The Stacks project

110.58 The lisse-étale site is not functorial

The lisse-étale site $X_{lisse,{\acute{e}tale}}$ of $X$ is the category of schemes smooth over $X$ endowed with (usual) étale coverings, see Cohomology of Stacks, Section 103.14. Let $f : X \to Y$ be a morphism of schemes. There is a functor

\[ u : Y_{lisse,{\acute{e}tale}} \longrightarrow X_{lisse,{\acute{e}tale}},\quad V/Y \longmapsto V \times _ Y X \]

which is continuous. Hence we obtain an adjoint pair of functors

\[ u^ s : \mathop{\mathit{Sh}}\nolimits (X_{lisse,{\acute{e}tale}}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (Y_{lisse,{\acute{e}tale}}), \quad u_ s : \mathop{\mathit{Sh}}\nolimits (Y_{lisse,{\acute{e}tale}}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (X_{lisse,{\acute{e}tale}}), \]

see Sites, Section 7.13. We claim that, in general, $u$ does not define a morphism of sites, see Sites, Definition 7.14.1. In other words, we claim that $u_ s$ is not left exact in general. Note that representable presheaves are sheaves on lisse-étale sites. Hence, by Sites, Lemma 7.13.5 we see that $u_ sh_ V = h_{V \times _ Y X}$. Now consider two morphisms

\[ \xymatrix{ V_1 \ar[rd] \ar@<1ex>[rr]^ a \ar@<-1ex>[rr]_ b & & V_2 \ar[ld] \\ & Y } \]

of schemes $V_1, V_2$ smooth over $Y$. Now if $u_ s$ is left exact, then we would have

\[ u_ s \text{Equalizer}(h_ a, h_ b : h_{V_1} \to h_{V_2}) = \text{Equalizer}(h_{a \times 1}, h_{b \times 1} : h_{V_1 \times _ Y X} \to h_{V_2 \times _ Y X}) \]

We will take the morphisms $a, b : V_1 \to V_2$ such that there exists no morphism from a scheme smooth over $Y$ into $(a = b) \subset V_1$, i.e., such that the left hand side is the empty sheaf, but such that after base change to $X$ the equalizer is nonempty and smooth over $X$. A silly example is to take $X = \mathop{\mathrm{Spec}}(\mathbf{F}_ p)$, $Y = \mathop{\mathrm{Spec}}(\mathbf{Z})$ and $V_1 = V_2 = \mathbf{A}^1_\mathbf {Z}$ with morphisms $a(x) = x$ and $b(x) = x + p$. Note that the equalizer of $a$ and $b$ is the fibre of $\mathbf{A}^1_\mathbf {Z}$ over $(p)$.

Lemma 110.58.1. The lisse-étale site is not functorial, even for morphisms of schemes.

Proof. See discussion above. $\square$

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