Solutions to The Rising Sea (Vakil)

Proof:

Suppose, to the contrary, that there is some other maximal ideal \( \mm' \neq \mm_p \) contained in \( \OO_p \). Then we can pick some germ \( (f, U) \in \mm' \) not in \( \mm_p \) — since \( (f, U) \) is not an element of \( \mm_p \), we have by definition that \( f(p) \neq 0 \). In particular, this implies that \( 1/f \) is differentiable in a (possibly smaller) neighborhood of \(p\), given explicitly by \( d(1/f)\vert_p = -f'(p)/ [f(p)]^2 \). However, this implies that \( (f, U) \) is a unit in the ring \( \OO_p \) so that \( \mm' = \OO_p \) — a contradiction. Thus, \( \mm_p \) is the unique maximal ideal so that \( \OO_p \) is by definition a local ring.

Proof:

First, the cotangent space is simply defined as the dual of the tangent space — in our case, if we denote our smooth (real) manifold \(M\), then \( T_p^*M = \Hom_\mathbb{R}(T_pM, \mathbb{R} ) \). If we assume without loss of generality \( p = 0 \) and let \( \frac{\partial}{\partial x_1}\vert_0, \dots, \frac{\partial}{\partial x_n}\vert_0 \) be basis for \( T_0M \), then the natural dual basis is denoted \( dx_1\vert_0, \dots, dx_n\vert_0 \). By Taylor's theorem, we have that for any \( f \in \OO_p \), \( f(x_1, \dots, x_n) = f(0) + \sum_{i=1}^n \frac{\partial f}{\partial x_i}(0) x_i + O(x^2) \), where the \( x_1, \dots, x_n \) are local coordinates of \(M\) near \(0\). Clearly for any \( f \in \mm_p \) the \( f(0) \) term vanishes; in particular, we get an \( \mathbb{R}\)-linear map

Thus, if \( g \in \ker \phi \), then the Taylor expansion only contains second order terms and higher — thus, pulling out a factor of \(x_i\) from the Taylor expansion, \(g\) can be represented as the product of two functions which vanish at \( p = 0 \). In other words, \( \ker \phi = \mm_p^2 \) so that by the first isomorphism theorem we have that \( T_p^* M \cong \mm_p / \mm_p^2 \).

Proof:

Let us denote the category of open sets of \(X\) as \( Top_X \), and let \( \FF : Top_X \to \Set \) be a contravariant functor. Then for each open set \( U \subset X \) (i.e. object in \( Top_X \)), we get a set \( \FF(U) \). In addition, let \( \iota : U \to V \) be a morphism in \( Top_X \) (i.e. \( U \subset V \) ) — since \( \FF \) is a contravariant functor, we also get a map \( \FF(\iota) : \FF(V) \to \FF(U) \) (which can simply be interpreted as the "restriction map"). By the axioms of a functor, we know that \( \FF \) must preserve the identity.

Lastly, in order to show that the restriction maps \( U \subset V \subset W \) commute, it is crucial that we note \( Top_X \) is defined (in Vakil Example 1.2.9) to only have a single morphism \( U \hookrightarrow V\) if \( U \subset V \) — thus, the inclusion map is always necessarily the unique inclusion map (this should distinguish inclusions from embeddings, if that was not already clear, as there can be many ways to embed a space.) Consequently, if \( i : U \to V \) and \( j : V \to W \) are inclusion maps, then \( j \circ i : U \to W \) is the unique inclusion map of \( U \subset W \). Then as \( \FF \) is contravariant, we have \( \FF(i) : \FF(V) \to \FF(U) \), \( \FF(j) : \FF(W) \to \FF(V) \), and

which gives the necessary commuting diagram.

Proof:

For (a), let \( \mathscr{B} : Top_X \to \Set \) denote the presheaf taking an open set \(U \subseteq \mathbb{C} \) to the set \( \mathscr{B}(U) \) of bounded functions on \( U \). Consider the open cover \( \{ U_R \}_{R > 0} = \{ B_R(0) : R > 0 \}\) of balls centered at the origin. For each \( R > 0 \), the identity function \( f_R(z) = z \) is trivially bounded on \( B_R(0) \) by \(R\); however, gluing all these \(f_R\) together gives us the identity on \(C\) which is clearly unbounded.

For (b), take the open cover of branch cuts of \( \mathbb{C} \) (i.e. \(\mathbb{C} \) minus some ray centered at the origin.) This open cover is clearly parameterized by \( S^1\), so we will refer to each open set as \( U_\theta = \mathbb{C} \backslash \{ te^{i \theta} \mid t \geq 0 \} \) — for instance, the principal branch is commonly given on \( U_\pi \). Now on each \( U_\theta \), there is a well-defined logarithm and thus \( k^{\textrm{th}} \) root — in particular on each \( U_\theta \) there exists a square root of the identity function which we will call \( g_\theta(z) : U_\theta \to \mathbb{C} \), so that \( g_\theta^2(z) = z \). However, this again does not satisfy the gluability axiom of a sheaf, since if \( g : \mathbb{C} \to \mathbb{C} \) were a global square root of the identity, then on the unit circle we have \( e^{i \pi} = 1 = e^{-i \pi} \) but \( i = e^{i \pi/2} = g(e^{i \pi}) \neq g( e^{- i \pi ) = e^{-i \pi / 2} = -i \).

Solution:

Notice that there is always a map \( U_j \hookrightarrow \bigcup_{i \in I} U_i \) whenever \( j \) is an element of the index set \( I \). Therefore, the union of an open cover may be interpreted as a certain colimit. As a contravariant functor, a presheaf \( \FF \) covariantly maps each morphism \( U_j \hookrightarrow \bigcup_{i \in I} U_i \) to a morphism \( \FF( \bigcup U_i) \to \FF(U_j) \). Thus, the sheaf condition may be considered as \( \FF \) taking certain colimits to limits. More specifically, since we always have morphisms \( \iota_1 : U_i \cap U_j \hookrightarrow U_i \), \( \iota_2 : U_i \cap U_j \to U_j \) which induce "restriction maps" \( \FF(\iota_1) : \FF( U_i) \to \FF(U_i \cap U_j) \) and \( \FF(U_j) \to \FF(U_i \cap U_j) \), the requirement that we have an \( f_i \in \FF(U_i) \) for each \(i \) satisfying \( \res{U_i}{ U_i \cap U_j} f_i = \res{U_j}{ U_i \cap U_j} f_j \) is equivalent to saying a certain cone commutes, so that the limit exists. By the universal property of our limit, this global section \( f \in \FF(U) \) is unique, which is equivalent to the identity axiom.

It is important to distinguish in the paragraph above that the sheaf condition does not mean the presheaf takes all colimits to limits — this holds only when the family \( \{ U_i \} \) is a subposet closed under intersections (otherwise, our construction above wouldn't work). This is further explained by this stackexchange answer.

Proof:

For both parts (a) and (b), we have presheaves of functions defined on some topological spaces that we wish to show satisfy the sheaf condition. For each case, the identity axiom is somewhat trivial, as two functions are equal if and only if they are equal at each point (which is independent of restriction to a neighborhood at that point). In other words, if \( \{ U_i \} \) is our open cover of \( U \) and we have two (continuous / differentiable / smooth / etc) functions \( f_1, f_2 \) defined on \( U \), then \( f_1 = f_2 \) iff \( f_1(p) = f_2(p) \) at each \( p \in U \). But for each \(p\), since \( \{ U_i \} \) covers \( U \) we must have \( p \) contained in some \( U_j \) — assuming the hypothesis of the identity axiom, since we know that \( f_1 \vert_{U_j} = f_2\vert_{U_j} \) then \( f_1(p) = f_2(p) \). Since \( p \in U \) was arbitrary, \( f_1 \) and \(f_2\) agree pointwise so they are necessarily the same function. Thus, it suffices to prove for both parts (a) and (b) that each presheaf satisfies that gluability axiom.

1. Let \( \{ U_i \} \) be an open cover of \( U \), \( \FF \) denote the presheaf of differentiable functions, and suppose for each \( i \) we have a function \( f_i \in \FF(U_i) \) such that the restrictions agree on intersections \( U_i \cap U_j \). Define a function \( f : U \to \mathbb{R} \) pointwise by \( f(p) = f_i(p) \) whenever \( p \in U_i\) — this is necessarily well-defined for each \( p \in U \) since \( \{ U_i \} \) covers \(U\), and if \( U_j \) is another set with \( p \in U_j \), then \( f_i \vert_{U_i \cap U_j} = f_j \vert_{ U_i \cap U_j } \) by hypothesis so that \( f_i(p) = f_j(p) = f(p) \). Therefore, if we are able to show that \( f \) is differentiable, then the gluability axiom will be satisfied (the argument looks completely identical if we had instead defined \( \FF \) to be the sheaf of smooth functions \( U \to \mathbb{R} \)).

   Recall \( f \) is differentiable on \( U \) if and only if, for each \( p \in U \) there exists some neighborhood \( W \subset U \) of \( p\) on which the derivative \( f'(x) \) exists. Now for arbitrary \( p \in U \) we know that \( p \) is an element of some \( U_i \); by construction, we have that \( f(p) = f_i(p) \). But since \( f_i(p) \) is differentiable, there exists some neighborhood \( W' \subset U_i \) of \(p\) such that \( f_i'(x) \) exists on \( W \). But since \( W' \subset U_i \subset U \), by commutivity of the restriction map it follows that \( f'(x) \) exists on \( W \) so that \( f \) is differentiable on \( U \).

2. Let \( X \) be a topological space, \( U \subseteq X \) open, and suppose \( \{ U_i \} \) is some open cover of \( U \). Let \( \FF \) denote the presheaf of continuous real-valued functions. By our first paragraph, we know that the identity axiom holds — we again use the construction of part (a) above to define a function \( f : U \to \mathbb{R}\) pointwise via \( f(p) = f_i(p) \) where \( U_i \) is some neighborhood of \(p\). This construction is well-defined by the previous argument. Therefore, it suffices to show that \(f\) is continuous at each point \( p \in U \). Fix some \( p \in U \), and let \( V \subset \mathbb{R}\) be an open neighborhood of \( f(p) (= f_j(p)\) for any \( j \) with \( p \in U_j \) ). Then it is easy to show from our construction of \(f\) that

   $$ f^{-1}(V) = \bigcup_{ i \in I } f^{-1}_i (V) $$

   Since each \( f_j \) is continuous, \( f_j^{-1} (V) \) is open in \(U_j\) (and therefore \(U\)). Thus, \( f^{-1}(V) \) is open in \( U \) since a topology is closed under arbitrary unions.

Proof:

To show the identity axiom, let \(U\) be open and suppose \( \{ U_i \} \) is an open cover. Suppose we fix some sections \( f_1, f_2 \in \FF(U) \) such that the restrictions \( f_1 \vert_{U_i} = f_2 \vert_{U_i} \) for all open sets \( U_i \) in our cover. Since \( f_1\) and \(f_2\) are considered as maps, then as before we have that \( f_1 = f_2 \) if and only if \( f_1(p) = f_2(p) \) for all points \( p \in U \). If we fix \( p \in U \), then as an open cover there must exist some \( U_j \) such that \( p \in U_j \). But since \( f_1 \vert_{U_j} = f_2 \vert_{U_j} \) we necessarily have that \( f_1(p) = f_2(p) \) as desired.

For the gluability condition, let \(f_i \in \FF(U_i) \) for each \(i\) and assume the hypothesis that \( f_i \vert_{U_i \cap U_j} = f_j \vert_{U_i \cap U_j} \) for all \( i, j\). Define a function \( f : U \to S \) pointwise by \( f(p) = f_i(p) \) whenever \( p \in U_i \). As before, this is well-defined since the \(\{ U_i\} \) cover \(U\), and if \( U_j \) is another neighborhood of \( p \) then \( p \in U_i \cap U_j \) so our supposition tells us \( f_i \vert_{U_i \cap U_j}(p) = f_j\vert_{U_i \cap U_j}(p)\). Therefore, it suffices to show that \( f \) is locally constant — however, this is obvious since \( f_i \) is locally constant in an open neighborhood \( W \subset U_i \) of \(p\), so that \( f_i(x) = f_i(p) \) for all \(x \in W\). But then by construction \( f(x) = f(p) \) for all \(x \in W\) as well and \( W \) is open in \(U\) as an open subset of an open subset, so \( f \) is locally constant.

Proof:

Our proof of Exercise 2.2.D(b) did not rely on any of the topological properties of \( \mathbb{R} \) (that is, we used the general definition of a continuous map between topological spaces). By replaceing \( \mathbb{R} \) with \(Y\) in the proof, the result follows.

Proof:

1. Use \( \FF \) to denote the sheaf of continuous sections of \( \mu \), and let \( U \) be an open set in \(X\) with \( \{ U_i \} \) as an open cover. Moreover, suppose we have \( s_i \in \FF(U_i) \) for each \( i \) such that \( s_i \vert_{U_i \cap U_j} = s_j \vert_{U_i \cap U_j} \) for all \( i, j \). As before, we define \( s : U \to Y \) pointwise by \( s(p) = s_i(p) \) whenever \( p \in U_i \). By the previous problem, we know that \( s \) is well-defined and continuous — therefore, it suffices to show that \( \mu \circ s = id \vert_U \). However, this is obvious since the two functions agree iff they agreee pointwise; by fixing \( p \in U \), we must have \( p \in U_j \) for some \(j\). Then

   $$ \mu \circ s(p) = \mu \circ s_i(p) = id\vert_{U_i}(p) = p $$

   Since \( p \) was arbitrary, the result follows.

2. Let \( \FF(U) \) denote the set of continuous maps \( U \to Y \). For any \( V \subset U \), the restriction map \( \FF(U) \to \FF(V) \) is clearly a homomorphism, since for any \( p \in V \) and \( f, g \in \FF(U) \)

   $$ (f + g)\vert_V (p) = f(p) + g(p) = f\vert_V (p) + g\vert_V(p) $$

   Each \( \FF(U) \) also has a group structure, since multiplication of \( f, g : U \to Y \) may be done pointwise, and if \(1_Y \) is the identity, then the map \( id_U : U \to Y \) given by \( p \mapsto 1_Y \) is continuous and necessarily the identity. Additionally, inverses may be taken pointwise — if \( \iota : Y \to Y \) represents the (continuous) inversion map on our topological group \(Y\), then for any continuous map \( f : U \to Y \), the inverse is the continuous map \( \iota \circ f \). Thus, \( \FF(U) \) carries the structure of a group, so the result follows from Exercise 2.2.F.

Proof:

We first need to show that \( \pi_* \FF \) is indeed a presheaf on \( Y \). From Exercise 2.2.A, it would also suffice to describe \( \pi_*\FF \) as a contravariant functor from the category of open subsets of \(Y\), \( Top_Y \). Proceeding with this line of logic, we define a functor \( \pi^{-1} : Top_Y \to Top_X \) as follows: for any open set \( V \subset Y \), \( \pi^{-1}(V) \) is an open subset of \(X\) since \(\pi\) is continuous. Moreover, if \( W \subseteq V\) as open subsets of \(Y\), then preimages preserve inclusion so that \( \pi^{-1}(W) \subset \pi^{-1}(V) \). Thus, our functor \( \pi^{-1} \) covariantly transforms inclusions \( j : W \hookrightarrow V \) to inclusions \( \pi^{-1} \circ j \circ \pi : \pi^{-1}(W) \hookrightarrow \pi^{-1}(V) \). Since \( \FF \) is a contravariant functor \( Top_X \to Set \) and \( pi^{-1} \) is a covariant functor \( Top_Y \to Top_X \), \( \pi_*\FF(V) := \FF ( \pi^{-1}(V) ) = \FF \circ \pi^{-1}(V) \) is a contravariant functor \( Top_Y \to \Set \) and therefore a presheaf.

Next suppose that \( \FF \) is a sheaf, and let \(V\) be an open subset of \(Y\) and let \( \{ V_i \} \) be an open cover of \(V\). Take \( U = \pi^{-1}(V) \), and for each \( i \) let \( U_i = \pi^{-1}(V_i) \) – since the preimage preserves inclusion and unions, \( \{ U_i \} \) covers \( U \). If \( f_1, f_2 \in \pi_* \FF(V) = \FF(U)\) with \( f_1 \vert_{V_i} = f_2\vert_{V_i} \) for each \(i\), then \( f_1 \circ \pi^{-1} \vert_{V_i} = f_2 \circ \pi^{-1}\vert_{V_i} \) for each \(i\) — by construction of our open cover \( \{ U_i \} \) this means \( f_1 \vert_{U_i} = f_2 \vert_{U_i}\) for all \(i\). Since \(\FF\) is a sheaf, we necessarily have that \( f_1 = f_2 \) Similarly, if we have some \( f_i \in \pi_*\FF(V_i) \) for each \( i \) with \( f_i \vert_{V_i \cap V_j} = f_j \vert_{V_i \cap V_j} \) for each \(i, j\), then \( f_i \in \FF(U_i) \) for each \( i \) with \( f_i \vert_{U_i \cap U_j} = f_j \vert_{U_i \cap U_j} \) since preimages preserve intersections. Thus, there exists some \( f \in \FF(U) \) with \( f\vert_{U_i} = f_i \).

Proof:

For the sake of brevity, we will tackle this problem using colimits. Suppose both colimits \( (\pi_*\FF)_q := \varinjlim_{W \ni q} \pi_* \FF(W) \) and \( \FF_p := \varinjlim_{U \ni p} \FF(U) \) exist, and let \( W \hookrightarrow V \) both be open neighborhoods of \( q \) in \(Y\). Then \( \pi^{-1}(W) \hookrightarrow \pi^{-1}(V) \) are open neighborhoods of \( p \) in \(X\), so we have canonical maps \( \FF( \pi^{-1}(V) ) \to \FF_p \) and \( \FF( \pi^{-1}(W) ) \to \FF_p \) that commute with the restriction map \( \FF( \pi^{-1}(V) ) \to \FF( \pi^{-1}(W) ) \) (by definition of the colimit). Since our neighborhoods \( W \hookrightarrow V\) of \( q \) were arbitrary, by the universal property of the colimit these maps must uniquely factor through

In other words, there is a natural morphism \( ( \pi_* \FF )_q \to \FF_p \) as desired.

Proof:

Unwinding our definitions in the language of colimits, \( \FF \) is a sheaf of abelian groups such that for each morphism \( U \hookrightarrow V \) the following diagram commutes

Assuming all of our necessary colimits exist, then we have canonical maps \( \FF(V) \to \FF_p \) and \( \FF(U) \to \FF_p \) which commute with the restriction map:

By composing with our action / \( \OO_X \)-module structure, we get maps \( \OO_X(V) \times \FF(V) \to \FF_p \) for each neighborhood \(V\) of \(p\) which commute with our restriction maps (shown below).

Then by the universal property of the colimit, these maps uniquely factor through \( \varinjlim \left( \OO_X(V) \times \FF(V) \right) \). By the dual of Exercise 1.6.J, colimits commute with colimits — in particular, colimits commute with coproducts. However, since the category of modules over a ring is an abelian category, our products and coproducts coincide. Thus, the composed maps above really factor through

In other words, there is a unique morphism \( \OO_{X, p} \times \FF_p \to \FF_p \) that commutes with our maps

for each open set \(V\). But these are simply the restriction of our \( \OO_X(V) \)-module structure to \( \FF_p \), so that the map \( \OO_{X,p} \times \FF_p \to \FF_p \) satisfies the axioms of a \( \OO_{X,p} \)-module structure (i.e. preserves identity, composition).

Proof:

As before, we consider \( \GG_p \) as the colimit \( \varinjlim_{U \ni p} \GG(U)\). In particular, for any open neighborhoods \( U \subset V \) of \(p\) (i.e. morphism \( U \hookrightarrow V \) ), we get a commuting diagram

As with our previous exercise, we may consider the composition maps

Since morphisms of sheaves are natural morphisms, the following diagram commutes (i.e. is independent of path from \( \FF(V) \) to \( \GG_p \)):

In particular, for every open neighborhood \( W \) of \( p \) we get a morphism \( \FF(W) \to \GG_p \) that commutes with restriction maps, so that by the universal property of the colimit there is a unique map \( \phi_p : \FF_p \to \GG_p \) so that \( \phi \) factors through \( \FF_p \).

Proof:

From Exercise 2.2.H, we know that \( \pi_* \) is at the very least a map of objects — that is, for any (pre)sheaf of sets over \(X\) \( \FF \in \Set_X \), \( \pi_*\FF \) is a (pre)sheaf of sets over \(Y\). It suffices to prove that \( \pi_* \) maps morphisms of sheaves on \(X\) to morphisms of sheaves on \(Y\) — that is, it preserves natural transformations.

Suppose \( \FF \) and \(\GG\) are sheaves of sets over \(X\) and \( \phi : \FF \to \GG \) is a morphism of sheaves; that is, for \( U \hookrightarrow V \) open in \(X\), we get a commuting diagram

Now if \( W \hookrightarrow W' \) are open sets in \( Y \), we know that the preimage \( \pi^{-1} \) preserves inclusion so that the restriction map \( \pi_*\FF(W') \to \pi_*\FF(W) \) is simply the restriction map \( \rho_{\pi^{-1}(W), \pi^{-1}(W') } \). For each open set \( W \subset Y \), we define \( \pi_*\phi(W) : \pi_*\FF(W) \hookrightarrow \pi_*\GG(W) \) by \( \pi_*(\phi)(W) = \phi( \pi^{-1}( W) ) \). Since our original diagram commutes, this yields a natural transformation so that \( \pi_*\phi \) is a morphism of sheaves over \(Y\). In particular, \( \pi_* \) preserves the identity by construction, and if

are morphisms, then \( \pi_*(\psi \circ \phi) (V) = \psi \circ \phi( pi^{-1}(V) ) = \psi(\pi^{-1}(V)) \circ \phi(\pi^{-1}(V)) = \pi_*\psi \circ \pi_* \phi (V) \) so that \( \pi_* \) is indeed a covariant functor.

Proof:

Before all else, we must show that \( \Hom(\FF, \GG) \) is a presheaf — for that, we must define the restriction map. Let \( U \subset V \) be open sets in \(X\), and label the inclusion morphism \( \iota : U \hookrightarrow V \). By the previous exercise, \( \iota_* \) gives a functor \( \Set_V \to \Set_U \) Additionally, we define \( \alpha : \FF \vert_U \to \FF \vert_V \) as follows: for any open \(W \subset U\), we also have \(W\) is an open set of \( V \) so for \( \sigma \in \FF\vert_U(W) \) we let \( \alpha(\sigma(W)) = \sigma(W) \). Thus, for any morphism of sheaves \( \tau : \FF \vert_V \to \GG \vert_V \) we define the restriction \( \res{V}{ U}(\tau) \) by \( \iota_* \circ \tau \circ \alpha \). In other words, the outside rectangle

commutes whenever the middle square commutes. It is easy to see that the restriction maps necessarily commute since the inclusions \( U' \hookrightarrow U \hookrightarrow V \) commute, so that \( (\iota \circ j)_* = \iota_* \circ j_* \) is the unique pushforward (where \( j : U' \hookrightarrow U \)).

To see that \( \Hom (\FF, \GG) \) is indeed a sheaf, suppose that \( U \subset X \) is open and \( \{ U_i \} \) is an open cover of \(U\). First tackling the identity axiom, let \( \sigma_1, \sigma_2 \in \Hom(\FF, \GG)(U) \) and suppose \( \sigma_1 \vert_{U_i} = \sigma_2 \vert_{U_i} \) for each \( i\) — now fix some input \( f \in \FF(U) \). Then for each \( i \), \( \sigma_1\vert_{U_i}(f\vert_{U_i}) = \sigma_2\vert_{U_i}(f\vert_{U_i}) \) in \( \GG\vert_{U_i}\) — since \( \GG \) is itself a sheaf, we have that \( \sigma_1(f) = \sigma_2(f) \). Since \( f \in \FF(U) \) was arbitrary, \( \sigma_1 = \sigma_2 \).

The gluability axiom effectively works the same, where we fix an argument \( f \in \FF(U) \) and use the sheaf structure on \(\GG\) to prove the sheaf condition on \( \Hom(\FF, \GG) \). In particular, let \( \sigma_i \in \Hom(\FF, \GG)(U_i) \) for each \(i\) such that \( \sigma_i \vert_{U_i \cap U_j} = \sigma_j \vert_{U_i \cap U_j} \) for all \( i, j\). Then by evaluating at \( f\vert_{U_i} \) for each \( i \), we have sections \( \sigma_i(f\vert_{U_i}) \) for each \(i\) which necessarily commute with intersections, so there exists some global \( \widetilde{\sigma}_f \) which agrees with \( \sigma_i(f\vert_{U_i} \) for each \( i \). Since \( f \in \FF(U) \) was arbitrary we may define \( \sigma \in \Hom(\FF, \GG)(U) \) by \( \sigma(f) := \widetilde{\sigma}_f \).

Proof:

1. Suppose \( \tau \in \Hom_{\Set_X}( \FF, \GG ) \) is a morphism of sheaves (of sets) and \( U \hookrightarrow V \). Then we get a commuting diagram $$ \require{amscd} \begin{CD} \underline{ \{ p \} }(V) @>{\tau(V)}>> \FF(V) \\ @VV{\res{V}{U}}V @VV{\res{V}{U}}V \\ \underline{ \{p\} }(U) @>{\tau(U)}>> \FF(U) \end{CD} $$ however, since \( \underline{ \{ p\} }(W) = \{ p \} \) for all open sets \( W \), this may really be interpreted as

   

   It is easy to see then that the data of \( \tau(W)(p) \) is independent of open set \(W \subset X\). Thus, a natural transformation \( \tau : \underline{ \{ p\} } \to \FF\) is uniquely determined by the image of \(p\), so the isomorphism is simply \( \tau \mapsto \tau(-)(p) \).

2. The argument is effectively the same as above, with the added fact that a group homomorphism \( \phi : \Z \to G \) is uniquely determined by its image \(\phi(1)\) (since 1 is a generator of \(\Z\) as an abelian group under addition). Thus, if \( \tau : \underline{\Z} \to \FF \) is a morphism of sheaves of abelian groups, then for any inclusion \( U \hookrightarrow V \) we get a commuting diagram

   

   where our \( \FF(V), \FF(U) \) are now abelian groups and \( \tau(V), \tau(U) \) are group homomorphisms. As before, our isomorphism is \( \tau \mapsto \tau(-)(1) \).

3. Though the argument is made slightly more abstract, \( (X, \OO_X) \) refers to a ringed space so that the structure sheaf \( \OO_X \) is a sheaf of rings. Letting \(1_U\) denote the multiplicative identity of \( \OO_X(U) \), we define a global section \( 1 \in \OO_X \) by \( 1\vert_U = 1_U \). Now for each \( U \subset X \) open, \( \OO_X(U) = (1_U) \) (as is the case for all rings with unity) so that for any \( \OO_X(U)\)-linear map \( \phi : \OO_X(U) \to M \), \( \phi(x) = \phi(x \cdot 1_U) = x \cdot \phi(1_U) \) — in other words, \( \phi \) is uniquely determined by the image \( \phi(1_U) \). Therefore, if \( \tau \in \Hom_{\Mod{\OO_X}}(\OO_X, \FF) \), then \( \tau(U) \) is \( \OO_X(U) \)-linear for each open set \(U\) and commutes with restrictions:

   $$ \require{amscd} \begin{CD} \OO_X(V) @>{\tau(V)}>> \FF(V) \\ @V{\res{V}{U}}VV @V{ \res{V}{U} }VV \\ \OO_X(U) @>{\tau(U)}>> \FF(U) \end{CD} $$

   The isomorphism looks the same in this case.

Proof:

Following the hint, if \( \phi : \FF \to \GG \) is a morphism of sheaves (i.e. natural transformation of functors), then we get a commuting diagram

If we let \( \iota(U) : \ker_{\textrm{pre}} \phi(U) \to \FF(U) \) and \( \iota(V) : \ker_{\textrm{pre}} \phi(V) \to \FF(V) \) denote the canonical maps, then by commutivity we have \( \phi(U) \circ \res{V}{U} \circ \iota(V) = \res{V}{U} \circ \phi(V) \circ \iota(V) = \res{V}{U} \circ 0 = 0\). Thus, by the universal property of the kernel, there exists a unique morphism \( \rho_{V,U} : \ker_{\textrm{pre}} \phi(V) \to \ker_{\textrm{pre}} \phi(U) \) so that \( \res{V}{U} \circ \iota(V) = \iota(U) \circ \rho_{V,U} \). In particular, this tells us the following diagram commutes:

This diagram also tells us that \( \iota \) is a natural transformation. To see that the restriction maps commute, let \( U \subset V \subset W \). Since \( \res{W}{V} \circ \res{V}{U} = \res{W}{U} \) and

commutes, then since \( \rho_{W,U} \) is defined to be the \( \textbf{unique} \) morphism such that \( \iota(U) \circ \rho_{W, U} = \res{W}{U} \circ \iota(W) = \res{W}{V} \circ \res{V}{U} \circ \iota(U) \), we necessarily have \( \rho_{W, U} = \rho_{W,V} \circ \rho_{V,U} \).

Proof:

We define the presheaf cokernel in the same way as \( \coker_{\textrm{pre}} \phi (U) = \coker ( \phi(U) ) \). If we let \( j(U) : \GG \to \coker ( \phi(U) ) \) denote the canonical map, then we get a diagram

where the left square commutes. But then \( j(U) \circ \res{V}{U} \circ \phi(V) = j(U) \circ \phi(U) \circ \res{V}{U} = 0 \circ \res{V}{U} = 0 \), so by the universal property of cokernels, there exists a unique map \( \rho_{V,U} : \coker ( \phi(V) ) \to \coker ( \phi(U) ) \) that makes the right square commute. Defining this as the restriction map, the same argument as the previous exercise shows that the restrictions commute so that \( \coker_{\textrm{pre}} \phi \) is indeed a presheaf.

To see that \( \coker_{ \textrm{pre} } \phi \) satisfies the universal property in \( \Set_X \), suppose \( \GG' \) is another presheaf with restriction map \( \textrm{res}'_{V,U} : \GG'(V) \to \GG'(U) \) and \( \tau : \GG \to \GG' \) is a natural map such that \( \tau \circ \phi = 0 \). Then \( \tau(U) \circ \phi(U) = 0 \) and \( \tau(V) \circ \phi(V) = 0 \), so by the universal property of the cokernel (in set, or whatever category our sheaves are over), we have unique maps \( f(U) \) and \( f(V) \) such that \( \tau(V) = f(V) \circ j(V) \) and \( \tau(U) = f(U) \circ j(U) \). That is, we get a commuting diagram:

We wish to show that \( f \) commutes with restriction maps so that it can be described as a natural transformation (i.e. a morphism of presheaves). To see this, notice

Since the cokernel map \( j(V) \) is always an epimorphism, we necessarily have \( \textrm{res}'_{V,U} \circ f(V) = f(U) \circ \rho_{V,U} \). Thus, \( f : \GG \to \GG'\) is a morphism of presheaves, and necessarily unique by the uniqueness of each \( f(U) \).

Proof:

Let us label the functor \( \textrm{Ev}_U : Ab_X^{\textrm{pre}} \to Ab \) (for evaluation at \(U\)). Suppose we have sheaves \( \mathscr{A}, \mathscr{B}, \mathscr{C} \) over \(X\), along with morphisms of presheaves \( \alpha : \mathscr{A} \to \mathscr{B} \) and \( \beta : \mathscr{B} \to \mathscr{C} \). Moreover, suppose the sequence

is exact. That is to say \( \textrm{im}_{\textrm{pre}}\,\alpha = \textrm{coker}_{ \textrm{pre} } ( \ker_{\textrm{pre} } \,\alpha) = \ker_{\textrm{pre}} \beta \), \( \ker_{\textrm{pre}}\alpha = \mathscr{A} \) and \( \textrm{im}_{\textrm{pre}} \beta = \mathscr{C} \). But then by our construction of the kernel and cokernel presheaves, \( (\textrm{im}_{\textrm{pre}}\alpha )(U) = \textrm{coker}_{\textrm{pre}}( \ker \alpha(U) ) = \textrm{im} (\alpha(U)) \), so \( \im \alpha(U) = \ker \beta(U) \), \( \ker \alpha(U) = \mathscr{A}(U) \) and lastly \( \im \beta(U) = \mathscr{C}\). Thus,

is an exact sequence so that \( \textrm{Ev}_U \) is exact.

Proof:

This is simply a generalization of the previous exercise — the data of a natural transformation is the data of all the maps at each index. Since the index category for a presheaf is the category of open sets of \(X\), the data of a morphism of sheaves \( \phi \) is the data of all the \( \phi(U) \) for \( U \) open. In addition, we defined the kernel (resp. cokernel) sheaf to be the data of \( \ker \phi(U) \) (resp. \( \coker \phi(U) \) ) for each \(U\). Hence it should be immediate from unwinding the definitions that \( \im_{\textrm{pre}} \phi = \ker_{\textrm{pre}} \phi \) if and only if \( \im \phi(U) = \ker \phi(U) \) for all \( U \), \( \im_{\textrm{pre}} \phi = \FF \) if and only if \( \im \phi(U) = \FF(U) \) for all \(U\), etc.

Proof:

First, let \( U \subset X \) be an open set and let \( \{ U_i \} \) be an open cover of \( U \). For the identity axiom, suppose that we fix some \( f_1, f_2 \in \ker \phi(U) \) with \( \rho_{U,U_i} f_1 = \rho_{U, U_i} f_2 \) for all \( i \). Then \( \iota(U_i) \circ \rho_{U, U_i} \circ f_1 = \iota(U_i) \circ \rho_{U, U_i} \circ f_2 \) in \( \FF(U_i) \). By by commutivity of the following diagram

we have that \( \res{U}{U_i} \circ \iota(U) \circ f_1 = \res{U}{U_i} \circ \iota(U) \circ f_2 \) for all \(i\). Now since \( \FF \) is assumed to be a sheaf, this tells us that \( \iota(U) \circ f_1 = \iota(U) \circ f_2 \). Since the kernel map \( \iota(U) \) is always a monomorphism, this implies that \( f_1 = f_2 \).

To see that the gluability axiom holds, we will display the full diagram that should be pictured:

Suppose \( f_i \in \ker \phi(U_i) \) for each \( i \), and \( \rho_{U_i, U_i \cap U_j} \circ f_i = \rho_{U_j, U_i \cap U_j} \circ f_j \) for all \( i, j \). Then

However, by commutivity this tells us that

for all \( (i, j) \) in \( \FF( U_i \cap U_j ) \). Since \( \FF \) is indeed a sheaf, this implies that there exists some section \( s \in \FF(U) \) such that \( \res{U}{U_i} \circ s = \iota(U_i) \circ f_i \) for all \( i \). But then by another commutivity argument / diagram chasing, \(\res{U}{U_i} \circ \phi(U) \circ s \) must vanish since \( \phi(U_i) \circ \iota(U_i) \circ f_i \) vanishes by the canonical property of the kernel map. Thus, there must exist some \( f \in \ker \phi(U) \) with \( \iota(U) f = s \). By uniqueness of the restriction map \( \rho \) from before, it must be the case that \( \rho_{U, U_i} f = f_i \) for all \(i\).

Proof:

Before proceeding, it is important to notice that since \( \underline{\Z} \) is the constant sheaf (not presheaf), sections are only required to be locally constant. We first wish to check exactness. Let us label our morphism \( \underline{\Z} \to \OO_X \) as \( \iota \) and the morphism \( \OO_X \to \FF \) as \( \phi \). To see that \( \iota \) is injective, we need to show that \( \ker_{\textrm{pre}} \iota = \underline{0}\) as presheaves. By construction, for any \( U \subset X \) open we have \( (\ker_{\textrm{pre}} \iota)(U) = \ker \iota(U) \). Now \( \iota(U) : \mathbb{Z} \to \OO_X(U) \) is simply the map taking locally constant sections \( n \in \Z \) to the locally constant function \( f(x) = n \) on \( U \) (which is necessarily holomorphic). Thus, if \( k \in \ker \iota(U) \) then we have by definition \( k = 0 \) on all connected components. Thus, \( \ker \iota(U) \) is trivial — since \( U \subset X \) was arbitrary, \( \ker_{\textrm{pre}} \iota \) is the trivial presheaf as needed.

To see that \( \phi \) is surjective, we need to show \( \im_{\textrm{pre}} \phi = \FF \) as presheaves. That is, for each \( U \subset X \) open we have that \( \im \phi(U) = \FF(U) \). Now by definition, \( \FF(U) \) is the set of holomorphic functions \( g : U \to \C \) that admit a holomorphic logarithm on \(g(U)\) (open by the open mapping theorem). Thus, for \( g : U \to \C \) arbitrary we have that \( f = \frac{1}{2\pi i} \log g \) is holomorphic as well on \( U \) and thus an element of \( \OO_X(U) \). Moreover, \( \phi(U)(f) = \textrm{exp}(2\pi i f) = \textrm{exp} \log g = g \) giving us \( \im \phi(U) = \FF(U) \) — since \( U \) was arbitrary, this tells us \( \im_{\textrm{pre}} \phi = \FF \) as presheaves.

To see exactness at the middle, we need only show \( \im_{\textrm{pre}} \iota = \ker_{\textrm{pre}} \phi \) — that is, \( \im \iota(U) = \ker \phi(U) \) for each open set \( U \). But this is fairly straightforward, as a function \( f : U \to \C \) satisfies \( \textrm{exp} (2 \pi i f) \) if and only if \( f \) is locally an integer on all connected components — that is to say \( f \in \underline{\Z} \).

Showing that \( \FF \) is not a sheaf is a fairly common exercise — consider the open covering of \( \C^*\) given by \( U_1 = \C - \mathbb{R}_{\leq 0}\), \( U_2 = \C - \mathbb{R}_{\geq 0} \). Then by considering the identity \( f(z) = z \), \( z = r e^{i\theta} \) has a logarithm \( \textrm{Log}(z) = \textrm{ln}(r) + i \textrm{Arg}(z) \) on \( U_1 \) with \( \textrm{Arg}(z) \) taking values in \( (-\pi, \pi) \), and logarithm \( \textrm{log}'(z) = \textrm{ln}(r) + i \textrm{arg}'(z) \) on \(U_2)\) with \( \textrm{arg}'(z) \) taking values on \( (0, 2\pi) \). Then the restrictions of \( f(z) = z \) are obviously the same, but \( f(z) = z \) cannot have a global holmorphic logarithm on \( \C^* \) since that would imply the existence of a well-defined inverse for \( e^w = e^{w + 2\pi i k} \).

Proof:

Let \( p \in U\) be arbitrary. Then by construction of the colimit \( \FF_p \), we have canonical maps \( \alpha_{p, U} : \FF(U) \to \FF_p \) that commute with the restriction maps:

But then by the universal property of the product, we get a unique map \( \psi_{U} : \FF(U) \to \prod_{q \in U} \FF_q \) such that \( \pi_p \circ \psi_{U} = \alpha_{p, U} \) (where \( \pi_p \) is the canonical projection map).

Now suppose \( f, g \in \FF(U) \) and \( \psi(f) = \psi(g) \). Then

for all \( p \in U \). Taking representatives \( [U_p, \phi \vert_{U_p}] \), \( [V_p, \psi\vert_{V_p}] \), we get an open cover \( \{ U_p \cap V_p \}_{p \in U} \) such that

By the identity axiom, it must then be the case that \( \phi = \psi \).

Proof:

It suffices to show that \( \textrm{Supp}(s)^C = \{ p \in X : s_p = 0\ \textrm{in} \ \FF_p \} \) is an open set — thus, let \(q \in \textrm{Supp}(s)^C \) be arbitrary. If we consider the definition of a germ as an equivalence class \(s_q = \{ (\sigma, U) \mid \sigma \in \FF(U), q \in U \} \) (recall this is equivalent to the colimit definition), then as \( \sigma_q \) is the zero section there exists some open set \( U \subset X \) and section \( \sigma \in \FF(U) \) which we may assume without loss of generality (by restricting to a possibly smaller set) is the zero section on \( \FF(U) \). But then we must have \( U \subseteq \textrm{Supp}(s)^C \) as well, since \(\sigma_x\) is the germ of the zero section for any point \( x \in U \). Thus, every point in \( \textrm{Supp}(s)^C\) has a neighborhood contained in \( \textrm{Supp}(s)^C \) so that \( \textrm{Supp}(s)^C \) is open.

Proof:

Suppose \( \prod_{p \in U} s_p \in \prod_{p \in U} \FF_p \) is a collection of compatible germs. Using the second of the two equivalent definitions given, this implies that there exists some open cover \( \{ U_i \} \) of \(U\) and sections \( f_i \in \FF(U_i) \) such that \( (f_i)_q = s_q \) for all \( q \in U_i \). Then the \( f_i \) must agree on intersections by Exercise 2.4.A, as they both have germs equal to \(s\) over the intersection — thus, by the gluability axiom, there exists some \( f \in \FF(U) \) with \( \res{U}{U_i} f = f_i \) for each \(i\). But then we have that for any \( x \in U \), \(x\) is in some \( U_i \) (by virtue of a cover) so that \( f_x = (f_i)_x = s_x \). If we let \( \varrho : \FF(U) \to \prod_{p \in U} \FF_p \) denote our map \( s \mapsto \prod_{p \in U} s_p \), since \( x \in U \) was arbitrary we necessarily have \( \varrho(f) = \prod_{p \in U} s_p \) as desired.

Proof:

Let \( s \in \FF(U) \) be arbitrary. Omitting the universal property nonsense, our morphisms \( \phi_1 \) and \( \phi_2 \) induce unique morphisms \( (\phi_1)_p \) and \( (\phi_2)_p \) over each stalk, which thus induce unique maps \( \overline{ \phi_1 } : \prod_{p \in U} \FF_p \to \prod_{p \in U} \GG_p \) and \( \overline{ \phi_2 } : \prod_{p \in U} \FF_p \to \prod_{p \in U} \GG_p \) (you may use the following diagrams as a reference for this if needed):

Now by the uniqueness of these morphisms, it follows that the original diagram (in the problem statement) commutes for both \( \phi_1 \) and \( \phi_2\). By hypothesis, we have that \( (\phi_1)_p(s_p) = (\phi_2)_p(s_p) \) for each \( p \in U \). By commutivity of the two above diagrams, this tells us that \( \overline{\phi_1}( \prod_{p \in U} s_p ) = \overline{\phi_2}( \prod_{p \in U} s_p ) \) — by commutivity of our original diagram (in the problem statement), this implies that \( \rho( \phi_1(s) ) = \rho(\phi_2(s)) \). Since \( \GG \) is in fact a sheaf (of sets), by Exercise 2.4.A we know that the map \( \rho : \GG(U) \to \prod_{p \in U} \GG_p \) is in fact injective — specifically a monomorphism, such that \( \phi_1 (s) = \phi_2(s) \). Since \( s \in \FF(U) \) was arbitrary, the result follows.

Proof:

For the forward direction, if we assume that \( f : \FF \to \GG \) is an isomorphism, then there exists some inverse morphism \( g : \GG \to \FF \) with \( g \circ f = 1_{\FF}\) and \( f \circ g = 1_{\GG} \). Consequently, we get the commuting diagram

for each open set \( U \). For \( p \) fixed, and section \( s_p \in \FF_p \), there exists some open neighborhood \( V \) of \(p\) and section \( \sigma \in \FF(V) \) such that \( \sigma_p \) restricts to \( s_p \) (though it need not be true that \( \sigma_q = s_q \) for other points \(q \in V\), as that would require the compatible germ condition). Then since \( f(V) : \FF(V) \to \GG(V) \) is an isomorphism and \( \pi_p \) (restricted to \(V\)) is surjective (since \( \pi_p( \prod_{p \in V} \sigma_p ) = s_p \)), the induced morphisms \( f_p \) and \( g_p \) necessarily satisfy \( g_p \circ f_p = 1_{\FF_p} \) and so forth.

Conversely, suppose that \( f_p : \FF_p \to \GG_p \) is an isomorphism for each \( p \in U \). It is a straightforward argument to see that \( f \) must be injective — let \( s, t \in \FF(U) \) and suppose \( f(s) = f(t) \). Then \( \overline{\varrho}(f(s)) = \overline{\varrho}(f(t)) \), so that by commutivity of our diagram we have \( f_p(\varrho (s)) = f_p(\varrho(t)) \). Since \( f_p \) is injective by supposition and \( \varrho \) was shown to be injective in Exercise 2.4.A, it follows that \( s = t \).

To see that \( f_U : \FF(U) \to \GG(U) \) (for an open set \(U\)) is surjective requires some diagram chasing, the use of the second definition of compatible germs. Fix some \( t \in \GG(U) \) — by Exercise 2.4.C, the image of \( t \) under \( \overline{\varrho} \) yields some open cover \( \{ U_i \} \) and sections \( t_i \in \GG(U_i) \) such that \( (t_i)_q = t_q \) for all \( q \in U_i \). Now since each \( f_p \) is surjective, we can find some \( s_p \in \FF_p \) with \( f_p(s_p) = t_p \). Now on each \( U_i \) we must have that the \( s_p \) glue together nicely by injectivity of \( f_p \). But then \( \prod_{p \in U} s_p \) is a collection of compatible germs, so that it is in the image of \( \rho \); by the commutivity of the original diagram, it follows that \( t \) is in the image of \(f\).

Proof:

1. Following the hint, let \( X = \{ a, b \} \) equipped with the discrete topology \( \tau = \{ \emptyset, \{ a\}, \{ b \}, X \} \). Now if we let \(A\) be any set, and define the presheaf \( \FF : \textrm{Top}_X \to \Set_X \) by

   $$ \FF(S) = \begin{cases} 0 & S \neq X \\ X & S = X \end{cases} $$

   then clearly the smallest set containing each point \( p \in X \) is the singleton set, so that each stalk \( \FF_p \) is trivial. However, \( X \to \{a\} \times \{b\} \) is clearly not injective since \( \| X\| > 1 \).

2. Using the same set-up as before, if we instead let

   $$ \GG(S) = \begin{cases} 0 & S = \emptyset \\ \{ a \} & S \ \textrm{is}\ \textrm{a}\ \textrm{singleton} \\ X & S = X \end{cases} $$

   then if \( \phi_1, \phi_2 : \FF \to \GG \) are morphisms of sheaves, \( (\phi_1)_a(s) \) and \( ( \phi_2)_b(s) \) are the constant map \( s \mapsto a \), then our maps \( \phi_1(X), \phi_2(X)\) may differ: in particular, we can still have \( \phi_1(X)(a) = a \) and \( \phi_2(X)(a) = b \) without the stalks being ill-defined.

3. Our Exercise 2.3.J works here.

Proof:

For the first statement, let \(\FF\) be a presheaf and \( \textrm{sh}_1 : \FF \to \FF^{sh}_1 \) and \( \textrm{sh}_2 : \FF \to \FF^{sh}_2 \) be sheafifications. Then by the universal property of the sheafification there exist unique morphisms \( f: \FF^{sh}_1 \to \FF^{sh}_2 \) and \( g : \FF^{sh}_2 \to \FF^{sh}_1 \) with \( \textrm{sh}_2 = f \circ \textrm{sh}_1 \) and \( \textrm{sh}_1 = g \circ \textrm{sh}_2 \), making the following diagram commute:

But then we clearly have that \( \textrm{sh}^1 = g \circ \textrm{sh}_2 = g \circ f \circ \textrm{sh}_1 \) and vice-versa, so that \( g \) and \(f\) are inverse. Thus, the two sheafifications are isomorphic.

For the second statement, if \( \GG \) is another sheaf and \( g : \FF \to \GG \), then the following diagram clearly commutes

Proof:

Assuming all the sheafifications exist, then using the universal property of \( \FF^{sh} \) as the sheafification of \( \FF \), since \( \textrm{sh}_\GG \circ \phi \) a morphism of presheaves from \( \FF \) to a sheaf, there exists a unique morphism \( \phi_{sh} : \FF^{sh} \to \GG^{sh} \) making the following diagram commute:

Proof:

\( \FF^{sh} \) is clearly a presheaf since \( \FF \) is (the restriction of restriction maps is a restriction map.) It suffices to show that \( \FF^{sh} \) satisfies the identity and gluability axioms. Let \( U \) be an open set with open cover \( \{ U_i \} \). If \( s_1, s_2 \in \FF^{sh}(U) \) with \( \res{U}{U_i} s_1 = \res{U}{U_i} s_2 \) on each \( U_i \), then by construction of \( \FF^{sh} \) we have \( \prod_{p \in U_i} (s_1)_p = \prod_{p \in U_i} (s_2)_p \) for each \( i \) — in particular since the \( \{ U_i \} \) cover \( U \), we have that the stalks are the same at each point. By Exercise 2.4.A, this implies \( s_1 = s_2 \).

For the gluability axiom, suppose \( s_i \in \FF(U_i) \) for each \( i \) which agree on restrictions. By simply taking the pointwise product \( s = \prod_{ p \in U } (s_i)_p \), since the \( s_i \) agree on restrictions then for \( q \in U_i \cap U_j \) there exist neighborhoods \( V_i \subset U_i \) and \( V_j \subset U_j\) of \(q\) and sections \( f_i \in \FF^{sh}(V_i), f_j \in \FF^{sh}(V_j) \) that have the same stalks as \( s_i \) and \(s_j\) everywhere. However, since the \( s_i \) and \( s_j \) are assumed to agree on the restrictions, the \( f_i\) and \(f_j\) agree on \( V_i \cap V_j \) — thus, we may take \( V_i \cap V_j \) to be our required neighborhood for the compatible germs. It is immediate from unwinding the definition that \( s \) restricts to the \(s_i\).

Proof:

Our construction of \( \FF^{sh}(U) \) can be interpreted as a natural map since it commutes with with restrictions. That is, if \(U \subset V\) we have \( \textrm{sh}( \FF(U) ) = \{ \prod_{p \in U} f_p \mid \dots \} \) and \( \textrm{sh}( \FF(V) = \{ \prod_{p \in V} f_p \mid \dots \} \), then the restriction \( \textrm{res}^{sh}_{V, U} ( \prod_{ p \in V } f_p ) = \prod_{p \in U} f_p \) clearly commutes. In other words, we are simply looking at the restriction of the natural map \( \FF(U) \to \prod_{p \in U} \FF_p \) from Exercise 2.4.A, which is still necessarily natural.

Proof:

By Exercise 2.4.I, \( \FF^{sh} \) is indeed a sheaf. Suppose \( \GG \) is also a sheaf and \( \phi : \FF \to \GG \) is a morphism of presheaves. Then for each \( p \in U \), we get induced maps of stalks \( \phi_p : \FF_p \to \GG_p \) — by considering the induced map on the product \( \prod \phi_p \), we necessarily have the following diagram commutes

Moreover, if \( \psi : \FF^{sh} \to \GG \) is another map that makes the diagram above commute, then the maps \( \phi_p \) and \( \psi_p \) agree on all stalks so that by Exercise 2.4.D we have \( \phi = \psi \) — in other words, the map \( \phi \) is unique.

Proof:

Let \( (-)^{pre} : \textrm{Sh}_X \to \textrm{PSh}_X \) denote the forgetful functor from sheaves on \(X\) to presheaves on \(X\) (we aren't using Vakil's notation \(\Set_X\) here for presheaves of sets over \(X\) since the base category for our sheaves isnt specified.) Now if \( f : \FF^{sh} \to \GG \) is a morphism of sheaves and \( \textrm{sh} : \FF \to \FF^{sh} \) denotes the sheafification presheaf morphism, then \( \tau_{\FF \GG} (f) = f \circ \textrm{sh} \) is the required morphism of presheaves. In the reverse direction, if \( \FF \) is a presheaf over \(X\) and \( \GG \) is a sheaf, and let \( g : \FF \to \GG \) be a presheaf morphism. Since \( \GG \) is in fact a sheaf, there is a unique map \( \widetilde{g} : \FF^{sh} \to \GG \) by the universal property of sheafification as desired. This gives us the bijection \( \tau_{\FF\GG} : \Mor( \FF^{sh}, \GG ) \to \Mor( \FF, \GG^{pre} ) \).

To see that the two are adjoint, let \( \FF' \) be another presheaf, \( f : \FF \to \FF' \) a presheaf morphism, and fix some \( \alpha \in \Mor_{Sh}( (\FF')^{sh}, \GG ) \). If we let \( \textrm{sh}, \textrm{sh}' \) denote the sheafification maps \( \FF \to (\FF)^{sh} \) and \( \FF' \to (\FF')^{sh} \), respectively, then since \( (\FF')^{sh} \) is indeed a sheaf, the universal property of sheafification tells us that there exists a unique map \( f^{sh} : \FF^{sh} \to (\FF' )^{sh} \) that makes the following diagram commute:

But the path from \(\FF\) to \( \GG \) along the top is precisely

while the path along the bottom is

Since \( \alpha : (\FF')^{sh} \to \GG \) was arbitrary, the following diagram commutes

(using the notation for the forgetful functor here was a bit unnecessary). The other commutative diagram is almost trivial since any morphism of sheaves is a morphism of presheaves — in particular, if \( \GG' \) is another sheaf and \( g : \GG \to \GG' \) a morphism of sheaves, then \( (g^{pre})^* = g^* \). Additionally, both \( \tau_{\FF\GG} \) and \( \tau_{\FF\GG'} \) agree, as they are both precomposition with \( \textrm{sh} : \FF \to \FF' \).

Proof:

We proceed in the order of the hint provided: to show (b) \( \Rightarrow \) (a), suppose \( \FF' \) is another category and \( \psi_1, \psi_2 : \FF' \to \FF \) are two additional morphisms such that \( \phi \circ \psi_1 = \phi \circ \psi_2 \). Evaluating at the level of stalks, we then have \( \phi_p \circ \psi_{1,p} = \phi_p \circ \psi_{2,p} \) for all \(p\). Now since \( \phi_p \) is injective, this implies \( \psi_{1,p} = \psi_{2,p} \) — by Exercise 2.4.D, this tells us \( \psi_1 = \psi_2 \).

In order to show that (a) \( \Rightarrow \) (c), we will refer to Mark Kamsma's answer on this StackExchange post, as it is probably the most straightforward proof you will find. Using the language of Vakil, suppose \( x, y \in \FF(U) \) with \( \phi(U)(x) = \phi(U)(y) \), and define the "indicator sheaf" \(H\) by

Define morphisms \( \alpha : H \to \FF \) and \( \beta : H \to \FF \) by \( \alpha(U)(\ast) = x \), \( \beta(U)(\ast) = y \) (since \(\alpha(U), \beta(U)\) set-theoretic maps, there is no need to define the image of the empty-set). Then by assumption \( \phi(U) \circ \alpha(U)(\ast) = \phi(U)(x) = \phi(U)(y) = \phi(U) \circ \beta(U)(\ast) \). Since \( \phi \) is mono, \( x = \alpha(U)(\ast) = \beta(U)(\ast) = y \).

Lastly, (c) \( \Rightarrow \) (b) follows from unwinding definitions. If we let \( p \in U \) be arbitrary and suppose \( \phi_p(s_p) = \phi_p(t_p) \) for some \( s_p, t_p \in \FF_p \), then there exists some open neighborhoods \( V, W \subset U\) of \( p \) and representatives \( s \in \FF(V), t \in \FF(W) \) with compatible germs. By considering the restriction to the neighborhood \( V \cap W \) of \(p\), we must necessarily have \( \phi(V \cap W)(s\vert_{V \cap W}) = \phi(V \cap W)(t\vert_{V \cap W}) \). But by hypothesis, this implies that \( s\vert_{V \cap W} = t\vert_{V \cap W} \) — in particular, \( s_p = t_p \).

Proof:

For the forward direction (i.e. (a) \( \Rightarrow\) (b)), suppose \( \phi \) is an epimorphism, and consider the skyscraper sheaf \( i_{p,\ast}\{0,1\} \):

Define a morphism \( \alpha : \GG \to i_{p,\ast}\{0,1\} \) by

and \(\beta : \GG \to i_{p,\ast}\{0,1\}\) as the constant map \( \beta(U)(s) = 1 \) (we have assumed up to this point \(U\) is a neighborhood of \(p\) so that the maps are well-defined on the skyscraper sheaf — in order to ensure this map stays well-defined on all open subsets of \(X\), we need only require \(\alpha(V) \) and \(\beta(V)\) be the zero map for open sets \(V\) not containing \(p\).) But then \( \alpha \circ \phi = \beta \circ \phi \) since for any \( f \in \FF \), the stalk of \( \phi(f) \) is clearly in \( \im(\phi_p) \). Since \( \phi \) is assumed to be an epimorphism, this tells us \( \alpha = \beta \) — in particular, for all \( s \in \GG \), \( s_p \in \im(\phi_p) \) so that \( \phi_p \) is surjective.

In the reverse direction, suppose that \( \phi_p : \FF_p \to \GG_p \) is surjective for all \(p\), and let \( \alpha, \beta : \GG \to \GG' \) (for some sheaf \( \GG'\)) with \( \alpha \circ \phi = \beta \circ \phi \). At the level of stalks, this implies that \( \alpha_p \circ \phi_p = \beta_p \circ \phi_p \) for each \(p\) — since \( \phi_p \) is surjective, this implies \( \alpha_p \) and \(\beta_p\) agree on all inputs \( g_p \in \GG_p \). Thus, \( \alpha_p = \beta_p \); by Exercise 2.4.D, this implies that \( \alpha = \beta \) as desired.

Proof:

Fix some point \( p \in \C \) and let \( f_p \in \OO_{X,p}^* \) be a locally nonvanishing germ. Then by taking a representative \( (U, f) \) for some open set \( U \) and \( f \in \OO_X^*(U) \), there exists a possibly smaller neighborhood \(V\) of \( p\) such that \( f \) has a local logarithm. Since \( f \) non-zero on \(V\), \( \log(f_p) \) maps to \(f_p\). Thus, \( \textrm{exp} \) is surjective at the level of stalks.

For the second statement, Exercise 2.3.J provides the simplest example.

Proof:

This simply follows from the fact that if \( U \) is an open set in some topological space \( X \), then \( U \) is a union of base elements \( U = \bigcup_i B_i \). But then \( \{ B_i \} \) may also be considered as an open cover of \( U \) — thus, if we know the information of \( \FF \) on base elements \( \FF(B_i) \), we may recover sections \( s \in \FF(U) \) via the identity and gluability axioms of \(\FF\).

Proof:

It is fairly easy to see that, over open sets \(B\) in the base and \( \prod_{p \in B} s_p \in \FF_p \), the condition that there exists a base element and locally compatible section over that base element is trivial (as we may take our original base element and same section). In particular, we are simply looking at the natural morphism

From section §2.4, which by Exercise 2.4.A is known to be injective, and surjective by the same reasoning as Exercise 2.4.C (by possibly replacing "gluability axiom" with "gluability over a base".) Another way of viewing this is that at the level of stalks, the natural map is the identity.

Proof:

1. Suppose \( \phi : F \to G \) is a morphism of sheaves on a base \( \mathcal{B} \) and \( \psi_1, \psi_2 : \FF \to \GG \) are two morphisms of sheaves whose induced morphisms of sheaves on a base agree with \( \phi \). By passing to the level of stalks, the induced morphism (as noted in the previous exercise) is simply the identity, so \( (\psi_1)_p = \phi_p = (\psi_2)_p \). Since morphisms are determined by stalks (c.f. Exercise 2-4-D), \( \psi_1 = \psi_2 \).

2. Let \( \phi : F \to G\) be a morphism of sheaves on a base, yielding a diagram

   $$ \require{amscd} \begin{CD} F(B_i) @>{\phi(B_i)}>> G(B_i) \\ @V{\res{B_i}{B_j}}VV @VV{\res{B_i}{B_j}}V \\ F(B_j) @>{\phi(B_j)}>> G(B_j) \end{CD} $$

   We define the induced morphism \( \widetilde{\phi} : \FF \to \GG \) as follows: for \( U \subset V \) open (not necessarily base elements) and \( \prod_{p \in V} f_p \in \FF(V) \), we take \( \widetilde{\phi}( \prod_{p \in V} f_p ) = \prod_{p \in V} \phi_p(f_p) \). To see that this is indeed well-defined, notice that since \( \prod_{p \in V} f_p \) has compatible germs, for every \( p \in V \) there exists some basis element \( B_i \subset V \) and \( s \in F(B_i) \) such that \( s_q = f_q \) for all \(q \in B_i \). But then we have \( \phi(s) \in G(B_i) \), and \( \phi_q(s_q) = \phi_q(f_q) \), so that \( \phi(s) \) is our desired representative making the germs of \( \prod_{p \in V} \phi_p(f_p) \in \im \widetilde{\phi} \) compatible.

   To see that this commutes with restrictions, notice that our restriction maps \( \res{V}{U} : \FF(V) \to \FF(U) \) on the induced stalks are simply \( \prod_{p \in V} f_p \mapsto \prod_{p \in U} f_p \). The difference this makes is that while our germs are still compatible, there may be some smaller \( B_j \subset U \) and \( s' \in F(B_j) \) with \( \phi_q(s_q') = \phi_q(f_q) \). But then this \( s' \in F(B_j) \) must necessarily be \(s' = \res{U_i}{U_j} s \) by the identity axiom since they agree on the level of stalks — similarly, \( \phi(s') \) is simply the restriction of \( \phi(s) \). Since our morphism of sheaves on a base \( \phi : F \to G \) commutes with restrictions, it follows that \( \widetilde{\phi} \) does as well.

Proof:

Consider the collection \( \mathcal{B} = \{ B \subseteq X\ \textrm{open} \mid B \subseteq U_i\ \textrm{for}\ \textrm{some}\ i \} \). We wish to show this is indeed a base of \(X\). Suppose \( W \subseteq X \) is open. Then for each \( i \), \( W \subseteq U_i \) is open in \( U_i \) and thus an element of \( \mathcal{B} \). Since \( \{ U_i \} \) cover \(X\), we necessarily have \( W = \bigcup_i W \cap U_i \) so that every open set is a union of elements of \( \mathcal{B} \) — in particular, this implies \( \mathcal{B} \) is a base.

We next wish to construct a sheaf over a base \( F\) for our new base \( \mathcal{B} \); it should be fairly intuitive how to construct \( F\) in such a way that \( F\vert_{U_i} \cong \FF_i \). Specifically, we know that every base element \( U \in \mathcal{B} \) is contained in some \( U_i \) — therefore, we set \( F(U) := \FF_i(U) \). To see that this is independent of choice of \( i\) (up to isomorphism), suppose that \( U \) is also contained in some \( U_j \) as well — particularly, \( U \subseteq U_i \cap U_j \). Since \( \FF_i \vert_{U_i \cap U_j} \cong \FF_j \vert_{U_i \cap U_j} \) (via \( \phi_{ij} \)), we have \( F(U) := \FF_i(U) \cong \FF_j(U) \). Furthermore, \( F \) is indeed a sheaf since each \( \FF_i \) is, so by Theorem 2.5.1 we have that \( F \) extends to some sheaf \( \FF \) that agrees with \( F \) on base elements; by construction of our base \( \mathcal{B} \), this tells us \( \FF\vert_{U_i} = F\vert_{U_i} \), since any open set fully contained in \(U_i\) is a base element. In particular, this gives the obvious equality \( \FF\vert_{U_i} = \FF_i \) by construction of our sheaf on a base \(F\).

Proof:

Let \( \phi : \FF \to \GG \) be a morphism of sheaves, and let \( \ker \phi \) be the kernel sheaf (c.f. Exercise 2.3.I) with morphism \( \iota : \ker \phi \hookrightarrow \FF \). Passing to the level of stalks, we get induced morphisms \( \iota_p : (\ker \phi)_p \to \FF_p \) and \( \phi_p : \FF_p \to \GG_p \) that make the following diagram commute:

We wish to show that \( (\ker \phi )_p \) satisfies the universal property of the kernel of \( \phi_p \) — since the kernel is unique up to unique isomorphism, the result will follow. To begin, we first wish to show that \( \phi_p \circ \iota_p = 0 \). Suppose that \( s \in (\ker \phi)_p \) is arbitrary; by the compatible germs condition from §2.4, we can find some open set \(V \) and section \( f \in \ker \phi(V) \) such that \( f_q = s_q \) for all \( q \in V \). Now \( \phi(V) \circ \iota(V) (f) = 0 \) by virtue of the kernel map, so by commutivity of the above diagram \( \phi_p \circ \iota_p (f_q) = \phi_p \circ \iota_p (s_q) = 0 \) for all \(q \in V\) — in particular, this holds at \(p\) as desired.

To show that \( (\ker \phi)_p \) satisfies the universal property in the category of sheaves, suppose that \( \mathscr{K} \) is another sheaf and \( j : \mathscr{K} \to \FF \) is another morphism of sheaves with \( \phi_p \circ j_p = 0 \). By a similar use of the compatible germs condition above, for any \( s \in \mathscr{K} \) we may find an open set \( V \) and \( f \in \mathscr{K}_V \) which restricts to \( s \) at the level of stalks. Since \( \phi_p \circ j_p (f) \) is trivial and \(\GG\) is a sheaf, there is a neighborhood (by restricting \( V \), we may assume without loss of generality that \(V\) works) on which \( \phi(V) \circ j(V) (f)\) is trivial (c.f. Exercise 2.4.B). By the universal property of the kernel \( \ker \phi(V) \), there exists a unique morphism \( \psi(V) : \mathscr{K}(V) \to \ker \phi(V) \) that commutes with restrictions (by a similar argument to Exercise 2.3.E). As a brief aside, this should not be considered as a morphism \( \psi : \mathscr{K} \to \FF\), but \( \psi : \mathscr{K}\vert_V \to \FF\vert_V \); regardless of this fact (since stalks are local), this gives us an induced morphism \( \psi_p : \mathscr{K}_p \to (\ker \phi)_p \) that makes the necessary diagram commute. Thus, \( \ker (\phi_p) = (\ker \phi)_p \) up to unique isomorphism.

Proof:

The proof is almost identical to that above; if we let \( \phi : \FF \to \GG \) denote our morphism of sheaves, then there is a natural map \( \coker ( \phi_p) \to (\coker_{pre}\phi)_p \) by the universal property of the cokernel. Now at the level of stalks, our map \( \textrm{sh} : \coker_{pre} \phi \to \coker \phi \) becomes the identity; however, it is only true in the sheafification \( \coker \phi \) that when our image of the stalk is trivial, there is a neighborhood that it is trivial (again, this comes from Exercise 2.4.B, and is true more generally when \( \coker_{pre}\phi \) is a separated presheaf.) Using this, the argument becomes the same (really dual) that \( (\coker \phi)_p \) satisfies the universal property of the cokernel of \( \phi_p \).

Proof:

The latter statement is immediate; recall that in any abelian category, the canonical image of a morphism \(f : A \to B\) is \( \im f := \ker \coker f \). By our previous two exercises, we have that \( \ker ( \coker \phi )_p = \ker ( (\coker \phi)_p ) = \ker \coker (\phi_p) \). Now our image presheaf is defined simply as \( \im_{pre} \phi := \ker ( \coker_{pre} \phi ) \). Since sheaves over an abelian category is also an abelian category, we would also expect \( \im \phi \cong \ker ( \coker \phi ) = \ker( \widetilde{sh} \circ \coker_{pre} \phi ) \) (where \( \widetilde{\textrm{sh}} : \coker_{pre} \phi \to \coker \phi \) is the sheafification) — we will show afterwards that the sheafification should commute with the kernel. To prove the statement constructively, recall from Exercise 1.6.I that kernels commute with limits (in an abelian category) — in particular products. Consider the sheafification of the image presheaf \( \textrm{sh} : \im_{pre} \phi (U) \mapsto \prod_{p \in U} (\im_{pre} \phi)_p \) where the germs must be compatible in \( \im_{pre} \phi \); this can be expanded for clarity as

As mentioned above, the stalks of the kernel are the kernel of the stalks and our product commutes with the kernel — thus, we get an induced map

Since we originally required the stalks to have compatible germs in \( \im_{pre} \phi \) and the kernel map is injective, it is easy to see that the elements of \( \prod_{p \in U} (\coker_{pre} \phi)_p \) necessarily have compatible germs in \( \coker_{pre} \phi \). But then our map is really the kernel of the sheafification of \( \coker_{pre} \phi \) (i.e. the cokernel sheaf), so that the diagram

commutes — in particular, the bottom morphism is the natural morphism induced from the universal property of the sheafification which is necessarily an isomorphism by the above argument.

Proof:

This is immediate from the previous three exercises (and I suppose Exercise 2.3.A if you still feel the need to refer to it). If

is exact for sheaves \( \mathscr{A}, \mathscr{B}, \mathscr{C} \in Ab_X \) and morphisms of sheaves \(f,g \), then \( \ker f = 0 \) and \( \im g = \mathscr{C} \). Now by exercises 2.4.N and 2.4.O, these statements hold iff they hold at the level of stalks, so that we necessarily have \( \ker f_p = 0 \), \( \im g_p = \mathscr{C}_p \). Showing that \( \im f_p = \ker g_p \) then follows from the previous exercises.

Proof:

By Exercise 2.4.P, we have that \( \OO_X^* \) is a quotient sheaf of \( \OO_X \) — in particular \( \OO_{X, p} \to \OO_{X,p}^* \) is surjective for all \( p \in P \). By Exercise 2.6.C, we have that the image is defined by the induced maps of stalks so that

Is exact. By Exercise 2.4.N, the first three terms are exact as well. Lastly, by Exercise 2.6.A and Exercise 2.6.D \( \im (\cdot 2\pi i) = \ker \exp\) holds at the level of stalks, so the desired sequence is exact.

Proof:

By Exercise 2.4.N, we know that

is exact. Let \( \phi : \FF \to \GG \) denote the first map and \( \psi : \GG \to \mathscr{H} \) denote the second; in the category of sheaves of abelian groups we have that \( \im \phi = \ker \psi \), where \( \im \phi \) denotes the sheafification of the image presheaf. To see that \( \im \phi(U) \subseteq \ker \psi(U) \), fix some \( t_U = \phi(U)(s_U) \). Since exactness at the level of morphisms of sheaves is equivalent to exactness at the level of stalks, we know that \( \im \phi_p = \ker \psi_p \) for each \( p \in U \); thus, by commuting the stalks and images we have that \( \psi(t_U)_p = 0 \) for all \( p \in U \). Since \( \mathscr{H} \) is a sheaf, by Exercise 2.4.A this implies that \( \psi(t_U) = 0 \).

In the reverse direction, assume that \( t_U \in \ker \psi(U) \) and \( p \in U \). At the level of stalks, we have that since \( \im \phi_p = \ker \psi_p \) there exists some \( s_p \in \FF_p \) with \( \phi_p(s_p) = (t_U)_p \). However, since we are considering the sheafification of the image presheaf and \( \im (\phi_p) = (\im \phi)_p \), there exists some neighborhood of compatible germs \( V_p \) of \(\phi(s_p)\) — since the \(V_p\) cover \(U\), the gluability axiom of the image sheaf and \(\FF\) tell us that there exist some \( s \in \im \phi(U) \) with \( \phi(U) (s) = t \).

The standard example of where this fails to be exact was given Exercise 2.4.P, since the complex logarithm cannot be defined on all of \( U = \C - \{0\} \).

Proof:

Recall that the pushforward sheaf \( \pi_* \FF \) is defined on \(Y\) by \( \pi_* \FF (V) := \FF( \pi^{-1}(V) ) \) for all \(V \subset Y\). By the previous exercise, we know that for any \( V \subset Y \), \( \Gamma(—, \pi^{-1}(V)) \) is an exact functor, so

is exact in the category of abelian groups. Now exactness at all open subsets implies exactness of sheaves, though the reverse is not true (this can be seen by a similar proof to that above by looking at the level of stalks) — thus, we necessarily have that

is exact.

Proof:

While it may be tempting to try to solve this problem using stalks (at least in the category of sheaves of abelian groups, though if we add some conditions for coherent sheaves of \( \OO_X \)-modules then stalks are fine), this post on Math StackExchange by Anton Geraschenko shows that stalk do not commute with the sheaf-hom in general.

Instead, we essentially mimic the proof of Exercise 1.6.F — if \( U \subset X\) is an arbitrary open set and

is an exact sequence of sheaves of abelian groups, then post-composition \( f_* : \mathcal{Hom}( \FF\vert_U, \GG_1 \vert_U ) \to \mathcal{Hom} ( \FF\vert_U, \GG_2 \vert_U ) \) is injective since \( f^*(\alpha) = f^*(\beta) \Rightarrow f \circ \alpha = f \circ \beta \Rightarrow \alpha = \beta \) (as an injective map is a monomorphism). In an identical manner to before, \( g \circ f = 0 \Rightarrow g^* \circ f^* = 0 \).

To prove the first result, it remains to show that \( \ker g_* \subset \im f_* \) — as before, if \( \alpha : \FF\vert_U \to \GG_2 \vert_U \) and \( g\vert_U \circ \alpha = 0 \), then as \( \GG_1 \vert_U \) may be considered as the kernel of \( f\vert_U \) (up to isomorphism), there exists a unique morphism of sheaves \( \iota_U : \FF\vert_U \to \GG_1 \vert_U \) with \( f \vert_U \circ \iota_U = \alpha \). The main difference here is that since \( U \) was arbitrary, then the sections of sheaf-hom are exact for all open sets \(U\) and thus the sequence is exact.

An almost identical proof ofExercise 1.6.F (d) shows the second result, utilizing the fact that \(U\) arbitrary.

Proof Outline:

Recall that if \( \FF \) is an \( \OO_X \)-module, then each \( \FF(U) \) is an \( \OO_X(U) \)-module in such a way that the canonical action commutes with restrictions:

It is easy to see that a morphism in the category of \( \OO_X \)-modules is simply a morphism of abelian groups that is \( \OO_X(U) \)-linear at each open set \(U\) (this should also be interpreted for reasons we will see in the following paragraph that \( \phi \) commutes with the action). By Exercise 2.2.J, we have that

commutes — in particular, the uniquely induced stalk map \( \phi_p : \FF_p \to \GG_p \) is necessarily \( \OO_{X,p} \)-linear. Then the proof must first require that the sheaves of abelian groups \( \ker \phi \) and \( \coker \phi \) (sheafification of cokernel presheaf) have an induced \( \OO_X \)-module structure — by definition this only needs to be checked at the level of open sets. It should not be surprising that the \( \OO_X \)-module structure of \( \ker \phi \) comes from the universal property of the kernel — if \( \iota : \ker \phi \to \FF \) denotes our morphism of sheaves (of abelian groups), the following diagram demonstrates our induced \( \OO_X(U) \)-structure on the kernel:

Note that we use the fact that \( \phi \) commutes with our module action in the diagram above to show that \( \phi(U) \circ \textrm{action} \circ \iota(U) = \textrm{action} \circ \phi(U) \circ \iota(U) = 0\). In a similar fashion, if we let \( \coker \phi\) denote the cokernel sheaf and \( j : \GG \to \coker \phi \) the canonical morphism, then

gives the induced \( \OO_X(U) \)-module structure (here we tacitly make note of the fact that \( A \times \coker f \) satisfies the universal property of \( id_A \times f \) .) We will not check that these induced maps necessarily satisfy the axioms of a module structure.

By an almost identical argument to Exercise 2.6.A and Exercise 2.6.C (by possibly adding the word \( \OO_{X,p} \)-linear where necessary), the fact that every monomorphism is the kernel of its cokernel and every epimorphism is the kernel of its cokernel can be checked at the level of stalks (i.e. the exact same argument made for abelian groups).

Proof:

1. To define categorically what we mean, let \( \FF, \GG \) be \( \OO_X \)-modules. We say that a morphism \( \phi : \FF \times \GG \to \mathscr{H} \) is \( \OO_X \)-bilinear if \( \phi(U) : \FF(U) \times \GG(U) \to \mathscr{H}(U) \) is \( \OO_X(U) \)-bilinear for each open set \(U\), and it commutes with restrictions in both components (e.g. for every \( U \subset V \) and fixed \( g \in \GG(V) \), \( \phi(—, g) \) commutes with restrictions). Then the tensor product of \( \OO_X \)-modules is the unique (up to isomorphism) \( \OO_X \)-module \( \FF \otimes_{\OO_X} \GG \), together with a map \( \phi : \FF \times \GG \to \FF \otimes_{\OO_X} \GG \), such that for each \( \OO_X \)-module \( \mathscr{H} \) and bilinear map \( \psi : \FF \times \GG \to \mathscr{H} \) there exists a unique map of \( \OO_X \)-modules \( \overline{\psi} : \FF \otimes_{\OO_X} \GG \to \mathscr{H} \) with \( \psi = \overline{\psi} \circ \phi \).

   Now from the above definitions, the presheaf tensor product \( (\FF \otimes_{\OO_X} \GG)_{pre} \) is fairly easy to define :

   $$ (\FF \otimes_{\OO_X} \GG)_{pre} (U) := \FF(U) \otimes_{\OO_X(U)} \GG(U) $$

   where the restriction maps are \( \textrm{res}_\FF \otimes \textrm{res}_\GG \). If we define \( \FF \otimes_{\OO_X} \GG := \textrm{sh}\left( (\FF \otimes_{\OO_X} \GG)_{pre} \right) \) and \( \phi : \FF \times \GG \to (\FF \otimes_{\OO_X} \GG )_{pre} \) denotes the canonical map at the level of presheaves, our canonical map at the level of sheaves would just be \( \textrm{sh} \circ \phi \). To see that this satisfies the universal property at the level of sheaves, we need only use the fact that it satisfies the universal property at the level of presheaves and then the universal property of sheafification. More specifically, if \( \mathscr{H} \) is an \( \OO_X \)-module and \( \psi : \FF \times \GG \to \mathscr{H} \) is an \( \OO_X \)-bilinear map, then interpreted as a presheaf of \( \OO_X \)-modules there exists a unique \( \OO_X \)-linear \( \overline{\psi} : ( \FF \otimes_{\OO_X} \GG )_{pre} \to \mathscr{H} \). Since \( \mathscr{H} \) is indeed a sheaf, the universal property of sheafification finally gives us the desired map \( \FF \otimes_{\OO_X} \GG \to \mathscr{H} \).

   

2. To see that the tensor product commutes with stalks, notice that we have ring maps

   

   so that by Exercise 1.5.E we may interpret \( \FF_p \otimes_{\OO_{X,p}} \GG_p \) as an \( \OO_X(U) \)-module. If we define a map \( \psi(U) : \FF(U) \times \GG(U) \to \FF_p \otimes_{\OO_{X,p}} \GG_p\) by \( (s, t) \mapsto s_p \otimes t_p \), this is \( \OO_X(U) \)-linear by the previous argument, so by the universal property of the tensor product there exists a unique map \( \overline{\psi(U)} : \FF(U) \otimes_{\OO_X(U)} \GG(U) \to \FF_p \otimes_{ \OO_{X,p} } \GG_p \). One may check that everything is constructed in such a way that the following pyramid commutes:

   

   By restricting our attention to the face on the right side, we get a commuting cone for every \( U \subset V \) so that there is at the very least a natural map \( ( \FF \otimes_{\OO_X} \GG)_p \to \FF_p \otimes_{\OO_{X,p}} \GG_p \). Instead of showing that this satisfies the universal property of the colimit, it is easier to simply construct an inverse \( \FF_p \otimes_{\OO_{X,p}} \GG_p \to \FF \otimes_{\OO_X} \GG)_p \) by taking compatible germs of some representative \( s \otimes t \) and showing that this map is well-defined (I am going to hand-wave this).

Proof:

To see that the preimage sheaf is indeed a presheaf, we essentially only need to give an explicit construction of the restriction map. Using subscripts to clarify which underlying set we are in, suppose \( U_Y \hookrightarrow V_Y \) are open sets in \(Y\). Then if \( W_X \) is an open set in \(X\) such that \( \pi(W_X) \subseteq U_Y \), then we clearly have that \( \pi(W_X) \subseteq V_Y\) by transitivity. Since \( W_X \subset X \) was arbitrary, by the universal property of the colimit we get a unique natural map

which we will call the restriction map. By uniqueness, it follows immediately that

are the same map. The fact that the restriction map from \( U \) to \(U\) is the identity is trivial by showing that the identity allows (co)cones to commute.

To give an example of where the preimage sheaf fails to be a sheaf, consider \( \pi : \{ a, b \} \to \{a\} \) (both with the discrete topology) and \( \FF = \underline{\Z} \). Then for every open set \( U \subset \{ a, b \} \), we have that the only open set containing \( \pi(U) \) is the singleton \( \{a\} \) — that is \( \pi^{-1} \FF(U) = \FF_a \cong \Z \) (notice that this means the global sections \( s \in \pi^{-1}\FF(X) \) are constant). Now if we define sections \( s\vert_a \in \pi^{-1} \FF( \{a\} ) \) by \( s\vert_a = 0 \) and \( s\vert_b \in \pi^{-1} \FF( \{ b \} ) \) by \( s\vert_b = 1 \), these vacuously agree on intersections (since \( \{a\} \cap \{b\} = \emptyset \) ), but cannot extend to a global section since global sections are constant.

Proof:

It suffices to give bijections

For the first bijection \( \Mor_X( \pi^{-1} \GG, \FF ) \leftrightarrow \Mor_{YX}(\GG, \FF ) \), let \( \phi : \pi^{-1} \GG \to \FF \) be a morphism of sheaves. Then for each \( U \subseteq X \), \( V \subset Y \) with \( \pi(U) \subset V \), define

which is indeed compatible when taking restriction maps. In the reverse direction, suppose \( \phi = ( \phi_{VU} ) \) is a compatible collection of morphisms, giving us a diagram

Now since \( \pi^{-1}_{pre} \GG(U) := \varinjlim_{V \supset \pi(U)} \GG(V) \) and our choice of \(U, V\) is arbitrary, by the universal property of the colimit there exists an induced morphism \( \alpha : \pi^{-1}_{pre} \GG \to \FF \). Since \( \FF\) is a sheaf, by the universal property of sheafification there is an induced map \( \overline{\alpha} : \pi^{-1} \GG \to \FF \). Thus, our map \( \Mor_{YX} ( \GG, \FF ) \to \Mor_Y(\pi^{-1}\GG, \FF) \) is given by \( \phi_{VU} \mapsto \overline{\alpha} \) — this is inverse to our original map by the following:

We next wish to construct a bijection \( \Mor_Y( \GG, \pi_* \FF ) \leftrightarrow \Mor_{YX}( \GG, \FF ) \); first, if \( \psi : \GG, \pi_* \FF \). Then for any \( U' \subset U \subset X \), \( V' \subset V \subset Y \) with \( \pi(U') \subset V' \), \( \pi(U) \subset V \), we have \( U \subset \pi^{-1}(V) \) as open maps so we have a natural map \( \res{ \pi^{-1}(V) }{ U} : \pi_* \FF \to \FF \) giving us the following diagram

Thus, we define the map in the forward direction as \( \phi \mapsto \{ \res{\pi_{-1}(V)}{U} \circ \psi(V) \} \). For the reverse direction, let \( \psi = \{ \psi_{VU} \} \) be a compatible collection. Then for each \( V' \subset V \subset Y \), we have maps \( \psi_{V, \pi^{-1}(V)} : \GG(V) \to \FF( \pi^{-1}(V) ) \), \( \psi_{V', \pi^{-1}(V')} : \GG(V') \to \FF( \pi^{-1}(V') ) \). Since \( \FF ( \pi^{-1}(V) ) = \pi_* \FF(V) \) by definition, we get the desired morphism which clearly commutes with restrictions. The two are clearly inverse to one another.

Proof:

By Exercise 2.4.M stalks are preserved by sheafification, so

Now if \( U \) is any arbitrary neighborhood of \( p \), we wish to define a map \( \varinjlim_{V \supset \pi(U)} \GG(V) \to \varinjlim_{ V \supset q} \GG(V) \). Now for \( V_1, V_2 \) open sets containing \( \pi(U) \), \( V_1 \) and \(V_2\) must also necessarily be neighborhoods of \(q\), so we get canonical maps \( \GG(V_1) \to \varinjlim_{V \ni q} \GG(V) \) that commute with restrictions. By the universal property of the colimit, there is a unique map \( \iota_U \) making the following diagram commute:

It is fairly easy to see this commutes with restrictions since \( U_1 \subset U_2 \) implies \( \pi(U_1) \subset \pi(U_2) \), defining the restriction maps in the same way as Exercise 2.7.A, and making use of the fact that the map obtained by the universal proprety is unique. But then for \( p \in U_1 \subset U_2 \) we get a (co)cone

From this, we immediately get a map \( ( \pi^{-1} \GG)_p \to \GG_q \) by the universal property of the colimit. To show that this is an isomorphism, it suffices to show \( \GG_p \) satisfies universal property. Now if \(Z\) is another object and \( j_{U_1}, j_{U_2} \) are maps to \(Z\) that make the above diagram commute and \(V\) is an open neighborhood of \(q\), we have by the continuity of \(\pi\) that \( U = \pi^{-1}(V) \) is an open neighborhood of \(p\). Therefore, we define a map \( \psi_V : \GG(V) \to Z \) by \( j_U \) — since \(V \ni q\) was arbitrary, taking the colimit we obtain a unique map \( \psi : \GG_q \to Z \) as desired.

Proof:

Now for all \(W \subset U\) open

so that \( i^{-1}_{pre} \GG = \GG\vert_U \) as presheaves. Since \( \GG \) is a sheaf, sheafification is the identity (Exercise 2.4.G ) which gives us the result.

Proof:

From the discussion in the previous section (and the hint), it suffices to check on the level of stalks. If

is exact sequence of sheaves on \(Y\), then it is necessarily exact at all stalks. Now if \( p \in X \) is arbitrary, take \( q = \pi(p) \) — by Exercise 2.7.C, \( (\pi^{-1} \FF)_p \cong \FF_q \) and so forth for the other sheaves. By the first two sentences it follows that

is exact for all \(p\) so that the result follows.