The Stacks project

Proof. Note that the assertion makes sense since we have seen that $(X_{\acute{e}tale}, \mathcal{O}_{X_{\acute{e}tale}})$ and $(Y_{\acute{e}tale}, \mathcal{O}_{Y_{\acute{e}tale}})$ are locally ringed sites, see Lemma 59.33.5. Moreover, we know that $X_{\acute{e}tale}$ has enough points, see Theorem 59.29.10 and Remarks 59.29.11. Hence it suffices to prove that $(f_{small}, f_{small}^\sharp )$ satisfies condition (3) of Modules on Sites, Lemma 18.40.8. To see this take a point $p$ of $X_{\acute{e}tale}$. By Lemma 59.29.12 $p$ corresponds to a geometric point $\overline{x}$ of $X$. By Lemma 59.36.2 the point $q = f_{small} \circ p$ corresponds to the geometric point $\overline{y} = f \circ \overline{x}$ of $Y$. Hence the assertion we have to prove is that the induced map of stalks

is a local ring map. Suppose that $a \in (\mathcal{O}_ Y)_{\overline{y}}$ is an element of the left hand side which maps to an element of the maximal ideal of the right hand side. Suppose that $a$ is the equivalence class of a triple $(V, \overline{v}, a)$ with $V \to Y$ étale, $\overline{v} : \overline{x} \to V$ over $Y$, and $a \in \mathcal{O}(V)$. It maps to the equivalence class of $(X \times _ Y V, \overline{x} \times \overline{v}, \text{pr}_ V^\sharp (a))$ in the local ring $(\mathcal{O}_ X)_{\overline{x}}$. But it is clear that being in the maximal ideal means that pulling back $\text{pr}_ V^\sharp (a)$ to an element of $\kappa (\overline{x})$ gives zero. Hence also pulling back $a$ to $\kappa (\overline{x})$ is zero. Which means that $a$ lies in the maximal ideal of $(\mathcal{O}_ Y)_{\overline{y}}$. $\square$

Proof. This means that $t : f^{-1}_{small} \to f^{-1}_{small}$ is a transformation of functors such that the diagram

is commutative. Suppose $V \to Y$ is étale with $V$ affine. By Morphisms, Lemma 29.39.2 we may choose an immersion $i : V \to \mathbf{A}^ n_ Y$ over $Y$. In terms of sheaves this means that $i$ induces an injection $h_ i : h_ V \to \prod _{j = 1, \ldots , n} \mathcal{O}_ Y$ of sheaves. The base change $i'$ of $i$ to $X$ is an immersion (Schemes, Lemma 26.18.2). Hence $i' : X \times _ Y V \to \mathbf{A}^ n_ X$ is an immersion, which in turn means that $h_{i'} : h_{X \times _ Y V} \to \prod _{j = 1, \ldots , n} \mathcal{O}_ X$ is an injection of sheaves. Via the identification $f_{small}^{-1}h_ V = h_{X \times _ Y V}$ of Lemma 59.36.2 the map $h_{i'}$ is equal to

(verification omitted). This means that the map $t : f_{small}^{-1}h_ V \to f_{small}^{-1}h_ V$ fits into the commutative diagram

The commutativity of the right square holds by our assumption on $t$ explained above. Since the composition of the horizontal arrows is injective by the discussion above we conclude that the left vertical arrow is the identity map as well. Any sheaf of sets on $Y_{\acute{e}tale}$ admits a surjection from a (huge) coproduct of sheaves of the form $h_ V$ with $V$ affine (combine Topologies, Lemma 34.4.12 with Sites, Lemma 7.12.5). Thus we conclude that $t : f_{small}^{-1} \to f_{small}^{-1}$ is the identity transformation as desired. $\square$

Proof. Let us argue this carefuly since it is a bit confusing. Let $t : a_{small}^{-1} \to b_{small}^{-1}$ be the $2$-isomorphism. Consider any open $V \subset Y$. Note that $h_ V$ is a subsheaf of the final sheaf $*$. Thus both $a_{small}^{-1}h_ V = h_{a^{-1}(V)}$ and $b_{small}^{-1}h_ V = h_{b^{-1}(V)}$ are subsheaves of the final sheaf. Thus the isomorphism

has to be the identity, and $a^{-1}(V) = b^{-1}(V)$. It follows that $a$ and $b$ are equal on underlying topological spaces. Next, take a section $f \in \mathcal{O}_ Y(V)$. This determines and is determined by a map of sheaves of sets $f : h_ V \to \mathcal{O}_ Y$. Pull this back and apply $t$ to get a commutative diagram

where the triangle is commutative by definition of a $2$-isomorphism in Modules on Sites, Section 18.8. Above we have seen that the composition of the top horizontal arrows comes from the identity $a^{-1}(V) = b^{-1}(V)$. Thus the commutativity of the diagram tells us that $a_{small}^\sharp (f) = b_{small}^\sharp (f)$ in $\mathcal{O}_ X(a^{-1}(V)) = \mathcal{O}_ X(b^{-1}(V))$. Since this holds for every open $V$ and every $f \in \mathcal{O}_ Y(V)$ we conclude that $a = b$ as morphisms of schemes. $\square$

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

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

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:

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

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

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

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

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

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

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

Proof. You can prove this theorem by carefuly adjusting the arguments of the proof of Lemma 59.40.4 to the global setting. However, we want to indicate how we can glue the result of that lemma to get a global morphism due to the rigidity provided by the result of Lemma 59.40.2. Unfortunately, this is a bit messy.

Let us prove existence when $Y$ is affine. In this case choose an affine open covering $X = \bigcup U_ i$. For each $i$ the inclusion morphism $j_ i : U_ i \to X$ induces a morphism of locally ringed topoi $(j_{i, small}, j_{i, small}^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (U_{i, {\acute{e}tale}}), \mathcal{O}_{U_ i}) \to (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{O}_ X)$ by Lemma 59.40.1. We can compose this with $(g, g^\sharp )$ to obtain a morphism of locally ringed topoi

see Modules on Sites, Lemma 18.40.10. By Lemma 59.40.4 there exists a unique morphism of schemes $f_ i : U_ i \to Y$ and a $2$-isomorphism

Set $U_{i, i'} = U_ i \cap U_{i'}$, and denote $j_{i, i'} : U_{i, i'} \to U_ i$ the inclusion morphism. Since we have $j_ i \circ j_{i, i'} = j_{i'} \circ j_{i', i}$ we see that

Hence by uniqueness (see Lemma 59.40.3) we conclude that $f_ i \circ j_{i, i'} = f_{i'} \circ j_{i', i}$, in other words the morphisms of schemes $f_ i = f \circ j_ i$ are the restrictions of a global morphism of schemes $f : X \to Y$. Consider the diagram of $2$-isomorphisms (where we drop the components ${}^\sharp $ to ease the notation)

The notation $\star $ indicates horizontal composition, see Categories, Definition 4.29.1 in general and Sites, Section 7.36 for our particular case. By the result of Lemma 59.40.2 this diagram commutes. Hence for any sheaf $\mathcal{G}$ on $Y_{\acute{e}tale}$ the isomorphisms $t_ i : f_{small}^{-1}\mathcal{G}|_{U_ i} \to g^{-1}\mathcal{G}|_{U_ i}$ agree over $U_{i, i'}$ and we obtain a global isomorphism $t : f_{small}^{-1}\mathcal{G} \to g^{-1}\mathcal{G}$. It is clear that this isomorphism is functorial in $\mathcal{G}$ and is compatible with the maps $f_{small}^\sharp $ and $g^\sharp $ (because it is compatible with these maps locally). This proves the theorem in case $Y$ is affine.

In the general case, let $V \subset Y$ be an affine open. Then $h_ V$ is a subsheaf of the final sheaf $*$ on $Y_{\acute{e}tale}$. As $g$ is exact we see that $g^{-1}h_ V$ is a subsheaf of the final sheaf on $X_{\acute{e}tale}$. Hence by Lemma 59.31.1 there exists an open subscheme $W \subset X$ such that $g^{-1}h_ V = h_ W$. By Modules on Sites, Lemma 18.40.12 there exists a commutative diagram of morphisms of locally ringed topoi

where the horizontal arrows are the localization morphisms (induced by the inclusion morphisms $V \to Y$ and $W \to X$) and where $g'$ is induced from $g$. By the result of the preceding paragraph we obtain a morphism of schemes $f' : W \to V$ and a $2$-isomorphism $t : (f'_{small}, (f'_{small})^\sharp ) \to (g', (g')^\sharp )$. Exactly as before these morphisms $f'$ (for varying affine opens $V \subset Y$) agree on overlaps by uniqueness, so we get a morphism $f : X \to Y$. Moreover, the $2$-isomorphisms $t$ are compatible on overlaps by Lemma 59.40.2 again and we obtain a global $2$-isomorphism $(f_{small}, (f_{small})^\sharp ) \to (g, (g)^\sharp )$. as desired. Some details omitted. $\square$