7.22 Cocontinuous functors which have a right adjoint
Let \mathcal{C} and \mathcal{D} be sites. Let u : \mathcal{C} \to \mathcal{D} and v : \mathcal{D} \to \mathcal{C} be functors of the underlying categories such that v is right adjoint to u. In this case, if v is continuous, then u is cocontinuous (Lemma 7.22.4). If u is cocontinuous, then it is often (but not always, see Example 7.22.5) the case that v is continuous, and if so, then v defines a morphism of sites whose associated morphism of topoi is the same as that defined by u.
Lemma 7.22.1. Let \mathcal{C} and \mathcal{D} be sites. Let u : \mathcal{C} \to \mathcal{D}, and v : \mathcal{D} \to \mathcal{C} be functors. Assume that u is cocontinuous and that v is a right adjoint to u. Let g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) be the morphism of topoi associated to u, see Lemma 7.21.1. Then
for a sheaf \mathcal{F} on \mathcal{C} the sheaf g_*\mathcal{F} is equal to the presheaf v^ p\mathcal{F}, in other words, (g_*\mathcal{F})(V) = \mathcal{F}(v(V)), and
for a sheaf \mathcal{G} on \mathcal{D} we have g^{-1}\mathcal{G} = (v_ p\mathcal{G})^\# .
Proof.
For \mathcal{F} as in (1) we have
g_*\mathcal{F} = {}_ su\mathcal{F} = {}_ pu\mathcal{F} = v^ p\mathcal{F} = \mathcal{F} \circ v
The first equality is Lemma 7.21.1. The second equality is Lemma 7.20.2. The third equality is Lemma 7.19.3. The final equality is the definition of v^ p in Section 7.5. This proves (1). For \mathcal{G} as in (2) we have
g^{-1}\mathcal{G} = (u^ p\mathcal{G})^\# = (v_ p\mathcal{G})^\#
The first equality is Lemma 7.21.1. The second equality is Lemma 7.19.3.
\square
Lemma 7.22.2. Notation and assumptions as in Lemma 7.22.1. If in addition v is continuous then v defines a morphism of sites f : \mathcal{C} \to \mathcal{D} whose associated morphism of topoi is equal to g.
Proof.
We will use the results of Lemma 7.22.1 without further mention. To prove that v defines a morphism of sites f as in the statement of the lemma, we have to show that v_ s is an exact functor (see Definition 7.14.1). Since v_ s\mathcal{G} = (v_ p\mathcal{G})^\# = g^{-1}\mathcal{G} this follows from the fact that g is a morphism of topoi. Then we see that f^{-1} = v_ s = g^{-1} and we find that f = g as morphisms of topoi.
\square
Example 7.22.3. This example continues the discussion of Example 7.14.3 from which we borrow the notation \mathcal{C}, \tau , \tau ', \epsilon . Observe that the identity functor v : \mathcal{C}_{\tau '} \to \mathcal{C}_\tau is a continuous functor and the identity functor u : \mathcal{C}_\tau \to \mathcal{C}_{\tau '} is a cocontinuous functor. Moreover u is left adjoint to v. Hence the results of Lemmas 7.22.1 and 7.22.2 apply and we conclude v defines a morphism of sites, namely
\epsilon : \mathcal{C}_\tau \longrightarrow \mathcal{C}_{\tau '}
whose corresponding morphism of topoi is the same as the morphism of topoi associated to the cocontinuous functor u.
Lemma 7.22.4. Let \mathcal{C} and \mathcal{D} be sites and let v : \mathcal{D} \to \mathcal{C} be a continuous functor. Assume v has a left adjoint u : \mathcal{C} \to \mathcal{D}. Then
u is cocontinuous,
the results of Lemmas 7.22.1 and 7.22.2 hold.
In particular, v defines a morphism of sites f : \mathcal{D} \to \mathcal{C}.
Proof.
Let U be an object of \mathcal{C} and let \{ V_ j \to u(U)\} _{j \in J} be a covering in \mathcal{D}. Then \{ v(V_ j) \to v(u(U))\} _{i \in I} is a covering in \mathcal{C}. Via the adjunction map U \to v(u(U)) we can base change this to a covering \{ W_ j \to U\} _{j \in J} with W_ j = v(V_ j) \times _{v(u(U))} U. Denoting p_ j : W_ j \to v(V_ j) the first projection, we obtain maps
u(W_ j) \xrightarrow {u(p_ j)} u(v(V_ j)) \longrightarrow V_ j
where the second arrow is the adjunction map. This determines a morphism \{ u(W_ j) \to u(U)\} \to \{ V_ j \to u(U)\} of families of maps with fixed target, showing that u is indeed cocontinuous. The other statements are immediate from Lemmas 7.22.1 and 7.22.2.
\square
Example 7.22.5. Let \mathcal{C} and \mathcal{D} be sites. Let u : \mathcal{C} \to \mathcal{D} and v : \mathcal{D} \to \mathcal{C} be functors of the underlying categories such that v is right adjoint to u. Lemma 7.22.4 shows that if v is continuous, then u is cocontinous. Conversely, if u is cocontinuous, then we can't conclude that v is continuous. We will give an example of this phenomenon using the big étale and smooth sites of a scheme, but presumably there is an elementary example as well. Namely, consider a scheme S and the sites (\mathit{Sch}/S)_{\acute{e}tale} and (\mathit{Sch}/S)_{smooth}. We may assume these sites have the same underlying category, see Topologies, Remark 34.11.1. Let u = v = \text{id}. Then u as a functor from (\mathit{Sch}/S)_{\acute{e}tale} to (\mathit{Sch}/S)_{smooth} is cocontinuous as every smooth covering of a scheme can be refined by an étale covering, see More on Morphisms, Lemma 37.38.7. Conversely, the functor v from (\mathit{Sch}/S)_{smooth} to (\mathit{Sch}/S)_{\acute{e}tale} is not continuous as a smooth covering is not an étale covering in general.
Comments (1)
Comment #9580 by Matthew Emerton on