7.26 Glueing sheaves

This section is the analogue of Sheaves, Section 6.33.

Lemma 7.26.1. Let $\mathcal{C}$ be a site. Let $\{ U_ i \to U\}$ be a covering of $\mathcal{C}$. Let $\mathcal{F}$, $\mathcal{G}$ be sheaves on $\mathcal{C}$. Given a collection

$\varphi _ i : \mathcal{F}|_{\mathcal{C}/U_ i} \longrightarrow \mathcal{G}|_{\mathcal{C}/U_ i}$

of maps of sheaves such that for all $i, j \in I$ the maps $\varphi _ i, \varphi _ j$ restrict to the same map $\varphi _{ij} : \mathcal{F}|_{\mathcal{C}/U_ i \times _ U U_ j} \to \mathcal{G}|_{\mathcal{C}/U_ i \times _ U U_ j}$ then there exists a unique map of sheaves

$\varphi : \mathcal{F}|_{\mathcal{C}/U} \longrightarrow \mathcal{G}|_{\mathcal{C}/U}$

whose restriction to each $\mathcal{C}/U_ i$ agrees with $\varphi _ i$.

Proof. The restrictions used in the lemma are those of Lemma 7.25.8. Let $V/U$ be an object of $\mathcal{C}/U$. Set $V_ i = U_ i \times _ U V$ and denote $\mathcal{V} = \{ V_ i \to V\}$. Observe that $(U_ i \times _ U U_ j) \times _ U V = V_ i \times _ V V_ j$. Then we have $\mathcal{F}|_{\mathcal{C}/U_ i}(V_ i/U_ i) = \mathcal{F}(V_ i)$ and $\mathcal{F}|_{\mathcal{C}/U_ i \times _ U U_ j}(V_ i \times _ V V_ j/U_ i \times _ U U_ j) = \mathcal{F}(V_ i \times _ V V_ j)$ and similarly for $\mathcal{G}$. Thus we can define $\varphi$ on sections over $V$ as the dotted arrows in the diagram

$\xymatrix{ \mathcal{F}(V) \ar@{=}[r] & H^0(\mathcal{V}, \mathcal{F}) \ar@{..>}[d] \ar[r] & \prod \mathcal{F}(V_ i) \ar[d]_{\prod \varphi _ i} \ar@<1ex>[r] \ar@<-1ex>[r] & \prod \mathcal{F}(V_ i \times _ V V_ j) \ar[d]_{\prod \varphi _{ij}} \\ \mathcal{G}(V) \ar@{=}[r] & H^0(\mathcal{V}, \mathcal{G}) \ar[r] & \prod \mathcal{G}(V_ i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod \mathcal{G}(V_ i \times _ V V_ j) }$

The equality signs come from the sheaf condition; see Section 7.10 for the notation $H^0(\mathcal{V}, -)$. We omit the verification that these maps are compatible with the restriction maps. $\square$

The previous lemma implies that given two sheaves $\mathcal{F}$, $\mathcal{G}$ on a site $\mathcal{C}$ the rule

$U \longmapsto \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)}( \mathcal{F}|_{\mathcal{C}/U}, \mathcal{G}|_{\mathcal{C}/U})$

defines a sheaf $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{F},\mathcal{G})$. This is a kind of internal hom sheaf. It is seldom used in the setting of sheaves of sets, and more usually in the setting of sheaves of modules, see Modules on Sites, Section 18.27.

Lemma 7.26.2. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$, $\mathcal{G}$ and $\mathcal{H}$ be sheaves on $\mathcal{C}$. There is a canonical bijection

$\mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{F}\times \mathcal{G},\mathcal{H}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{F},\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G},\mathcal{H}))$

which is functorial in all three entries.

Proof. The lemma says that the functors $-\times \mathcal{G}$ and $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G},-)$ are adjoint to each other. To show this, we use the notion of unit and counit, see Categories, Section 4.24. The unit

$\eta _\mathcal {F} : \mathcal{F} \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G},\mathcal{F}\times \mathcal{G})$

sends $s \in \mathcal{F}(U)$ to the map $\mathcal{G}|_{\mathcal{C}/U} \to \mathcal{F}|_{\mathcal{C}/U}\times \mathcal{G}|_{\mathcal{C}/U}$ which over $V/U$ is given by

$\mathcal{G}(V) \longrightarrow \mathcal{F}(V)\times \mathcal{G}(V), \quad t \longmapsto (s|_{V},t).$

The counit

$\epsilon _{\mathcal{H}} : \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G}, \mathcal{H}) \times \mathcal{G} \longrightarrow \mathcal{H}$

is the evaluation map. It is given by the rule

$\mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)}( \mathcal{G}|_{\mathcal{C}/U}, \mathcal{H}|_{\mathcal{C}/U}) \times \mathcal{G}(U) \longrightarrow \mathcal{H}(U),\quad (\varphi , s) \longmapsto \varphi (s).$

Then for each $\varphi : \mathcal{F} \times \mathcal{G} \to \mathcal{H}$, the corresponding morphism $\mathcal{F} \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G},\mathcal{H})$ is given by mapping each section $s \in \mathcal{F}(U)$ to the morphism of sheaves on $\mathcal{C}/U$ which on sections over $V/U$ is given by

$\mathcal{G}(V) \longrightarrow \mathcal{H}(V),\quad t \longmapsto \varphi (s|_ V, t).$

Conversely, for each $\psi : \mathcal{F} \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G}, \mathcal{H})$, the corresponding morphism $\mathcal{F} \times \mathcal{G} \to \mathcal{H}$ is given by

$\mathcal{F}(U) \times \mathcal{G}(U) \longrightarrow \mathcal{H}(U),\quad (s, t) \longmapsto \psi (s)(t)$

on sections over an object $U$. We omit the details of the proof showing that these constructions are mutually inverse. $\square$

Lemma 7.26.3. Let $\mathcal{C}$ be a site and $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (h_ U^\# , \mathcal{F}) = j_*(\mathcal{F}|_{\mathcal{C}/U})$ for $\mathcal{F}$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

Proof. This can be shown by directly constructing an isomorphism of sheaves. Instead we argue as follows. Let $\mathcal{G}$ be a sheaf on $\mathcal{C}$. Then

\begin{align*} \mathop{\mathrm{Mor}}\nolimits (\mathcal{G}, j_*(\mathcal{F}|_{\mathcal{C}/U})) & = \mathop{\mathrm{Mor}}\nolimits (\mathcal{G}|_{\mathcal{C}/U}, \mathcal{F}|_{\mathcal{C}/U}) \\ & = \mathop{\mathrm{Mor}}\nolimits (j_!(\mathcal{G}|_{\mathcal{C}/U}), \mathcal{F}) \\ & = \mathop{\mathrm{Mor}}\nolimits (\mathcal{G} \times h_ U^\# , \mathcal{F}) \\ & = \mathop{\mathrm{Mor}}\nolimits (\mathcal{G}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (h_ U^\# , \mathcal{F})) \end{align*}

and we conclude by the Yoneda lemma. Here we used Lemmas 7.26.2 and 7.25.7. $\square$

Let $\mathcal{C}$ be a site. Let $\{ U_ i \to U\} _{i \in I}$ be a covering of $\mathcal{C}$. For each $i \in I$ let $\mathcal{F}_ i$ be a sheaf of sets on $\mathcal{C}/U_ i$. For each pair $i, j \in I$, let

$\varphi _{ij} : \mathcal{F}_ i|_{\mathcal{C}/U_ i \times _ U U_ j} \longrightarrow \mathcal{F}_ j|_{\mathcal{C}/U_ i \times _ U U_ j}$

be an isomorphism of sheaves of sets. Assume in addition that for every triple of indices $i, j, k \in I$ the following diagram is commutative

$\xymatrix{ \mathcal{F}_ i|_{\mathcal{C}/U_ i \times _ U U_ j \times _ U U_ k} \ar[rr]_{\varphi _{ik}} \ar[rd]_{\varphi _{ij}} & & \mathcal{F}_ k|_{\mathcal{C}/U_ i \times _ U U_ j \times _ U U_ k} \\ & \mathcal{F}_ j|_{\mathcal{C}/U_ i \times _ U U_ j \times _ U U_ k} \ar[ru]_{\varphi _{jk}} }$

We will call such a collection of data $(\mathcal{F}_ i, \varphi _{ij})$ a glueing data for sheaves of sets with respect to the covering $\{ U_ i \to U\} _{i \in I}$.

Lemma 7.26.4. Let $\mathcal{C}$ be a site. Let $\{ U_ i \to U\} _{i \in I}$ be a covering of $\mathcal{C}$. Given any glueing data $(\mathcal{F}_ i, \varphi _{ij})$ for sheaves of sets with respect to the covering $\{ U_ i \to U\} _{i \in I}$ there exists a sheaf of sets $\mathcal{F}$ on $\mathcal{C}/U$ together with isomorphisms

$\varphi _ i : \mathcal{F}|_{\mathcal{C}/U_ i} \to \mathcal{F}_ i$

such that the diagrams

$\xymatrix{ \mathcal{F}|_{\mathcal{C}/U_ i \times _ U U_ j} \ar[d]_{\text{id}} \ar[r]_{\varphi _ i} & \mathcal{F}_ i|_{\mathcal{C}/U_ i \times _ U U_ j} \ar[d]^{\varphi _{ij}} \\ \mathcal{F}|_{\mathcal{C}/U_ i \times _ U U_ j} \ar[r]^{\varphi _ j} & \mathcal{F}_ j|_{\mathcal{C}/U_ i \times _ U U_ j} }$

are commutative.

Proof. Let us describe how to construct the sheaf $\mathcal{F}$ on $\mathcal{C}/U$. Let $a : V \to U$ be an object of $\mathcal{C}/U$. Then

$\mathcal{F}(V/U) = \{ (s_ i)_{i \in I} \in \prod _{i \in I} \mathcal{F}_ i(U_ i \times _ U V/U_ i) \mid \varphi _{ij}(s_ i|_{U_ i \times _ U U_ j \times _ U V}) = s_ j|_{U_ i \times _ U U_ j \times _ U V} \}$

We omit the construction of the restriction mappings. We omit the verification that this is a sheaf. We omit the construction of the isomorphisms $\varphi _ i$, and we omit proving the commutativity of the diagrams of the lemma. $\square$

Let $\mathcal{C}$ be a site. Let $\{ U_ i \to U\} _{i \in I}$ be a covering of $\mathcal{C}$. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}/U$. Associated to $\mathcal{F}$ we have its canonical glueing data given by the restrictions $\mathcal{F}|_{\mathcal{C}/U_ i}$ and the canonical isomorphisms

$\left(\mathcal{F}|_{\mathcal{C}/U_ i}\right)|_{\mathcal{C}/U_ i \times _ U U_ j} = \left(\mathcal{F}|_{\mathcal{C}/U_ j}\right)|_{\mathcal{C}/U_ i \times _ U U_ j}$

coming from the fact that the composition of the functors $\mathcal{C}/U_ i \times _ U U_ j \to \mathcal{C}/U_ i \to \mathcal{C}/U$ and $\mathcal{C}/U_ i \times _ U U_ j \to \mathcal{C}/U_ j \to \mathcal{C}/U$ are equal.

Lemma 7.26.5. Let $\mathcal{C}$ be a site. Let $\{ U_ i \to U\} _{i \in I}$ be a covering of $\mathcal{C}$. The category $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)$ is equivalent to the category of glueing data via the functor that associates to $\mathcal{F}$ on $\mathcal{C}/U$ the canonical glueing data.

Proof. In Lemma 7.26.1 we saw that the functor is fully faithful, and in Lemma 7.26.4 we proved that it is essentially surjective (by explicitly constructing a quasi-inverse functor). $\square$

Let $\mathcal{C}$ be a site. We are going to discuss a version of glueing sheaves on the entire site $\mathcal{C}$. For each object $U$ in $\mathcal{C}$, let $\mathcal{F}_ U$ be a sheaf on $\mathcal{C}/U$. Recall that there is a functor $j_ f : \mathcal{C}/V \rightarrow \mathcal{C}/U$ associated to each morphism $f : V \rightarrow U$ in $\mathcal{C}$, given by $(a : W \to V) \mapsto (f \circ a : W \to U)$. For each such $f$, let

$c_ f : j_ f^{-1} \mathcal{F}_ U \to \mathcal{F}_ V$

be an isomorphism of sheaves. Assume that given any two arrows $f : V \to U$ and $g : W \to V$ in $\mathcal{C}$, the composition $c_ g \circ j_{g}^{-1} c_ f$ is equal to $c_{f \circ g}$. We will call such a collection of data $(\mathcal{F}_{U}, c_ f)$ an absolute glueing data for sheaves of sets on $\mathcal{C}$. A morphism of absolute glueing data $(\mathcal{F}_{U}, c_ f) \to (\mathcal{G}_{U}, c'_ f)$ is given by a collection $(\varphi _ U)$ of morphisms of sheaves $\varphi _ U : \mathcal{F}_ U \to \mathcal{G}_ U$, such that

$\xymatrix{ j_ f^{-1} \mathcal{F}_ U \ar[r]_{c_ f} \ar[d]_{j_ f^{-1} \varphi _ U} & \mathcal{F}_ V \ar[d]^{\varphi _ V} \\ j_ f^{-1} \mathcal{G}_ U \ar[r]^{c'_ f} & \mathcal{G}_ V }$

commutes for every morphism $f : V \to U$ in $\mathcal{C}$.

Associated to any sheaf $\mathcal{F}$ on $\mathcal{C}$ is its canonical absolute glueing data $(\mathcal{F}|_{\mathcal{C}/U},c_ f)$, where the canonical isomorphisms $c_ f : j_ f^{-1} \mathcal{F}|_{\mathcal{C}/U} \to \mathcal{F}|_{\mathcal{C}/V}$ for $f : V \to U$ come from the relation $j_ V = j_ U \circ j_ f$ as in Lemma 7.25.8. Any morphism $\varphi : \mathcal{F} \to \mathcal{G}$ of sheaves of $\mathcal{C}$ induces a morphism $(\varphi |_{\mathcal{C}/U})$ of canonical absolute glueing data.

Lemma 7.26.6. Let $\mathcal{C}$ be a site. The category $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is equivalent to the category of absolute glueing data via the functor that associates to $\mathcal{F}$ on $\mathcal{C}$ the canonical absolute glueing data.

Proof. Given an absolute glueing data $(\mathcal{F}_ U, c_ f)$, we construct a sheaf $\mathcal{F}$ on $\mathcal{C}$ by setting $\mathcal{F}(U) = \mathcal{F}_ U(U)$, where restriction along $f : V \to U$ given by the commutative diagram

$\xymatrix{ \mathcal{F}_ U(U) \ar[r] \ar@{=}[d] & \mathcal{F}_ U(V) \ar[r]^{c_ f} & \mathcal{F}_ V(V) \ar@{=}[d] \\ \mathcal{F}(U) \ar[rr] & & \mathcal{F}(V) }$

The compatibility condition $c_ g \circ j_{g}^{-1} c_ f = c_{f \circ g}$ ensures that $\mathcal{F}$ is a presheaf, and also ensures that the maps $c_ f : \mathcal{F}_ U(V) \to \mathcal{F}(V)$ define an isomorphism $\mathcal{F}_ U \to \mathcal{F}|_{\mathcal{C}/U}$ for each $U$. Since each $\mathcal{F}_ U$ is a sheaf, this implies that $\mathcal{F}$ is a sheaf as well. The functor $(\mathcal{F}_ U, c_ f) \mapsto \mathcal{F}$ just constructed is quasi-inverse to the functor which takes a sheaf on $\mathcal{C}$ to its canonical glueing data. Further details omitted. $\square$

Remark 7.26.7. There is a variant of Lemma 7.26.6 which comes up in algebraic geometry. Namely, suppose that $\mathcal{C}$ is a site with all fibre products and for each $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we are given a full subcategory $U_\tau \subset \mathcal{C}/U$ with the following properties

1. $U/U$ is in $U_\tau$,

2. for $V/U$ in $U_\tau$ and covering $\{ V_ j \to V\}$ of $\mathcal{C}$ we have $V_ j/U$ in $U_\tau$ and

3. for a morphism $U' \to U$ of $\mathcal{C}$ and $V/U$ in $U_\tau$ the base change $V' = V \times _ U U'$ is in $U'_\tau$.

In this setting $U_\tau$ is a site for all $U$ in $\mathcal{C}$ and the base change functor $U_\tau \to U'_\tau$ defines a morphism $f_\tau : U_\tau \to U'_\tau$ of sites for all morphisms $f : U' \to U$ of $\mathcal{C}$. The glueing statement we obtain then reads as follows: A sheaf $\mathcal{F}$ on $\mathcal{C}$ is given by the following data:

1. for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ a sheaf $\mathcal{F}_ U$ on $U_\tau$,

2. for every $f : U' \to U$ in $\mathcal{C}$ a map $c_ f : f_\tau ^{-1}\mathcal{F}_ U \to \mathcal{F}_{U'}$.

These data are subject to the following conditions:

1. given $f : U' \to U$ and $g : U'' \to U'$ in $\mathcal{C}$ the composition $c_ g \circ g_\tau ^{-1}c_ f$ is equal to $c_{f \circ g}$, and

2. if $f : U' \to U$ is in $U_\tau$ then $c_ f$ is an isomorphism.

If we ever need this we will precisely state and prove this here. (Note that this result is slightly different from the statements above as we are not requiring all the maps $c_ f$ to be isomorphisms!)

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