Lemma 59.40.4. Let $X$, $Y$ be affine schemes. Let

$(g, g^\# ) : (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{O}_ X) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}), \mathcal{O}_ Y)$

be a morphism of locally ringed topoi. Then there exists a unique morphism of schemes $f : X \to Y$ such that $(g, g^\# )$ is $2$-isomorphic to $(f_{small}, f_{small}^\sharp )$, see Modules on Sites, Definition 18.8.1.

Proof. In this proof we write $\mathcal{O}_ X$ for the structure sheaf of the small étale site $X_{\acute{e}tale}$, and similarly for $\mathcal{O}_ Y$. Say $Y = \mathop{\mathrm{Spec}}(B)$ and $X = \mathop{\mathrm{Spec}}(A)$. Since $B = \Gamma (Y_{\acute{e}tale}, \mathcal{O}_ Y)$, $A = \Gamma (X_{\acute{e}tale}, \mathcal{O}_ X)$ we see that $g^\sharp$ induces a ring map $\varphi : B \to A$. Let $f = \mathop{\mathrm{Spec}}(\varphi ) : X \to Y$ be the corresponding morphism of affine schemes. We will show this $f$ does the job.

Let $V \to Y$ be an affine scheme étale over $Y$. Thus we may write $V = \mathop{\mathrm{Spec}}(C)$ with $C$ an étale $B$-algebra. We can write

$C = B[x_1, \ldots , x_ n]/(P_1, \ldots , P_ n)$

with $P_ i$ polynomials such that $\Delta = \det (\partial P_ i/ \partial x_ j)$ is invertible in $C$, see for example Algebra, Lemma 10.143.2. If $T$ is a scheme over $Y$, then a $T$-valued point of $V$ is given by $n$ sections of $\Gamma (T, \mathcal{O}_ T)$ which satisfy the polynomial equations $P_1 = 0, \ldots , P_ n = 0$. In other words, the sheaf $h_ V$ on $Y_{\acute{e}tale}$ is the equalizer of the two maps

$\xymatrix{ \prod \nolimits _{i = 1, \ldots , n} \mathcal{O}_ Y \ar@<1ex>[r]^ a \ar@<-1ex>[r]_ b & \prod \nolimits _{j = 1, \ldots , n} \mathcal{O}_ Y }$

where $b(h_1, \ldots , h_ n) = 0$ and $a(h_1, \ldots , h_ n) = (P_1(h_1, \ldots , h_ n), \ldots , P_ n(h_1, \ldots , h_ n))$. Since $g^{-1}$ is exact we conclude that the top row of the following solid commutative diagram is an equalizer diagram as well:

$\xymatrix{ g^{-1}h_ V \ar[r] \ar@{..>}[d] & \prod \nolimits _{i = 1, \ldots , n} g^{-1}\mathcal{O}_ Y \ar@<1ex>[r]^{g^{-1}a} \ar@<-1ex>[r]_{g^{-1}b} \ar[d]^{\prod g^\sharp } & \prod \nolimits _{j = 1, \ldots , n} g^{-1}\mathcal{O}_ Y \ar[d]^{\prod g^\sharp }\\ h_{X \times _ Y V} \ar[r] & \prod \nolimits _{i = 1, \ldots , n} \mathcal{O}_ X \ar@<1ex>[r]^{a'} \ar@<-1ex>[r]_{b'} & \prod \nolimits _{j = 1, \ldots , n} \mathcal{O}_ X \\ }$

Here $b'$ is the zero map and $a'$ is the map defined by the images $P'_ i = \varphi (P_ i) \in A[x_1, \ldots , x_ n]$ via the same rule $a'(h_1, \ldots , h_ n) = (P'_1(h_1, \ldots , h_ n), \ldots , P'_ n(h_1, \ldots , h_ n))$. that $a$ was defined by. The commutativity of the diagram follows from the fact that $\varphi = g^\sharp$ on global sections. The lower row is an equalizer diagram also, by exactly the same arguments as before since $X \times _ Y V$ is the affine scheme $\mathop{\mathrm{Spec}}(A \otimes _ B C)$ and $A \otimes _ B C = A[x_1, \ldots , x_ n]/(P'_1, \ldots , P'_ n)$. Thus we obtain a unique dotted arrow $g^{-1}h_ V \to h_{X \times _ Y V}$ fitting into the diagram

We claim that the map of sheaves $g^{-1}h_ V \to h_{X \times _ Y V}$ is an isomorphism. Since the small étale site of $X$ has enough points (Theorem 59.29.10) it suffices to prove this on stalks. Hence let $\overline{x}$ be a geometric point of $X$, and denote $p$ the associate point of the small étale topos of $X$. Set $q = g \circ p$. This is a point of the small étale topos of $Y$. By Lemma 59.29.12 we see that $q$ corresponds to a geometric point $\overline{y}$ of $Y$. Consider the map of stalks

$(g^\sharp )_ p : (\mathcal{O}_ Y)_{\overline{y}} = \mathcal{O}_{Y, q} = (g^{-1}\mathcal{O}_ Y)_ p \longrightarrow \mathcal{O}_{X, p} = (\mathcal{O}_ X)_{\overline{x}}$

Since $(g, g^\sharp )$ is a morphism of locally ringed topoi $(g^\sharp )_ p$ is a local ring homomorphism of strictly henselian local rings. Applying localization to the big commutative diagram above and Algebra, Lemma 10.153.12 we conclude that $(g^{-1}h_ V)_ p \to (h_{X \times _ Y V})_ p$ is an isomorphism as desired.

We claim that the isomorphisms $g^{-1}h_ V \to h_{X \times _ Y V}$ are functorial. Namely, suppose that $V_1 \to V_2$ is a morphism of affine schemes étale over $Y$. Write $V_ i = \mathop{\mathrm{Spec}}(C_ i)$ with

$C_ i = B[x_{i, 1}, \ldots , x_{i, n_ i}]/(P_{i, 1}, \ldots , P_{i, n_ i})$

The morphism $V_1 \to V_2$ is given by a $B$-algebra map $C_2 \to C_1$ which in turn is given by some polynomials $Q_ j \in B[x_{1, 1}, \ldots , x_{1, n_1}]$ for $j = 1, \ldots , n_2$. Then it is an easy matter to show that the diagram of sheaves

$\xymatrix{ h_{V_1} \ar[d] \ar[r] & \prod _{i = 1, \ldots , n_1} \mathcal{O}_ Y \ar[d]^{Q_1, \ldots , Q_{n_2}}\\ h_{V_2} \ar[r] & \prod _{i = 1, \ldots , n_2} \mathcal{O}_ Y }$

is commutative, and pulling back to $X_{\acute{e}tale}$ we obtain the solid commutative diagram

$\xymatrix{ g^{-1}h_{V_1} \ar@{..>}[dd] \ar[rrd] \ar[r] & \prod _{i = 1, \ldots , n_1} g^{-1}\mathcal{O}_ Y \ar[dd]^{g^\sharp } \ar[rrd]^{Q_1, \ldots , Q_{n_2}} \\ & & g^{-1}h_{V_2} \ar@{..>}[dd] \ar[r] & \prod _{i = 1, \ldots , n_2} g^{-1}\mathcal{O}_ Y \ar[dd]^{g^\sharp } \\ h_{X \times _ Y V_1} \ar[r] \ar[rrd] & \prod \nolimits _{i = 1, \ldots , n_1} \mathcal{O}_ X \ar[rrd]^{Q'_1, \ldots , Q'_{n_2}} \\ & & h_{X \times _ Y V_2} \ar[r] & \prod \nolimits _{i = 1, \ldots , n_2} \mathcal{O}_ X }$

where $Q'_ j \in A[x_{1, 1}, \ldots , x_{1, n_1}]$ is the image of $Q_ j$ via $\varphi$. Since the dotted arrows exist, make the two squares commute, and the horizontal arrows are injective we see that the whole diagram commutes. This proves functoriality (and also that the construction of $g^{-1}h_ V \to h_{X \times _ Y V}$ is independent of the choice of the presentation, although we strictly speaking do not need to show this).

At this point we are able to show that $f_{small, *} \cong g_*$. Namely, let $\mathcal{F}$ be a sheaf on $X_{\acute{e}tale}$. For every $V \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$ affine we have

\begin{align*} (g_*\mathcal{F})(V) & = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})}(h_ V, g_*\mathcal{F}) \\ & = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})}(g^{-1}h_ V, \mathcal{F}) \\ & = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})}(h_{X \times _ Y V}, \mathcal{F}) \\ & = \mathcal{F}(X \times _ Y V) \\ & = f_{small, *}\mathcal{F}(V) \end{align*}

where in the third equality we use the isomorphism $g^{-1}h_ V \cong h_{X \times _ Y V}$ constructed above. These isomorphisms are clearly functorial in $\mathcal{F}$ and functorial in $V$ as the isomorphisms $g^{-1}h_ V \cong h_{X \times _ Y V}$ are functorial. Now any sheaf on $Y_{\acute{e}tale}$ is determined by the restriction to the subcategory of affine schemes (Topologies, Lemma 34.4.12), and hence we obtain an isomorphism of functors $f_{small, *} \cong g_*$ as desired.

Finally, we have to check that, via the isomorphism $f_{small, *} \cong g_*$ above, the maps $f_{small}^\sharp$ and $g^\sharp$ agree. By construction this is already the case for the global sections of $\mathcal{O}_ Y$, i.e., for the elements of $B$. We only need to check the result on sections over an affine $V$ étale over $Y$ (by Topologies, Lemma 34.4.12 again). Writing $V = \mathop{\mathrm{Spec}}(C)$, $C = B[x_ i]/(P_ j)$ as before it suffices to check that the coordinate functions $x_ i$ are mapped to the same sections of $\mathcal{O}_ X$ over $X \times _ Y V$. And this is exactly what it means that the diagram

$\xymatrix{ g^{-1}h_ V \ar[r] \ar@{..>}[d] & \prod \nolimits _{i = 1, \ldots , n} g^{-1}\mathcal{O}_ Y \ar[d]^{\prod g^\sharp } \\ h_{X \times _ Y V} \ar[r] & \prod \nolimits _{i = 1, \ldots , n} \mathcal{O}_ X }$

commutes. Thus the lemma is proved. $\square$

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