# Solutions to The Rising Sea (Vakil)

### Section 2.1: Motivating Example: The Sheaf of Differentiable Functions

###### Exercise 2.1.A:
Show that this is the only maximal ideal of $$\OO_p$$. (Hint: Show that every element of $$\OO_p \backslash \mm_p$$ is invertible.)

###### Exercise 2.1.B:
Notice that $$\mm_p / \mm_p^2$$ is a module over $$\OO_p / \mm_p \cong \mathbb{R}$$, i.e., it is a real vector space. It turns out to be naturally (whatever that means) the cotangent space to the smooth manifold at $$p$$. This insight will prove handy later, when we define tangent and cotangent spaces of schemes. Prove this. (Rhetorical question for experts: what goes wrong if the sheaf of continuous functions is substituted for the sheaf of differentiable functions?)

### Section 2.2: Definition of a Sheaf and Presheaf

###### Exercise 2.2.A:
Given any topological space, we have a "category of open sets" (Example 1.2.9), where the objects are the open sets and the morphisms are inclusions. Verify that the data of a presheaf is precisely the data of a contravariant functor from the category of open sets of $$X$$ to the category of sets. (This interpretation is surprisingly useful.)

###### Exercise 2.2.B:
Show that the following are presheaves on $$\mathbb{C}$$ (with the classical topology), but not sheaves: (a) bounded functions, (b) holomorphic functions admitting a holomorphic square root.

###### Exercise 2.2.C:
The identity and gluability axioms may be interpreted as saying that $$\FF ( \bigcup_{i \in I} U_i )$$ is a certain limit. What is this limit?

###### Exercise 2.2.D:
1. Verify that the examples of §2.1 are indeed sheaves of differentiable function, or continuous functions, or smooth functions, or of functions on a manifold of $$\mathbb{R}^n$$)
2. Show that real-valued continuous functions on (open sets of) a topological space $$X$$ form a sheaf.

###### Exercise 2.2.E:
Now let $$\FF(U)$$ be the maps to $$S$$ that are locally constant, i.e., for any point $$p \in U$$, there is an open neighborhood of $$p$$ where the function is constant. Show that this is a sheaf. (A better description is this: endow $$S$$ with the discrete topology, and let $$\FF(U)$$ be the continuous maps $$U \to S$$.) This is called the constant sheaf (associated to $$S$$); do not confuse it with the constant presheaf. We denote this sheaf $$\underline{S}$$.

###### Exercise 2.2.F:
Suppose $$Y$$ is a topological space. Show that "continuous maps to $$Y$$" form a sheaf of sets on $$X$$. More precisely, to each open set $$U$$ of $$X$$, we associate the set of continuous maps of $$U$$ to $$Y$$. Show that this forms a sheaf. (Exercise 2.2.D(b), with $$Y = \mathbb{R}$$, and Exercise 2.2.E, with $$Y = S$$ with the discrete topology are both special cases.)

###### Exercise 2.2.G:
This is a fancier version of the previous exercise.
1. (sheaf of sections of a map) Suppose we are given a continuous map $$\mu : Y \to X$$. Show that "sections of $$\mu$$" form a sheaf. More precisely, to each open set $$U$$ of $$X$$, associate the set of continuous maps $$s : U \to Y$$ such that $$\mu \circ s = id\vert_U$$. Show that this forms a sheaf. (For those who have heard of vector bundles, these are a good example.) This is motivation for the phrase "section of a sheaf".
2. (This exercise is for those who know what a topological group is. If you don't know what a topological group is, you might be able to guess.) Suppose that $$Y$$ is a topological group. Show that continuous maps to $$Y$$ form a sheaf of $$groups.$$

###### Exercise 2.2.H:
Suppose $$\pi : X \to Y$$ is a continuous map, and $$\FF$$ is a presheaf on $$X$$. Then define $$\pi_* \FF$$ by $$\pi_*\FF (V) := \FF( \pi^{-1}(V) )$$, where $$V$$ is an open subset of $$Y$$. Show that $$\pi_* \FF$$ is a presheaf on $$Y$$, and is a sheaf if $$\FF$$ is. This is called the $$\textbf{pushforward}$$ or $$\textbf{direct}\ \textbf{image}$$ of $$\FF$$. More precisely, $$\pi_* \FF$$ is called the $$\textbf{pushforward}\ \textbf{of}\ \FF \ \textbf{by}\ \pi$$.

###### Exercise 2.2.I:
Suppose $$\pi : X \to Y$$ is a continuous map, and $$\FF$$ is a sheaf of sets (or rings or $$A$$-modules) on $$X$$. If $$\pi(p) = q$$, describe the natural morphism of stalks $$(\pi_* \FF)_q \to \FF_p$$. (You can use the explicit definition of stalk using representatives, §2.2.4, or the universal property, §2.2.5. If you prefer one way, you should try the other).

###### Exercise 2.2.J:
If $$( X, \OO_X )$$ is a ringed space and $$\FF$$ is an $$\OO_X$$-module, describe how for each $$p \in X$$, $$\FF_p$$ is an $$\OO_{X, p}$$-module.

### Section 2.3: Morphisms of Presheaves and Sheaves

###### Exercise 2.3.A:
If $$\phi : \FF \to \GG$$ is a morphism of presheaves on $$X$$ and $$p \in X$$, describe an induced morphism of stalks $$\phi_p : \FF_p \to \GG_p$$. Translation: taking the stalk at $$p$$ induces a functor $$\Set_X \to \Set$$. (Your proof will extend in obvious ways. For example, if $$\phi$$ is a morphism of $$\OO_X$$-modules, then $$\phi_p$$ is a map of $$\OO_{X,p}$$-modules. )

###### Exercise 2.3.B:
Suppose $$\pi : X \to Y$$ is a continuous map of topological spaces (i.e. a morphism in the category of topological spaces). Show that the pushforward gives a functor $$\pi_* : \Set_X \to \Set_Y$$. Here $$\Set$$ can be replaced by other categories. (Watch out for some possible confusion: a presheaf is a functor, and presheaves form a category. It may be best to forget that presheaves are a functor for now.)

###### Exercise 2.3.C:
Suppose $$\FF$$ and $$\GG$$ are two sheaves of sets on $$X$$. (In fact, it will suffice that $$\FF$$ is a presheaf.) Let $$\Hom (\FF , \GG )$$ be the collection of data $$\Hom(\FF, \GG)(U) := \Mor (\FF\vert_U, \GG\vert_U )$$ (Recall the notation $$\FF\vert_U$$, the restriction of the sheaf to the open set $$U$$, Example 2.2.8.) Show that this is a sheaf of sets on $$X$$. (To avoid a common confusion: the right side does not say $$\Mor (\FF(U),\GG(U))$$.) This sheaf is called “sheaf Hom”. (Strictly speaking, we should reserve Hom for when we are in an additive category, so this should possibly be called “sheaf Mor”. But the terminology “sheaf Hom” is too established to uproot.) It will be clear from your construction that, like Hom, Hom is a contravariant functor in its first argument and a covariant func- tor in its second argument.

###### Exercise 2.3.D:
1. If $$\FF$$ is a sheaf of sets on $$X$$, then show that $$\Hom(\{p\},\FF) \cong \FF$$, where $$\{p\}$$ is the constant sheaf associated to the one element set $$\{p\}$$.
2. If $$\FF$$ is a sheaf of abelian groups on $$X$$, then show that $$\Hom_{Ab_X} (\underline{\Z}, \FF ) \cong \FF$$ (an isomorphism of sheaves of abelian groups).
3. If $$\FF$$ is an $$\OO_X$$-module, then show that $$\Hom_{\Mod{\OO_X}} (\OO_X, f\FF) \cong \FF$$ (an isomorphism of $$\OO_X$$-modules).
A key idea in (b) and (c) is that 1 “generates” (in some sense) $$\Z$$ (in (b)) and $$\OO_X$$ (in (c)).

###### Exercise 2.3.E:
Show that $$\ker_{\textrm{pre}} \phi$$ is a presheaf. (Hint: if $$U \hookrightarrow V$$, define the restriction maps by chasing the following diagram: $$\require{amscd} \begin{CD} 0 @>>> \ker_{\textrm{pre}} \phi (V) @>>> \FF(V) @>>> \GG(V) \\ @. @VV{\exists !}V @VV{\res{V}{U} }V @VV{ \res{V}{U} }V \\ 0 @>>> \ker_{\textrm{pre}} \phi (U) @>>> \FF(U) @>>> \GG(U) \end{CD}$$

###### Exercise 2.3.F:
Show that the presheaf cokernel satisfies the universal property of cokernels (Definition 1.6.3) in the category of presheaves.

###### Exercise 2.3.G:
Show (or observe) that for a topological space $$X$$ with open set $$U$$, $$\FF \mapsto \FF(U)$$ gives a functor from presheaves of abelian groups on $$X$$, $$Ab_X^{\textrm{pre}}$$ to abelian groups, $$Ab$$. Then show that this functor is exact.

###### Exercise 2.3.H:
Show that a sequence of presheaves $$0 \to \FF_1 \to \FF_2 \to \dots \to \FF_n \to 0$$ is exact if and only if $$0 \to \FF_1(U) \to \FF_2(U) \to \dots \to \FF_n(U) \to 0$$ is exact for all $$U$$.

###### Exercise 2.3.I:
Suppose that $$\phi : \FF \to \GG$$ is a morphism of sheaves. Show that the presheaf kernel $$\ker_{\textrm{pre}} \phi$$ is in fact a sheaf. Show that it satisfies the universal property of kernels (Definition 1.6.3). (Hint: the second question follows immediately from the fact that $$\ker_{\textrm{pre}} \phi$$ satisfies the universal property in the category of presheaves.)

###### Exercise 2.3.J:
Let $$X$$ be $$\C$$ with the classical topology, let $$\underline{\Z}$$ be the constant sheaf on $$X$$ associated to $$\Z$$, $$\OO_X$$ the sheaf of holomorphic functions, and $$\FF$$ the presheaf of functions admitting a holomorphic logarithm. Describe an exact sequence of presheaves on $$X$$: $$0 \to \underline{\Z } \to \OO_X \to \FF \to 0$$ where $$\underline{\Z} \to \OO_X$$ is the natural inclusion and $$\OO_X \to \FF$$ is given by $$f \mapsto \textrm{exp}(2\pi i f)$$. (Be sure to verify exactness.) Show that $$\FF$$ is not a sheaf. (Hint: $$\FF$$ does not satisfy the gluability axiom. The problem is that there are functions that don’t have a logarithm but locally have a logarithm.) This will come up again in Example 2.4.10.

### Section 2.4: Properties Determined at the Level of Stalks, and Sheafification

###### Exercise 2.4.A:
Prove that a section of a sheaf of sets is determined by its germs, i.e., the natural map $$\FF(U) \to \prod_{p \in U} \FF_p$$ is injective. Hint 1: you won’t use the gluability axiom, so this is true for separated presheaves. Hint 2: it is false for presheaves in general, see Exercise 2.4.F, so you will use the identity axiom. (Your proof will also apply to sheaves of groups, rings, etc. — to categories of “sets with additional structure”. The same is true of many exercises in this section.)

###### Exercise 2.4.B:
Show that $$\textrm{Supp}(s)$$ is a closed subset of $$X$$.

###### Exercise 2.4.C:
Prove that any choice of compatible germs for a sheaf of sets $$\FF$$ over $$U$$ is the image of a section of $$\FF$$ over $$U$$. (Hint: you will use gluability.)

###### Exercise 2.4.D:
If $$\phi_1$$ and $$\phi_2$$ are morphisms from a presheaf of sets $$\FF$$ to a sheaf of sets $$\GG$$ that induce the same maps on each stalk, show that $$\phi_1 = \phi_2$$. Hint: consider the following diagram. $$\require{amscd} \begin{CD} \FF(U) @>>> \GG(U) \\ @VVV @VVV \\ \prod_{p \in U} \FF_p @>>> \prod_{p \in U} \GG_p \end{CD}$$

###### Exercise 2.4.E:
Show that a morphism of sheaves of sets is an isomorphism if and only if it induces an isomorphism of all stalks. Hint: Use (2.4.4.1). Once you have injectivity, show surjectivity, perhaps using Exercise 2.4.C, or gluability in some other way; this is more subtle.

###### Exercise 2.4.F:
1. Show that Exercise 2.4.A is false for general presheaves.
2. Show that Exercise 2.4.D is false for general presheaves.
3. Show that Exercise 2.4.E is false for general presheaves.
(General hint for finding counterexamples of this sort: consider a 2-point space with the discrete topology.)

###### Exercise 2.4.G:
Show that sheafification is unique up to unique isomorphism, assuming it exists. Show that if $$\FF$$ is a sheaf, then the sheafification is $$\textrm{id} : \FF \to \FF$$. (This should be second nature by now.)

###### Exercise 2.4.H:
Assume for now that sheafification exists. Use the universal property to show that for any morphism of presheaves $$\phi : \FF \to \GG$$, we get a natural induced morphism of sheaves $$\phi_{sh} : \FF^{sh} \to \GG^{sh}$$. Show that sheafification is a functor from presheaves on $$X$$ to sheaves on $$X$$.

###### Exercise 2.4.I:
Show that $$\FF^{sh}$$ (using the tautological restriction maps) forms a sheaf.

###### Exercise 2.4.J:
Describe a natural map of presheaves $$\textrm{sh} : \FF \to \FF^{sh}$$.

###### Exercise 2.4.K:
Show that the map sh satisfies the universal property of sheafification (Definition 2.4.6). (This is easier than you might fear.)

###### Exercise 2.4.L:
Show that the sheafification functor is left-adjoint to the forgetful functor from sheaves on $$X$$ to presheaves on $$X$$. This is not difficult — it is largely a restatement of the universal property. But it lets you use results from §1.6.12, and can "explain" why you don’t need to sheafify when taking kernel (why the presheaf kernel is already the sheaf kernel), and why you need to sheafify when taking cokernel and (soon, in Exercise 2.6.J) $$\otimes$$.

###### Exercise 2.4.M:
Show $$\FF \to \FF^{sh}$$ induces an isomorphism of stalks. (Possible hint: Use the concrete description of the stalks. Another possibility once you read Remark 2.7.3: judicious use of adjoints.)

###### Exercise 2.4.N:
Suppose $$\phi : \FF \to \GG$$ is a morphism of sheaves of sets on a topological space $$X$$. Show that the following are equivalent:
1. $$\phi$$ is a monomorphism in the category of sheaves.
2. $$\phi$$ is injective on the level of stalks: $$\phi_p : \FF_p \to \GG_p$$ is injective for all $$p \in X$$.
3. $$\phi$$ is injective on the level of open sets: $$\phi(U) : \FF(U) \to \GG(U)$$ is injective for all open $$U \subset X$$.
(Possible hints: for (b) implies (a), recall that morphisms are determined by stalks, Exercise 2.4.D. For (a) implies (c), use the "indicator sheaf" with one section over every open set contained in $$U$$, and no section over any other open set.) If these conditions hold, we say that $$\FF$$ is a subsheaf of $$\GG$$ (where the "inclusion" $$\phi$$ is sometimes left implicit).

###### Exercise 2.4.O:
Continuing the notation of the previous exercise, show that the following are equivalent.
1. $$\phi$$ is an epimorphism in the category of sheaves
2. $$\phi$$ is surjective on the level of stalks: $$\phi_p : \FF_p \to \GG_p$$ is surjective for all $$p \in X$$

###### Exercise 2.4.P:
Show that $$\textrm{exp} : \OO_X \to \OO^*_X$$ describes $$\OO_X^*$$ as a quotient sheaf of $$\OO_X$$. Find an open set on which this map is not surjective.

### Section 2.5: Recovering Sheaves from a "Sheaf on a Base"

###### Exercise 2.5.A:
Make this precise. How can you recover a sheaf $$\FF$$ from this partial information?

###### Exercise 2.5.B:
Verify that $$F(B) \to \FF(B)$$ is an isomorphism, likely by showing that it is injective and surjective (or else by describing the inverse map and verifying that it is indeed inverse). Possible hint: elements of $$\FF(B)$$ are determined by stalks, as are elements of $$F(B)$$.

###### Exercise 2.5.C:
Suppose $$\{ B_i \}$$ is a base for the topology of $$X$$. A morphism $$F \to G$$ of sheaves on the base is a collection of maps $$F(B_k) \to G(B_k)$$ such that the diagram $$\require{amscd} \begin{CD} F(B_i) @>>> G(B_i) \\ @V{\res{B_i}{B_j}}VV @VV{\res{B_i}{B_j}}V \\ F(B_j) @>>> G(B_j) \end{CD}$$ commutes for all $$B_j \hookrightarrow B_i$$.
1. Verify that a morphism of sheaves is determined by the induced morphism of sheaves on the base.
2. Show that a morphism of sheaves on the base gives a morphism of the induced sheaves. (Possible hint: compatible stalks.)

###### Exercise 2.5.D:
Suppose $$X = \bigcup U_i$$ is an open cover of $$X$$, and we have sheaves $$\FF_i$$ on $$U_i$$ along with isomorphisms $$\phi_{ij} : \FF_i\vert_{U_i \cap U_j} → \FF_j\vert_{ U_i \cap U_j}$$ (with $$\phi_{ii}$$ the identity) that agree on triple overlaps, i.e., $$\phi_{jk} \circ \phi_{ij} = \phi_{ik}$$ on $$U_i \cap U_j \cap Uk$$ (this is called the cocycle condition, for reasons we ignore). Show that these sheaves can be glued together into a sheaf $$\FF$$ on $$X$$ (unique up to unique isomorphism), such that $$\FF_i \cong \FF\vert_{U_i}$$, and the isomorphisms over $$U_i \cap U_j$$ are the obvious ones. (Thus we can "glue sheaves together", using limited patching information.) Warning: we are not assuming this is a finite cover, so you cannot use induction. For this reason this exercise can be perplexing. (You can use the ideas of this section to solve this problem, but you don’t necessarily need to. Hint: As the base, take those open sets contained in some $$U_i$$. Small observation: the hypothesis on $$\phi_{ii}$$ is extraneous, as it follows from the cocycle condition.)

###### Exercise 2.5.E:
Suppose a morphism of sheaves $$\FF \to \GG$$ on a base $$\{ B_i\}$$ is surjective for all $$B_i$$ (i.e., $$\FF(B_i) \to \GG(B_i)$$ is surjective for all $$i$$). Show that the corresponding morphism of sheaves (not on the base) is surjective (or more precisely: an epimorphism). The converse is not true, unlike the case for injectivity. This gives a useful sufficient criterion for “surjectivity”: a morphism of sheaves is an epimorphism (“surjective”) if it is surjective for sections on a base. You may enjoy trying this out with Example 2.4.10 (dealing with holomorphic functions in the classical topology on $$X = \C$$), showing that the exponential map $$\textrm{exp}: \OO_X → \OO_X^*$$ is surjective, using the base of contractible open sets.

### Section 2.6: Sheaves of Abelian Groups and $$\OO_X$$-Modules form Abelian Categories

###### Exercise 2.6.A:
Show that the stalk of the kernel is the kernel of the stalks: for all $$p \in X$$, there is a natural isomorphism $$( \ker (\FF \to \GG ) )_p \cong \ker (\FF_p \to \GG_p)$$

###### Exercise 2.6.B:
Show that the stalk of the cokernel is naturally isomorphic to the cokernel of the stalk.

###### Exercise 2.6.C:
Suppose $$\phi : \FF \to \GG$$ is a morphism of sheaves of abelian groups. Show that the image sheaf $$\im \phi$$ is the sheafification of the image presheaf. (You must use the definition of image in an abelian category. In fact, this gives the accepted definition of image sheaf for a morphism of sheaves of sets.) Show that the stalk of the image is the image of the stalk.

###### Exercise 2.6.D:
Show that taking the stalk of a sheaf of abelian groups is an exact functor. More precisely, if $$X$$ is a topological space and $$p \in X$$ is a point, show that taking the stalk at $$p$$ defines an exact functor $$Ab_X \to Ab$$.

###### Exercise 2.6.E:
Check that the exponential exact sequence (2.4.10.1) is exact.

###### Exercise 2.6.F:
Suppose $$U \subset X$$ is an open set, $$0 \to \FF \to \FF \to \mathscr{H}$$ is an exact sequence of sheaves of abelian groups. Show that $$\require{amscd} \begin{CD} 0 @>>> \FF(U) @>>> \GG(U) @>>> \mathscr{H}(U) \end{CD}$$ is exact. (You should do this "by hand", even if you realize there is a very fast proof using the left-exactness of the "forgetful" right adjoint to the sheafification functor.) Show that the section functor need not be exact: show that if $$0 \to \FF \to \GG \to \mathscr{H} \to 0$$ is an exact sequence of abelian groups, then $$\require{amscd} \begin{CD} 0 @>>> \FF(U) @>>> \GG(U) @>>> \mathscr{H}(U) @>>> 0 \end{CD}$$ need not be exact. (Hint: the exponential exact sequence (2.4.10.1). But feel free to make up a different example.)

###### Exercise 2.6.G:
Suppose $$0 \to \FF \to \GG \to \mathscr{H} \to 0$$ is an exact sequence of sheaves of abelian groups on $$X$$. If $$\pi : X \to Y$$ is a continuous map, show that $$\require{amscd} \begin{CD} 0 @>>> \pi_* \FF @>>> \pi_* \GG @>>> \pi_*\mathscr{H} \end{CD}$$ is exact. (The previous exercise, dealing with the left-exactness of the global section functor can be interpreted as a special case of this, in the case where $$Y$$ is a point.)

###### Exercise 2.6.H:
Suppose $$\FF$$ is a sheaf of abelian groups on a topological space $$X$$. Show that $$\mathcal{Hom}(\FF, \cdot)$$ is a left-exact covariant functor $$Ab_X \to Ab$$. Show that $$\mathcal{Hom}(\cdot, \FF)$$ is a left-exact contravariant functor $$Ab_X \to Ab$$.

###### Exercise 2.6.I:
Show that if $$( X, \OO_X )$$ is a ringed space, then $$\OO_X$$ form an abelian category. (There is a fair bit to check, but there aren’t many new ideas.)

###### Exercise 2.6.J:
1. Suppose $$\OO_X$$ is a sheaf of rings on $$X$$. Define (categorically) what we should mean by tensor product of two $$\OO_X$$-modules. Give an explicit construction, and show that it satisfies your categorical definition. Hint: take the "presheaf tensor product" — which needs to be defined — and sheafify. Note: $$\otimes_{\OO_X}$$ is often written $$\otimes$$ when the subscript is clear from the context. (An example showing sheafification is necessary will arise in Example 14.1.1.)
2. Show that the tensor product of stalks is the stalk of the tensor product. (If you can show this, you may be able to make sense of the phrase "colimits commute with tensor products".)

### Section 2.7: The Inverse Image Sheaf

###### Exercise 2.7.A:
Show that this defines a presheaf on $$X$$. Show that it needn’t form a sheaf. (Hint: map 2 points to 1 point.)

###### Exercise 2.7.B:
If $$\pi : X \to Y$$ is a continuous map, and $$\FF$$ is a sheaf on $$X$$ and $$\GG$$ is a sheaf on $$Y$$, describe a bijection $$\Mor_X( \pi^{-1} \GG, \FF ) \leftrightarrow \Mor_Y( \GG, \pi_* \FF )$$ Observe that your bijection is "natural" in the sense of the definition of adjoints (i.e., functorial in both $$\FF$$ and $$\GG$$ ). Thus Construction 2.7.2 satisfies the universal property of Definition 2.7.1. Possible hint: Show that both sides agree with the following third construction, which we denote $$\Mor_{YX} (\GG , \FF )$$. A collection of maps $$\phi_{VU} : \GG(V) → \FF(U)$$ (as $$U$$ runs through all open sets of $$X$$, and $$V$$ runs through all open sets of $$Y$$ containing $$\pi(U)$$) is said to be compatible if for all $$U' \subset U \subset X$$ and all open $$V' \subset V \subset Y$$ with $$\pi(U) \subset V$$, $$\pi(U') \subset V'$$, the diagram $$\require{amscd} \begin{CD} \GG(V) @>{ \phi_{VU}}>> \FF(U) \\ @V{ \res{V}{V'} }VV @VV{\res{U}{U'}}V \\ \GG(V') @>>{ \phi_{V'U'} }> \FF(U') \end{CD}$$ commutes. Define $$\Mor_{YX} ( \GG, \FF )$$ to be the set of all compatible collections $$\phi = \{\phi_{VU}\}$$.

###### Exercise 2.7.C:
Show that the stalks of $$\pi^{-1} \GG$$ are the same as the stalks of $$\GG$$. More precisely, if $$\pi(p) = q$$, describe a natural isomorphism $$\GG_q \cong (\pi^{-1}\GG)_p$$. (Possible hint: use the concrete description of the stalk, as a colimit. Recall that stalks are preserved by sheafification, Exercise 2.4.M. Alternatively, use adjointness.) This, along with the notion of compatible germs, may give you a simple way of thinking about (and perhaps visualizing) inverse image sheaves. (Those preferring the "espace étalé" or "space of sections" perspective, §2.2.11, can check that the pullback of the "space of sections" is the "space of sections" of the pullback.)

###### Exercise 2.7.D:
If $$U$$ is an open subset of $$Y$$, $$i: U \to Y$$ is the inclusion, and $$\GG$$ is a sheaf on $$Y$$, show that $$i^{-1} \GG$$ is naturally isomorphic to $$\GG\vert_U$$ (the restriction of $$\GG$$ to $$U$$, §2.2.8).

###### Exercise 2.7.E:
Show that $$\pi^{-1}$$ is an exact functor from sheaves of abelian groups on $$Y$$ to sheaves of abelian groups on $$X$$ (cf. Exercise 2.6.D). (Hint: exactness can be checked on stalks, and by Exercise 2.7.C, the stalks are the same.) Essentially the same argument will show that $$\pi^{-1}$$ is an exact functor from $$\OO_Y$$-modules (on $$Y$$) to $$(\pi^{-1}\OO_Y )$$-modules (on $$X$$), but don’t bother writing that down. (Remark for experts: $$\pi^{-1}$$ is a left adjoint, hence right-exact by abstract nonsense, as discussed in §1.6.12. Left-exactness holds because colimits of abelian groups over filtered index sets are exact, Exercise 1.6.K.)