Sheaves

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

A presheaf contains all local information, but since there is no way to glue this into global information, we cannot expect a local-to-global property. A sheaf is a presheaf that satisfies a gluing condition as in §Presheaves, ⁋Lemma 1.

Definition 1 A presheaf \(\mathcal{F}\) defined on a topological space \(X\) is called a sheaf if the following two conditions hold.

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

For example, 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 \(X\), the category \(\Sh(X,\mathcal{A})\) of \(\mathcal{A}\)-valued sheaves is defined 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 stalk of a sheaf.

Proposition 2 For a topological space \(X\), a sheaf \(\mathcal{F}\) defined on it, and any open set \(U\) of \(X\), the function

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

is injective.

Proof

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

From the description of the stalk \(\mathcal{F}_x\) that we saw in §Presheaves, §§Stalks, 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 \(V_x\) forms an open covering of \(U\), and therefore from the first condition of Definition 1 we know \(s=t\).

If we let \(\mathcal{F}\) be the sheaf of continuous functions from \(X\) to \(Y\), the above proposition roughly means that two continuous functions with the same value at every point are equal. On the other hand, it is obvious that a function obtained by arbitrarily specifying values at each point need not be continuous, so it is clear that the above function is not surjective.

Definition 3 For a topological space \(X\), a sheaf \(\mathcal{F}\) defined on it, 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 a suitable 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

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

of \(\mathcal{F}(V_{x_1}\cap V_{x_2})\) become the same element by Proposition 2. Therefore, using the second condition of Definition 1, we can find a suitable \(\tilde{s}\in \mathcal{F}(U)\) whose stalk 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 Suppose a morphism \(\phi:\mathcal{F}\rightarrow \mathcal{G}\) between sheaves defined on \(X\) is given. Then \(\phi\) being an isomorphism is equivalent to the morphisms \(\phi_x:\mathcal{F}_x \rightarrow \mathcal{G}_x\) on stalks all being isomorphisms.

Proof

First, if \(\phi\) is an isomorphism then 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 \(\phi\vert_U\) is an isomorphism for every open set \(U\).

First we prove that \(\phi\vert_U\) is always injective. Choose \(s\in \mathcal{F}(U)\) satisfying \(\phi(s)=0\). Then for all \(p\in U\),

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

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

Now we prove that \(\phi(U)\) is surjective. To do this, take arbitrary \(t\in \mathcal{G}(U)\) and construct \(s\in \mathcal{F}(U)\) satisfying \((\phi\vert_U)(s)=t\). Since \(\phi\) is surjective at the stalk level, for each \(p\in U\) there exist \(s_p\) satisfying \(\phi_p(s_p)=t_p\). Then there exists a suitable open neighborhood \(W\subseteq U\) of \(p\) such that we can arrange the germ of a section \(s\) defined on it at \(p\in U\) to be \(s_p\). Now since

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

if necessary we may restrict \(W\) to a smaller open neighborhood of \(p\) so that \((\phi\vert_W)(s)=t\vert_W\). Our claim is that the sections constructed pointwise in this way satisfy the gluing condition well, so we can produce a section defined over all of \(U\), and 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 it to \(p'\in U\). Then on \(W\cap W'\),

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

holds, so by the injectivity of \(\phi\vert_{W\cap W'}\) we have \(s\vert_{W\cap W'}=s'\vert_{W\cap W'}\). That the section formed by gluing these \(s\) maps to \(t\) under \(\phi\) is again clear from the identity axiom of \(\mathcal{G}\).

The Abelian Category of Sheaves

As with presheaves, in order to do diagram chasing in the category of (abelian) sheaves, kernels, cokernels, images, etc. of morphisms between sheaves must themselves be sheaves. However, since these are generally not sheaves except for the kernel, new definitions are needed.

Definition 5 For any topological space \(X\) and category \(\mathcal{A}\), the left adjoint of 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, the following lemma is needed.

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

The proof of this is somewhat long, but the essence can be summarized as constructing the sheaf of compatible germs from a presheaf \(\mathcal{F}\).

Proof of Lemma 6

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

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

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

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

The gluability axiom is trivially satisfied because \(\mathcal{F}^\dagger\) was defined as a presheaf of functions from the beginning. The presheaf morphism \(\theta:\mathcal{F}\rightarrow \mathcal{F}^\dagger\) is of course given for any open set \(U\) by

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

For the universal property, if an arbitrary morphism \(\phi: \mathcal{F}\rightarrow \mathcal{G}\) is given, then for each open set \(U\) we define \(\psi\vert_U\) by taking a function \(s\in \mathcal{F}^\dagger(U)\), picking for each point \(p\in U\) an open set \(V\) and \(t\in \mathcal{F}(V)\) satisfying the second condition above, mapping to \((\phi\vert_V)(t)\in \mathcal{G}(V)\), and gluing these to produce a section over \(U\).

Moreover, the above proof still works if we replace \(\Set\) with \(\Ab\), \(\Ring\), etc. Also, in all these cases, if \(\mathcal{F}\) was a sheaf to begin with, 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 \(\mathcal{F}^\dagger\), the stalks \(\mathcal{F}_x\) and \(\mathcal{F}^\dagger_x\) at any point \(x\in X\) are isomorphic. This is clear from the fact that the stalk is a colimit and \((-)^\dagger\) is a left adjoint functor. (§Adjoint Functors, ⁋Theorem 9) Then by definition we can 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 Suppose a sheaf \(\mathcal{F}\) defined on \(X\) and a sheaf morphism \(\phi:\mathcal{F}\rightarrow \mathcal{G}\) are given.

1. A subsheaf of \(\mathcal{F}\) is a sheaf \(\mathcal{F}'\) satisfying \(\mathcal{F}'(U)\subseteq \mathcal{F}(U)\) for each open set \(U\).
2. The kernel of \(\phi\) is the kernel of \(\phi\) as a presheaf morphism. If \(\ker\phi=0\), we call \(\phi\) injective.
3. The image of \(\phi\) is the sheaf obtained by applying sheafification to the image of \(\phi\) as a presheaf morphism, i.e., 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 (not the image presheaf as a presheaf morphism). In this case, \(\phi\) is surjective if \(\im\phi=\mathcal{G}\).

In a similar way, cokernels, quotients, and anything else that does not take values in the category of sheaves can all be defined using sheafification, and the fact that these objects satisfy the corresponding universal property in the sheaf category \(\Sh(X,\mathcal{A})\) can be shown using only the universal property of Definition 5 (without a construction like Lemma 6).

Sheaves Defined on a Base

On the other hand, applying Lemma 6, we can show that to describe an arbitrary sheaf \(\mathcal{F}\) defined on a topological space \(X\), it suffices to know how this sheaf is defined on a base \(\mathcal{B}\) of the topology.

Proposition 8 Suppose a topological space \(X\) and a base \(\mathcal{B}\) for it are given, and for each \(B\in \mathcal{B}\) we are given \(\mathcal{F}(B)\) and restriction maps

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

defined whenever \(B_i,B_j\in \mathcal{B}\) satisfy \(B_i\subseteq B_j\), and suppose they satisfy the following two conditions:

1. (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 \(s=t\).
2. (Gluability axiom) For any \(B\in \mathcal{B}\) and open covering \(\{B_i\}_{i\in I}\), suppose \(s_i\in\mathcal{F}(B_i)\) are given and they satisfy \(s_i\vert_{B_i\cap B_j}=s_j\vert_{B_i\cap B_j}\) for all \(i,j\). Then there exists some \(s\in \mathcal{F}(B)\) such that \(s\vert_{B_i}=s_i\) for all \(i\).

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

Proof

First set

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

Now for any open set \(U\), define

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

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

We do not write out the proofs separately.

Inverse Image of a Sheaf

We previously saw how to restrict a sheaf \(\mathcal{F}\) to an open set \(U\) of \(X\); using the inverse image sheaf, we can extend this to arbitrary subsets. First, since 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 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 is denoted by \(f^{-1}\).

Concretely, for a sheaf \(\mathcal{G}\) defined on \(Y\), the sheaf \(f^{-1}\mathcal{G}\) is given by first considering the presheaf

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

and then taking the sheafification of this presheaf:

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

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

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

Proof

First, by §Sheaves, ⁋Definition 5, giving a sheaf morphism \(f^{-1}\mathcal{G} \rightarrow \mathcal{F}\) is the same as giving a presheaf morphism \(f^{-1}_\text{pre} \mathcal{G} \rightarrow \mathcal{F}\), which in turn is the same as giving morphisms \(f^{-1}_\text{pre}\mathcal{G}(U) \rightarrow \mathcal{F}(U)\) for each open set \(U\) of \(X\) that preserve restrictions. On the other hand, a morphism

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

is determined by

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

via the universal property of the direct limit.

Now suppose a sheaf morphism \(\phi: \mathcal{G} \rightarrow f_\ast \mathcal{F}\) is given; then defining the corresponding \(\phi^\sharp: f^{-1}\mathcal{G} \rightarrow \mathcal{F}\) is the same as defining the above \(\mathcal{G}(V) \rightarrow \mathcal{F}(U)\), and this can be defined by the formula

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

Conversely, suppose a morphism \(\psi:f^{-1}\mathcal{G} \rightarrow \mathcal{F}\) is given and define \(\psi^\flat: \mathcal{G}\rightarrow f_\ast\mathcal{F}\). Compose this with \(f^{-1}_\text{pre} \rightarrow f^{-1}\mathcal{G}\) to view it as \(f^{-1}_\text{pre} \mathcal{G} \rightarrow \mathcal{F}\), and then evaluate this presheaf morphism at the open set \(f^{-1}(V)\) of \(X\) for an arbitrary open set \(V\) of \(Y\); we obtain for each \(V\)

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

Since in general \(f(f^{-1}(V))\subseteq V\) holds, this data includes \(\mathcal{G}(V) \rightarrow \mathcal{F}(f^{-1}(V))\), and thus \(\psi^\flat:\mathcal{G} \rightarrow f_\ast \mathcal{F}\) is defined from this.

Now \(\phi^{\sharp\flat}=\phi\) and \(\psi^{\flat\sharp}=\psi\) hold, and that this bijection is natural can be shown by a straightforward computation.

Let us look at the following two particularly noteworthy cases.

Example 12 Consider the case where the function \(f\) is an open embedding \(i:V\hookrightarrow Y\) sending an open set \(V\) of \(Y\) into \(Y\). Then by §Subspaces, ⁋Lemma 2, any open set \(U\) of \(V\) is also open in all of \(Y\), so

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

and therefore \(i^{-1}_\text{pre} \mathcal{G}\) is already a sheaf and

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

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

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

However,

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

so \(i_y^{-1}\mathcal{F}\) and \(\mathcal{G}\) can be identified with the set (or object of the category in which the sheaf takes values) \(i_y^{-1}\mathcal{F}(\{y\})\) and \(\mathcal{G}(\{y\})\) respectively.

On the other hand, \(i^{-1}\mathcal{F}\) is by its definition

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

and if we replace \(\mathcal{G}\) with a suitable object \(G\), then \((i_y)_\ast \mathcal{G}\) on the right-hand side is the skyscraper sheaf at \(y\in Y\) made from \(G\). That is, the following adjunction exists:

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

Example 14 In Example 13 we dealt with the inclusion of a single point \(i_y:\{y\}\rightarrow Y\). Similarly, for an open embedding \(j:U\hookrightarrow X\) we can define a sheaf functor \(j_!\) called extension by zero. For a sheaf \(\mathcal{F}\) defined on \(U\), 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 generally not a sheaf. For example, if \(X=\mathbb{R}^1\), \(U=\mathbb{R}^1\setminus\{0\}\), and \(\mathcal{F}\) is the constant sheaf \(\underline{\mathbb{Z}}\) on \(U\), then taking \(X=V_1\cup V_2\) with \(0\in V_1\) and \(V_2=X\setminus\{0\}\), even if two elements \((a,b)\) of \(\mathbb{Z}\oplus\mathbb{Z}\) agree well on \(V_1\cap V_2\), since \(j_{!,\text{pre}}(\mathcal{F})(X)=0\) the gluing axiom fails. Therefore sheafification is needed, and we define \(j_!\mathcal{F}\) as the sheafification of this presheaf.

In the adjunction of Lemma 11, 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 arises naturally from the setup where \(j_!\) is the left adjoint of \(j^{-1}\).

From this observation, one direction of the following proposition is clear.

Proposition 15 Suppose a sheaf morphism \(\phi: \mathcal{F}\rightarrow \mathcal{G}\) defined on a topological space \(X\) is given. Then the following are equivalent.

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

Proof

Taking stalks is a left adjoint functor, so it preserves colimits and hence cokernels; thus condition 1 implies condition 2 is clear. The converse direction follows basically from the fact that the image of a sheaf morphism comes from the sheafification of the stalks.

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

Proposition 16 Suppose a sheaf morphism \(\phi: \mathcal{F}\rightarrow \mathcal{G}\) defined on a topological space \(X\) is given. Then the following are all equivalent.

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

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References

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

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