathscr{Hom}(\FF, \GG) \) defined by \( \lambda \otimes g \mapsto ( f \mapsto \lambda(f) g ) \). On (sufficiently small) affine open sets \( U = \spec A \) where \( \FF\vert_U = \widetilde{A^{\oplus n}} \) and \( \GG\vert_U \cong \widetilde{N} \), which by standard algebra is always injective and surjective since \( \FF\) is finite rank. As usual, the inverse map is \( \psi^{-1} : \mathscr{Hom}(\FF, \GG) \to \FF^\vee \otimes \GG \) is defined on affine open subsets by taking the standard basis \( \{ e_i \} \) for \( A^{\oplus n} \) and setting

Proof:

As indicated in the hint, we assume \( X = \spec A \) and write \( \mathscr{H} = \widetilde{ A^{\oplus n} } \). As \( A^{\oplus n} \) is free and hence projective, the sequence

Let \( f : M \to N \) be our injection and \( s : A^{\oplus n} \to N \) be the lifting; by splitting we have \( \textrm{Im}(f) = \textrm{Im}(s)^\perp \) so that \( N \cong \textrm{Im}(f) \oplus \textrm{Im}(s) \). Then as \( N = M \oplus A^{m} \), any morphism \( \phi : N \to E \) restricts to \( M \) so that the map \( \mathscr{Hom}(\GG, \mathscr{E}) \to \mathscr{Hom}(\FF, \mathscr{E}) \) is surjective, which is equivalent to exactness on the right.

Proof:

Following the hint, suppose \( X = \spec A \) and \( \FF = \widetilde{M} \) for some finite-type \(A \)-module \( M \). Let \( m_1, \dots, m_n \) be a set of generators for \( M \). Then the images of our generators \( (m_i)_p \) in \( \FF_p \) necessarily generate \( \FF_p \), so \( \FF_p = 0 \) if and only if \( (m_1)_p, \dots, (m_n)_p \) vanish. Therefore, we may describe \( \textrm{Supp} \FF = \bigcup_i \textrm{Supp}(m_i) = \bigcup_i V(\textrm{Ann}(m_i)) = V \Big( \bigcap_i \textrm{Ann}(m_i) \Big) = V(\textrm{Ann} \, M)\) and thus it is a closed subset.

The hint gives a somewhat unintuitive example of when the support is not closed; instead, consider extension by zero for \( j : U \hookrightarrow X \) given by the sheafification of

Since sheafification does not affect stalks, \( (j_! \FF)_x = 0 \) iff \( x \in U \) so that \( \textrm{Supp} \, \FF = U \) which is open.

In the example given, the punchline is that by Exercise 1.3.F localization commutes with arbitrary direct sums, but not necessarily arbitrary products. Thus, for any prime \( \pp \subset \C[t] \), if we set \( M = \bigoplus_{a \in \C} \C[t]/(t- a) \) then

when \( \pp \) is maximal of the form \( (t - b) \), this obviously has nonempty support since it is supported somewhere in the arbitrary direct sum, so the support is the union of all closed points which itself is not closed (it is not the zero locus of any polynomial).

Proof:

Since the question is local by nature, we may simply reduce to the case that \( X \) is already affine, say \( X = \spec A \) and \( p = [\pp] \). Since \( \FF \) is finite-type quasicoherent, there exists some finitely generated \( A \)-module \( M \) such that \( \FF \cong \widetilde{M} \). If we let \( m_1, \dots, m_r \) denote the generators of \( M \) as an \(A\)-module, then they must also generate \( M_\qq\) for any point \( [\qq] \) so we must have

for some \( c_{ij} \in A_\pp \) (as the \( a_1\vert_p, \dots, a_n\vert_p \) generate \( \FF_p \) by assumption). If we let \( f \) denote the product of the denominators of the \( c_{ij} \), then we may continue to write

for some \( d_{ij} \in A \) so that the \( a_1, \dots, a_n \) continue to generate \( \FF \) in the neighborhood \(D(f)\).

Proof:

Similar to the previous exercise, we may simply reduce to the case that \( X = \spec A \) with \( p = [\pp] \). Since \( \FF \) is quasicoherent of finite type, we may write \( \FF = \widetilde{M} \) for some \( A \)-module \( M \). Now by assumption, \( M_\pp \cong A^{\oplus n}_\pp \) for some \(n \), so we may take \( a_1, \dots, a_n \in A \) such that their images generate the stalk \( M_\pp \). By the previous exercise, these elements remain generators on some distinguished open subset \( Y = D(f) \subset \spec A \), giving us a surjection \(\phi\vert_Y : \OO^{\oplus n}_Y \to \FF\vert_Y \). By assumption, \( \FF \) is a finitely presented sheaf so that the \( \ker \phi\vert_Y \) is finite type quasicoherent sheaf (the category of quasicoherent sheaves is closed under kernels). By Exercise 13.7.D, \( Z = \textrm{Supp}\,\ker \phi\vert_Y \) is closed — since the map is by assumption an isomorphism at \( p \), \( \ker (\phi\vert_Y)_\pp = 0 \) so that \( p \notin Z \). Thus, \( FF \) is locally free on \( Y \backslash Z \)

Proof:

1. Following the hint in the text in the paragraph below the question, we note that by the results of §12.5 the local rings \( \OO_{X, p} \) of a regular curve are necessarily discrete valuation rings and thus PIDs. By Important remark 12.5.14, the structure theorem for finitely generated modules over PIDs tells us that every such module is a finite direct sum of cyclic modules. Thus, if we let \( p \in X \) be a point on our curve and \( U = \spec A \) an affine neighborhood with \( p = [\pp] \), then as our torsion free sheaf \( \FF \) is assumed to be coherent, \( \FF_p \cong M_\pp \) is a finitely generated, torsion free \( A_\pp \)-module. By the structure theorem above, it may be decomposed as a finite sum of \( A_\pp \) and \( A_\pp / (t^r) \) (for our uniformizer \( t \)) — by torsion freeness there can be no such \( A_\pp / (t^r) \) terms so that \( \FF_p \) is in fact free. By the previous exercise, it is necessarily free in an open neighborhood and thus locally free.

2. Since regular curves are implicitly locally Noetherian, quasicompactness of the curve implies that is in fact Noetherian. Since closed subsets of Noetherian schemes have a finite number of irreducible components, the claim should follow (by Exercise 13.7.D to show that it is closed in the first place of course).

3. Similar to part (a), we know that the stalks of \( \OO_{X, p} \) for a regular curve \(X\) must be a PID; therefore, by the structure theorem for finitely generated modules over a PID we get a split exact sequence

   $$ 0 \to (\FF_p)_{tors} \to \FF_p \to (\FF_p)_{lf} \to 0 $$

   Again we may assume without loss of generality that our curve is in fact affine, say \( X = \spec A \) so that \( \FF = \widetilde{M} \) for some finitely presented \( A \)-module \( M \). Then \( \FF_p \cong A_\pp \oplus \dots \oplus A_\pp \oplus A_\pp / (t^{r_1}) \oplus \dots \oplus A_\pp / (t^{r_n})\), so we simpy set \( (\FF_p)_{lf} \) to be the sum of the \( A_\pp \oplus \dots \oplus A_\pp \), notice that this gives us a surjective morphism on some neighborhood of \( p \) and then take the kernel of this morphism which at the stalk \( p\) is simply \( A_\pp / (t^{r_1}) \oplus \dots \oplus A_\pp / (t^{r_n})\). One can show that this definition of \( \FF_{tors} \) as the kernel agrees with the usual torsion part of a sheaf.

Proof:

From Exercise 13.7.D we recall that the sheaf associated to \( M = \C[t] / (t) \) is only supported at the origin. To be precise, if \( \pp \subset \C[t] \) is any other prime ideal then \( M_\pp \) is the zero module precisely when there exists some \( s \notin \pp \) such that \( s \cdot f = 0 \) in \( M \) for every \( f \in M \); in general, the prudent choice for \( s \) would be \( t \). Thus, at every closed point other than \( (t) \), we have that the rank of \( \widetilde{ k[t]/(t) } \) is 0. More so, by considering the generic point \( [(0)] \) we certainly have that \( t \notin (0) \) as it is a free generator, so that \( t \cdot f = 0 \) in \( M \) for every \( f \in M \) showing that the rank of \( \FF\) is zero everywhere except the closed point corresponding to the origin.

Proof:

As expected, since rank is a stalk-local notion we may restrict our attention to \( X = \spec R \), and write \( \FF = \widetilde{M} \) and \( \GG = \widetilde{N} \) for some \( R \)-modules \( M, N \) (as they are quasicoherent). We will not prove the above relations as they are standard algebra facts / exercises, but using them we simply have

so that \( \rank \( \FF \oplus \GG \) = \rank \FF + \rank \GG \) follows by taking the dimension of the above \( \kappa(p) \)-vector spaces. The second equality is similar.

Proof:

Fixing a point \( p \in X \), we know that since \(\FF\) is finite type \( n = \dim_{\kappa(p)} \FF_p \otimes_{\OO_{X,p}} \kappa(p) \) is finite. By restricting to a sufficiently small affine open set \( U = \spec A \) with \( \FF\vert_{\spec A} \cong \widetilde{M} \) for some \(A \)-module \( M \), suppose we let \( a_1, \dots, a_n \in \FF(U) \) be such that their images \( a_1\vert_p, \dots, a_n\vert_p \) in \( \FF\vert_p \) generate the fibre. By Exercise 13.7.E, there is a possibly smaller open affine \( V \subset U \) such that \( a_1\vert_V, \dots, a_n \vert_V\) are generators of \( \FF\vert_V \). Then \( a_1\vert_q, \dots, a_n\vert_q \) form a spanning set for \( \FF\vert_q \) for every \( q \in V \) so that \( \rank(\FF_q) = \dim_{\kappa(q)} \FF\vert_q \leq n \) — thus if we let \( f : X \to \Z \) be the function \( q \mapsto \rank \FF_q \), \( f^{-1}(-\infty, n] \) contains the neighborhood \( V \ni p \) so that it is open.

Proof:

1. We follows the proof almost verbatim; since local freeness is an affine-local notion, we simply need to reduce to the case that \(X = \spec B\). Fixing some \( p = [\pp] \), we let \( m_1, \dots, m_n \) be (global in this case) sections of \( \FF(X) \) that generate \( \FF\vert_p \) as a \( \kappa(p) = K (B / \pp) \) vector space, then by Geometric Nakayama’s Lemma 13.7.E there is some open neighborhood \( V = \spec A \) in which \( m_1\vert_V, \dots, m_n\vert_V \) are generators — furthermore, let \( M \) be the \(A\) module with \( \FF\vert_V \cong \widetilde{M} \). Taking \( \phi : A^{\oplus n} \to M\vert_V \) denote the surjection \( (r_1, \dots, r_n) \mapsto \sum r_i m_i \), suppose to the contrary that \( \phi \) is not an isomorphism ( i.e. \( \ker \phi \neq 0 \)). Then we can find some nonzero element \( (r_1', \dots, r_n') \in \ker \phi \subset A^{\oplus n} \); without loss of generality (by possibly applying a change of basis), we may assume \( r_1' \neq 0 \). Then as \( r_1' \) is not the zero function on \( \spec A \) and \( \spec A \) is assumed to be reduced, we can find some point \( q = [\qq] \) such that \( r_1'(q) \neq 0 \), i.e. \( r_1' \notin \qq \) ( recall that in the reduced case there are nonzero functions that vanish at every point, e.g. \( \epsilon \) in \( k[\epsilon] / (\epsilon^2) \) ). Then \( r_1 \) is invertible in \( A_\qq \) so that \( (r_2, \dots, r_n) \mapsto m_1 + \sum_{i=2}^n \frac{r_i}{r_1} m_i\) is a surjection and thus \( \rank (\FF_q) \leq n-1 \) which contradicts the assumption of constant rank. Thus, \( \phi \) is an isomorphism on our affine neighborhood \( \spec A \) so that \( \FF \) is indeed locally free.

   Now suppose \( X \) is integral, \( \FF\) is finite type quasicoherent, and consider the subsets \( U_k = \{ p \in X \mid \rank \FF_p \leq k \} \) which are open by the previous exercise. Let \( \eta \) denote the unique generic point of \( X \) and suppose \( n = \rank \FF_\eta \). Then \( \eta \in U_n \backslash U_{n-1}\), which itself is an integral locally closed subscheme on which \( \FF \) has constant rank ( by construction). Then by the paragraph above, we know that \( \FF \) is locally free on \( U_n \backslash U_{n-1} \) — since any open subsets are always dense on irreducible spaces, the claim follows.

2. As indicated above, on non-reduced schemes it is possible for a function to have value \( 0 \) at all points without being the zero function (in particular this tells us the function is in the nilradical). In the example given, consider \( \spec A \) which consists of the double point \( p = [(x)] \). Then \( \kappa(p) = K(A / (x)) = k\) and \( M_{(x)} = \{ \frac{a}{b + cx} \mid a, b, c \in k, b \neq 0\} \). However, since \( M \) is an \(A\)-module by sending \( x \mapsto 0 \), this is really just isomorphic to \( k \) as an \(A\)-module. Then \( \rank \widetilde{M} \) is clearly 1 over all points (since there is one point) but it is not locally free since \( k \neq k[x] / (x^2)\).

Proof:

Since finite morphisms are implicitly affine, we may assume without loss of generality that \( X = \spec A \) and \( Y = \spec B\). Let \( \pi^\sharp : B \to A \) denote the morphism of global sections, and suppose \( \pp \subset A \) pulls back via \( \pi^\sharp \) to \( \qq \subset B \), i.e. \( \pi([\pp]) = [\qq] \). Then \( (\pi_\ast \OO_X)_\qq \) is simply \( A_\pp \) which we now interpret as a \( B \)-module via \( B \to A \to A_\pp \). In particular, \( A_\pp \otimes_{B_\qq} B_\qq / \qq B_\qq \) is a finite dimensional \( B_\qq / \qq B_\qq \)-vector space whose dimension is the same as the finite rank of \( A \) as a \(B\)-module.