Sheaves

Although a presheaf contains all local data, there is no way to glue them together to obtain global data, so we cannot expect a local-to-global property. A sheaf is a presheaf that satisfies a gluing condition such as §Presheaves, ⁋Lemma 1.

Definition 1 A presheaf $\mathcal{F}$ defined on a topological space $X$ is a sheaf if the following two conditions hold:

(Identity axiom) For any open set $U$ and open covering $\{U_i\}_{i\in I}$, if $s,t\in\mathcal{F}(U)$ satisfy $s\vert_{U_i}=t\vert_{U_i}$ for all $i\in I$, then $s=t$.
(Gluability axiom) For any open set $U$ and open covering $\{U_i\}_{i\in I}$, suppose we are given $s_i\in\mathcal{F}(U_i)$ such that $s_i\vert_{U_i\cap U_j}=s_j\vert_{U_i\cap U_j}$ for all $i,j$. Then there exists $s\in \mathcal{F}(U)$ such that $s\vert_{U_i}=s_i$ for all $i$.

For instance, by §Presheaves, ⁋Lemma 1, the presheaf of continuous functions from a topological space $X$ to $Y$ is a sheaf. (§Presheaves, ⁋Example 3)

For a fixed topological space $X$, we define the category $\Sh(X;\mathcal{A})$ of $\mathcal{A}$-valued sheaves as the full subcategory of $\PSh(X;\mathcal{A})$ whose objects are sheaves. That is, the inclusion functor $\Sh(X; \mathcal{A})\hookrightarrow \PSh(X;\mathcal{A})$ is the forgetful functor.

Now let us examine the stalks of a sheaf.

Proposition 2 For a topological space $X$, a sheaf $\mathcal{F}$ defined on $X$, and any open set $U$ of $X$, the map

\[\mathcal{F}(U) \rightarrow \prod_{x\in U} \mathcal{F}_x\]

is injective.

Proof

The given map is $s\mapsto (s_x)_{x\in U}$. Therefore, it suffices to show that if two elements $s,t\in \mathcal{F}(U)$ satisfy $s_x=t_x$ for all $x\in U$, then $s=t$.

From the expression of the stalk $\mathcal{F}_x$ examined in §Presheaves, §§Stalks, we know that for each $x\in U$, there exists an open set $x\in V_x\subseteq U$ such that $s\vert_{V_x}=t\vert_{V_x}$. The collection of such $V_x$ is an open covering of $U$, and therefore from the first condition of Definition 1, we have $s=t$.

If $\mathcal{F}$ is the sheaf of continuous functions from $X$ to $Y$, the above proposition roughly means that two continuous functions having the same value at every point are equal. On the other hand, since a function obtained by arbitrarily specifying values at each point is of course not necessarily continuous, it is clear that the above map is not surjective.

Definition 3 For a topological space $X$, a presheaf $\mathcal{F}$ defined on $X$, and any open set $U$ of $X$, an element $(s_x)_{x\in U}$ of the set $\prod_{x\in U} \mathcal{F}_x$ is said to consist of compatible germs if for each $x\in U$, there exists an open neighborhood $V_x\subseteq U$ such that, writing $s_x=[(\tilde{s}_x, V_x)]$, the equation

\[(\tilde{s}_x)_y=s_y\in \mathcal{F}_y\]

holds for all $y\in V_x$.

If $(s_x)_{x\in U}$ consists of compatible germs, then for any $x_1,x_2\in U$, the two elements of $\mathcal{F}(V_{x_1}\cap V_{x_2})$

\[\tilde{s}_{x_1}\vert_{V_{x_1}\cap V_{x_2}},\quad \tilde{s}_{x_2}\vert_{V_{x_1}\cap V_{x_2}}\]

are equal by Proposition 2. Therefore, using the second condition of Definition 1, we can find a suitable $\tilde{s}\in \mathcal{F}(U)$ such that the stalk of this element is $s_x$.

Then, for a presheaf $\mathcal{F}$ and a sheaf $\mathcal{G}$ defined on $X$, a presheaf morphism $\phi$ from $\mathcal{F}$ to $\mathcal{G}$ is completely determined by the morphisms $\phi_x: \mathcal{F}_x \rightarrow \mathcal{G}_x$ at each point $x$. Moreover, the following holds.

Proposition 4 On $X$, let $\phi:\mathcal{F}\rightarrow \mathcal{G}$ be a morphism between sheaves. Then $\phi$ is an isomorphism if and only if the morphisms $\phi_x:\mathcal{F}_x \rightarrow \mathcal{G}_x$ between the stalks are all isomorphisms.

Proof

First, if $\phi$ is an isomorphism, it is clear that the $\phi_x$ are isomorphisms. Thus, it suffices to show the converse. That is, if all $\phi_x$ are isomorphisms, we must prove that for every open set $U$, $\phi\vert_U$ is an isomorphism.

First, we show that $\phi\vert_U$ is always injective. Suppose $\phi(s)=0$ for some $s\in \mathcal{F}(U)$. Then for all $p\in U$,

\[0=(\phi\vert_U)(s)_p=\phi_p(s_p)\]

holds, and since we assumed that $\phi$ is an isomorphism at the stalk level, for all $p$ we have $s_p=0$. That is, for each $p\in U$, there exists a suitable open neighborhood of $p$, say $W\subseteq U$, such that $s\vert_W=0$ holds. Since the $W$ cover $U$, by the identity axiom we must have $s=0$.

We now prove that $\phi(U)$ is surjective. To this end, we take any $t\in \mathcal{G}(U)$ and need to make $(\phi\vert_U)(s)=t$ hold for some $s\in \mathcal{F}(U)$. Since $\phi$ is surjective at the stalk level, for each $p\in U$, there exists a germ satisfying $\phi_p(s_p)=t_p$, denoted $s_p$. Then there exists a suitable open neighborhood of $p$, say $W\subseteq U$, such that the germ of a section $s$ defined on it at $p\in U$ becomes $s_p$. Now,

\[(\phi\vert_W)(s)_p=\phi_p(s_p)=t_p\]

and so, by shrinking $W$ to a smaller open neighborhood of $p$ if necessary, we can make $(\phi\vert_W)(s)=t\vert_W$. Our claim is that the sections constructed in this way for each point $p$ satisfy the gluing condition, allowing us to produce a section defined on all of $U$; this can be proved as follows. Let $s\in \mathcal{F}(W)$ be the section obtained by applying the above argument to $p\in U$, and let $s'\in \mathcal{F}(W')$ be the section obtained by applying the above argument to $p'\in U$. Then on $W\cap指示 W'$,

\[(\phi\vert_{W\cap W'})(s\vert_{W\cap W'})=t\vert_{W\cap W'}=(\phi\vert_{W\cap W'})(s'\vert_{W\cap W'})\]

holds, so by the injectivity of $\phi\vert_{W\cap W'}$, we have $s\vert_{W\cap W'}=s'\vert_{W\cap W'}$. That sending the section obtained by gluing these $s$ through $\phi$ yields $t$ is again clear from the identity axiom of $\mathcal{G}$.

Abelian Category of Sheaves

Just as in the case of presheaves, to perform diagram chasing in the category of (abelian) sheaves, the kernels, cokernels, images, etc., of morphisms between sheaves must be sheaves. However, except for kernels, these are generally not sheaves, so a new definition is needed.

Definition 5 For any topological space $X$ and category $\mathcal{A}$, the left adjoint to the forgetful functor $\Sh(X;\mathcal{A})\rightarrow \PSh(X;\mathcal{A})$ is called the sheafification functor, and is denoted by $(-)^\dagger$.

To complete this definition, we need the following lemma.

Lemma 6 $(-)^\dagger:\PSh(X;\Set) \rightarrow \Sh(X;\Set)$ exists.

Although the proof of this is rather long, the core idea can be summarized as constructing a sheaf of compatible germs from the presheaf $\mathcal{F}$.

Proof of Lemma 6

As explained above, $\mathcal{F}^\dagger(U)$ is given by

\[\mathcal{F}^\dagger(U)=\{(s_x)_{x\in U}\mid\text{$(s_x)\in\prod \mathcal{F}_x$ consists of compatible germs}\}\]

First, $\mathcal{F}^\dagger$ is a sheaf. Let an open covering $U=\bigcup U_i$ be given. If $s\in \mathcal{F}^\dagger(U)$ satisfies, for all $i$, that $s\vert_{U_i}=0$, then $s=0$. This is clear because for any $p\in U$, if we suppose $p\in U_i$, then

\[s(p)=(s\vert_{U_i})(p)=0\]

holds. The gluability axiom is trivially satisfied since $\mathcal{F}^\dagger$ was defined from the start as a presheaf of functions. The presheaf morphism $\theta:\mathcal{F}\rightarrow \mathcal{F}^\dagger$ is naturally given, for any open set $U$, by

\[\theta\vert_U: \mathcal{F}(U) \rightarrow \mathcal{F}^\dagger(U);\qquad s\mapsto (p\mapsto s_p)\]

As for the universal property, if an arbitrary sheaf $\mathcal{G}$ and a morphism $\phi: \mathcal{F}\rightarrow \mathcal{G}$ are given, then for each open set $U$, we can define $\psi\vert_U$ by taking a function $s\in \mathcal{F}^\dagger(U)$, choosing for each point $p\in U$ an open set $V$ and $t\in \mathcal{F}(V)$ satisfying the condition of Definition 3, sending it to $(\phi\vert_V)(t)\in \mathcal{G}(V)$, and gluing these to define a section on $U$.

Moreover, the above proof still holds if we replace $\Set$ by $\Ab$ or $\Ring$. In addition, in all these cases, if $\mathcal{F}$ was already a sheaf from the beginning, the morphism $\mathcal{F}\rightarrow \mathcal{F}^\dagger$ obtained from Lemma 6 is an isomorphism.

As can be seen from the proof, for a presheaf $\mathcal{F}$ and its sheafification $\mathcal{F}^\dagger$, at any point $x\in X$, the stalks $\mathcal{F}_x$ and $\mathcal{F}^\dagger_x$ are isomorphic. This is clear from the fact that stalks are colimits and $(-)^\dagger$ is a left adjoint functor ([Category Theory] §Adjoint Functors, ⁋Theorem 9). Then by definition, we see that $\mathcal{F}^\dagger$ is exactly the sheaf of sections of the canonical projection $\Spe(\mathcal{F})\rightarrow X$ (§Presheaves, ⁋Example 3).

Definition 7 On $X$, let a sheaf $\mathcal{F}$ and a sheaf morphism $\phi:\mathcal{F}\rightarrow \mathcal{G}$ be given.

A subsheaf of $\mathcal{F}$ means that for each open set $U$, we have $\mathcal{F}'(U)\subseteq \mathcal{F}(U)$, and these inclusions form a presheaf morphism $\mathcal{F}'\hookrightarrow \mathcal{F}$, for a sheaf $\mathcal{F}'$.
The kernel of $\phi$ is the kernel of $\phi$ as a presheaf morphism. If $\ker\phi=0$, we call $\phi$ injective.
The image of $\phi$ refers to taking the image of $\phi$ as a presheaf morphism and applying sheafification to obtain the sheaf $(U\mapsto \im\phi\vert_U)^\dagger$; when $\phi$ is a sheaf morphism, $\im\phi$ is understood to be the sheaf obtained in this way (rather than the image presheaf as a presheaf morphism). In this case, that $\phi$ is surjective means that $\im\phi=\mathcal{G}$.

In a similar way, problematic notions that do not take values in the category of sheaves, such as cokernels or quotients, can all be defined using sheafification; that these objects satisfy the corresponding universal properties in the sheaf category $\Sh(X;\mathcal{A})$ can be shown (without constructions such as Lemma 6) using only the universal property of Definition 5.

Sheaves Defined on a Base

Meanwhile, applying Lemma 6, we can show that for a topological space $X$, in order to describe an arbitrary sheaf $\mathcal{F}$ defined on it, it suffices to know how this sheaf is defined on a base $\mathcal{B}$ of the topological space.

Proposition 8 Suppose we are given a topological space $X$ and its base $\mathcal{B}$, and for each $B\in \mathcal{B}$, an object $\mathcal{F}(B)$, and whenever $B_i,B_j\in \mathcal{B}$ satisfy $B_i\subseteq B_j$, restriction maps

\[\rho_{ji}: \mathcal{F}(B_j) \rightarrow \mathcal{F}(B_i)\]

are given, satisfying $\rho_{ii}=\id$, and whenever $B_i\subseteq B_j\subseteq B_k$, $\rho_{ki}=\rho_{ji}\circ \rho_{kj}$, as well as the following two conditions:

(Identity axiom) For any $B\in \mathcal{B}$ and open covering $\{B_i\in \mathcal{B}\}_{i\in I}$, if $s,t\in\mathcal{F}(B)$ satisfy $s\vert_{B_i}=t\vert_{B_i}$ for all $i\in I$, then $s=t$.
(Gluability axiom) For any $B\in \mathcal{B}$ and open covering $\{B_i\}_{i\in I}$, suppose we are given elements $s_i\in\mathcal{F}(B_i)$ such that for all $i,j$ and each subset of $B_i\cap B_j$ belonging to the base, $B'\in \mathcal{B}$, they satisfy $s_i\vert_{B'}=s_j\vert_{B'}$. Then there exists some $s\in \mathcal{F}(B)$ such that for all $i$, $s\vert_{B_i}=s_i$.

Then $\mathcal{F}$ can be uniquely extended to a sheaf on $X$.

Proof

First, we may set

\[\mathcal{F}_x=\varinjlim_{x\in B\in \mathcal{B}} \mathcal{F}(B)\]

Now for any open set $U$, if we define

\[\mathcal{F}(U)=\{(s:U \rightarrow \bigcup_{x\in U} \mathcal{F}_x)\mid s(x)\in \mathcal{F}_x,\quad\text{for $x\in U$, there exists $x\in B\subseteq U$ and $t\in \mathcal{F}(B)$ s.t. $s(y)=t_y$ for all $y\in B$}\}\]

then for the same reason as in Lemma 6, this defines a sheaf.

Examples of Sheaves

In the previous post, we examined examples of presheaves, some of which are also sheaves.

Example 9 The following are examples of sheaves:

Skyscraper sheaf $(i_x)_\ast A$ (§Presheaves, ⁋Example 5 (Skyscraper sheaf))
If $\mathcal{F}$ is a sheaf, the restriction $\mathcal{F}\vert_U$ and the pushforward $f_\ast \mathcal{F}$ are also sheaves. (§Presheaves, ⁋Example 7, §Presheaves, ⁋Example 8 (Pushforward))
Sheafification $\underline{A}$ of the constant presheaf (§Presheaves, ⁋Example 6 (Constant presheaf)), then $(i_x)_\ast A$ is the pushforward, via the inclusion $i_x:\{x\}\rightarrow X$, of the constant sheaf on $\{x\}$, $\underline{A}=A$.
Sheaf of sections (§Presheaves, ⁋Example 3)
If $\mathcal{G}$ is a sheaf, then $\sHom(\mathcal{F},\mathcal{G})$ is also a sheaf. (§Presheaves, ⁋Example 12 (Sheaf Hom)) Similarly, the product of sheaves is also a sheaf.

We do not provide separate proofs for these.

Inverse Image of a Sheaf

Earlier, we saw how to restrict a sheaf $\mathcal{F}$ on $X$ to an open subset $U$; by using the inverse image sheaf, we can extend this to an arbitrary subset. First, since the pushforward $f_\ast$ also preserves presheaf morphisms, $f_\ast:\PSh(X)\rightarrow \PSh(Y)$ is a functor, and the same holds for $f_\ast: \Sh(X) \rightarrow \Sh(Y)$ regarded as a functor defined on the category of sheaves.

Definition 10 The left adjoint of $f_\ast:\Sh(X) \rightarrow \Sh(Y)$ is called the inverse image sheaf and denoted by $f^{-1}$.

Concretely, for a sheaf on $Y$, say $\mathcal{G}$, $f^{-1}\mathcal{G}$ is given by considering the presheaf

\[f^{-1}_\text{pre}\mathcal{G}(U)=\varinjlim_{V\supseteq f(U)} \mathcal{G}(V)\]

and then taking the sheafification

\[f^{-1}\mathcal{G}=(f^{-1}_\text{pre}\mathcal{G})^\dagger=\left(U\mapsto \varinjlim_{V\supseteq f(U)} \mathcal{G}(V)\right)^\dagger\tag{$\ast$}\]

of this presheaf.

Lemma 11 The $f^{-1}$ defined by the equation ($\ast$) above satisfies the adjunction in Definition 10:

\[\Hom_{\Sh(X)}(f^{-1} \mathcal{G}, \mathcal{F})\cong \Hom_{\Sh(Y)}(\mathcal{G}, f_\ast \mathcal{F})\]

Proof

First, by Definition 5, giving a sheaf morphism $f^{-1}\mathcal{G} \rightarrow \mathcal{F}$ is equivalent to giving a presheaf morphism $f^{-1}_\text{pre} \mathcal{G} \rightarrow \mathcal{F}$, which is equivalent to giving, for any open subset of $X$, $U$, morphisms $f^{-1}_\text{pre}\mathcal{G}(U) \rightarrow \mathcal{F}(U)$ that properly respect restrictions. Meanwhile, by the universal property of the direct limit, the morphism

\[\varinjlim_{V\supseteq f(U)} \mathcal{G}(V) \rightarrow \mathcal{F}(U)\]

is determined by the morphisms

\[\mathcal{G}(V) \rightarrow \mathcal{F}(U), \qquad\text{$V$ open set in $Y$ containing $f(U)$}\]

Now suppose that a sheaf morphism $\phi: \mathcal{G} \rightarrow f_\ast \mathcal{F}$ is given. Then defining the corresponding morphism $\phi^\sharp: f^{-1}\mathcal{G} \rightarrow \mathcal{F}$ is equivalent to defining the morphisms $\mathcal{G}(V) \rightarrow \mathcal{F}(U)$ above, which can be defined by the following composition:

\[\mathcal{G}(V)\overset{\phi(V)}{\longrightarrow}f_\ast\mathcal{F}(V)=\mathcal{F}(f^{-1}(V)) \overset{\rho_{f^{-1}(V), U}}{\longrightarrow} \mathcal{F}(U)\]

Conversely, suppose that a morphism $\psi:f^{-1}\mathcal{G} \rightarrow \mathcal{F}$ is given, and let us define $\psi^\flat: \mathcal{G}\rightarrow f_\ast\mathcal{F}$. Composing this with $f^{-1}_\text{pre}\mathcal{G} \rightarrow f^{-1}\mathcal{G}$ to view it as $f^{-1}_\text{pre} \mathcal{G} \rightarrow \mathcal{F}$, if for any open set of $Y$, $V$, we evaluate this presheaf morphism on the open set of $X$, $f^{-1}(V)$, then for each $V$ we obtain

\[f^{-1}_\text{pre} \mathcal{G}(f^{-1}(V))=\varinjlim_{W\supseteq f(f^{-1}(V))}\mathcal{G}(W) \rightarrow \mathcal{F}(f^{-1}(V))\]

. However, since $f(f^{-1}(V))\subseteq V$ holds in general, this data contains $\mathcal{G}(V) \rightarrow \mathcal{F}(f^{-1}(V))$, and therefore $\psi^\flat:\mathcal{G} \rightarrow f_\ast \mathcal{F}$ is defined from this.

Now, a straightforward calculation shows that $\phi^{\sharp\flat}=\phi$ and $\psi^{\flat\sharp}=\psi$ hold, and that this bijection is natural.

Let us consider the following two cases of particular interest.

Example 12 Consider the case where the map $f$ is given by including an open subset of $Y$, $V$, into $Y$ via the open embedding $i:V\hookrightarrow Y$. Then, by §Subspaces, ⁋Lemma 2, any open subset of $V$, $U$, is also an open subset of $Y$, so

\[i^{-1}_\text{pre} \mathcal{G}(U)=\varinjlim_{W\supseteq i(U)} \mathcal{G}(W)=\mathcal{G}(i(U))\]

, and thus $i^{-1}_\text{pre} \mathcal{G}$ is already a sheaf and

\[i^{-1}\mathcal{G}=i^{-1}_\text{pre}\mathcal{G}=\mathcal{G}\vert_V\]

Example 13 Consider the inclusion of a point of $Y$, $i_y:\{y\}\rightarrow Y$. Then the adjunction of Lemma 11 gives

\[\Hom_{\Sh(\{y\})}(i_y^{-1} \mathcal{F}, \mathcal{G})\cong \Hom_{\Sh(Y)}(\mathcal{F},(i_y)_\ast \mathcal{G})\]

. Since

\[\Sh(\{y\})=\Set\qquad(\text{or, more generally $\Sh(\{y\}; \mathcal{A})=\mathcal{A}$)}\]

, we can identify $i_y^{-1}\mathcal{F}$ and $\mathcal{G}$ with the sets (or objects in the category in which the sheaves take values) $i_y^{-1}\mathcal{F}(\{y\})$ and $\mathcal{G}(\{y\})$, respectively.

Meanwhile, by definition, $i^{-1}\mathcal{F}$ is

\[\varinjlim_{V\supseteq \{y\}}\mathcal{F}(V)=\mathcal{F}_y\]

, and after replacing $\mathcal{G}$ with an appropriate object $G$, the right-hand side $(i_y)_\ast \mathcal{G}$ is the skyscraper sheaf formed from $G$ at $y\in Y$. That is, there exists the following adjunction:

\[\Hom_\mathcal{A}(\mathcal{F}_y, G)\cong \Hom_{\Sh(Y)}(\mathcal{F},(i_y)_\ast G)\]

Example 14 In Example 13, we dealt with the inclusion of a point $i_y:\{y\}\rightarrow Y$. Similarly, for an open embedding $j:U\hookrightarrow X$, we can define a sheaf functor $j_!$ called the extension by zero. On $U$, for a sheaf $\mathcal{F}$, consider the presheaf

\[j_{!,\text{pre}}(\mathcal{F})(V)=\begin{cases}\mathcal{F}(V) & V\subseteq U \\ 0 & V\not\subseteq U\end{cases}\]

However, this presheaf is not in general a sheaf. For instance, when $X=\mathbb{R}^1$, $U=\mathbb{R}^1\setminus\{0\}$, and $\mathcal{F}$ on $U$ is the constant sheaf $\underline{\mathbb{Z}}$, if we take $X=V_1\cup V_2$ with $0\in V_1$ and $V_2=X\setminus\{0\}$, then even though two elements of $\mathbb{Z}\oplus\mathbb{Z}$, $(a,b)$, agree on $V_1\cap V_2$, we have $j_{!,\text{pre}}(\mathcal{F})(X)=0$, so the gluing axiom fails. Therefore, sheafification is necessary, and we define $j_!\mathcal{F}$ to be the sheafification of this presheaf.

In the adjunction of Lemma 11, the pushforward $j_\ast$ is the right adjoint of $j^{-1}$. Symmetrically, $j_!$ is the left adjoint of $j^{-1}$. That is,

\[\Hom_{\Sh(X)}(j_!\mathcal{F}, \mathcal{G})\cong \Hom_{\Sh(U)}(\mathcal{F}, j^{-1}\mathcal{G})\]

holds. This bijection is obtained naturally in the setting where $j_!$ is the left adjoint of $j^{-1}$.

From the observation in Example 13, one direction of the following proposition is clear.

Proposition 15 On a topological space $X$, let a sheaf morphism $\phi: \mathcal{F}\rightarrow \mathcal{G}$ be given. Then the following are equivalent.

$\phi$ is surjective.
For any $x\in X$, $\phi_x: \mathcal{F}_x \rightarrow \mathcal{G}_x$ is surjective.

Proof

Since taking stalks is a left adjoint functor, it preserves colimits and thus also preserves cokernels, so it is clear that condition 1 implies condition 2. The converse direction is obtained essentially because the image of a sheaf morphism arises from the sheafification of stalks.

In a similar vein, the following holds, and additionally, since the sheaf kernel is defined as the presheaf kernel, the following holds.

Proposition 16 On a topological space $X$, let a sheaf morphism $\phi: \mathcal{F}\rightarrow \mathcal{G}$ be given. Then the following are all equivalent.

$\phi$ is injective.
For any $x\in X$, $\phi_x: \mathcal{F}_x \rightarrow \mathcal{G}_x$ is injective.
For any open set $U$, $\phi(U)$ is injective.

References

[Har] R. Hartshorne, Algebraic geometry. Graduate texts in mathematics. Springer, 1977.
[Vak] R. Vakil, The rising sea: Foundation of algebraic geometry. Available online.

K

Sheaves

Abelian Category of Sheaves

Sheaves Defined on a Base

Examples of Sheaves

Inverse Image of a Sheaf

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