The Stacks project

Lemma 66.28.2. Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $t$ be a $2$-morphism from $(f_{small}, f^\sharp )$ to itself, see Modules on Sites, Definition 18.8.1. Then $t = \text{id}$.

Proof. Let $X'$, resp. $Y'$ be $X$ viewed as an algebraic space over $\mathop{\mathrm{Spec}}(\mathbf{Z})$, see Spaces, Definition 65.16.2. It is clear from the construction that $(X_{small}, \mathcal{O})$ is equal to $(X'_{small}, \mathcal{O})$ and similarly for $Y$. Hence we may work with $X'$ and $Y'$. In other words we may assume that $S = \mathop{\mathrm{Spec}}(\mathbf{Z})$.

Assume $S = \mathop{\mathrm{Spec}}(\mathbf{Z})$, $f : X \to Y$ and $t$ are as in the lemma. This means that $t : f^{-1}_{small} \to f^{-1}_{small}$ is a transformation of functors such that the diagram

\[ \xymatrix{ f_{small}^{-1}\mathcal{O}_ Y \ar[rd]_{f^\sharp } & & f_{small}^{-1}\mathcal{O}_ Y \ar[ll]^ t \ar[ld]^{f^\sharp } \\ & \mathcal{O}_ X } \]

is commutative. Suppose $V \to Y$ is ├ętale with $V$ affine. Write $V = \mathop{\mathrm{Spec}}(B)$. Choose generators $b_ j \in B$, $j \in J$ for $B$ as a $\mathbf{Z}$-algebra. Set $T = \mathop{\mathrm{Spec}}(\mathbf{Z}[\{ x_ j\} _{j \in J}])$. In the following we will use that $\mathop{\mathrm{Mor}}\nolimits _{\mathit{Sch}}(U, T) = \prod _{j \in J} \Gamma (U, \mathcal{O}_ U)$ for any scheme $U$ without further mention. The surjective ring map $\mathbf{Z}[x_ j] \to B$, $x_ j \mapsto b_ j$ corresponds to a closed immersion $V \to T$. We obtain a monomorphism

\[ i : V \longrightarrow T_ Y = T \times Y \]

of algebraic spaces over $Y$. In terms of sheaves on $Y_{\acute{e}tale}$ the morphism $i$ induces an injection $h_ i : h_ V \to \prod _{j \in J} \mathcal{O}_ Y$ of sheaves. The base change $i' : X \times _ Y V \to T_ X$ of $i$ to $X$ is a monomorphism too (Spaces, Lemma 65.5.5). Hence $i' : X \times _ Y V \to T_ X$ is a monomorphism, which in turn means that $h_{i'} : h_{X \times _ Y V} \to \prod _{j \in J} \mathcal{O}_ X$ is an injection of sheaves. Via the identification $f_{small}^{-1}h_ V = h_{X \times _ Y V}$ of Lemma 66.19.9 the map $h_{i'}$ is equal to

\[ \xymatrix{ f_{small}^{-1}h_ V \ar[r]^-{f^{-1}h_ i} & \prod _{j \in J} f_{small}^{-1}\mathcal{O}_ Y \ar[r]^{\prod f^\sharp } & \prod _{j \in J} \mathcal{O}_ X } \]

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

\[ \xymatrix{ f_{small}^{-1}h_ V \ar[r]^-{f^{-1}h_ i} \ar[d]^ t & \prod _{j \in J} f_{small}^{-1}\mathcal{O}_ Y \ar[r]^-{\prod f^\sharp } \ar[d]^{\prod t} & \prod _{j \in J} \mathcal{O}_ X \ar[d]^{\text{id}}\\ f_{small}^{-1}h_ V \ar[r]^-{f^{-1}h_ i} & \prod _{j \in J} f_{small}^{-1}\mathcal{O}_ Y \ar[r]^-{\prod f^\sharp } & \prod _{j \in J} \mathcal{O}_ X } \]

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 Lemma 66.18.6 with Sites, Lemma 7.12.5). Thus we conclude that $t : f_{small}^{-1} \to f_{small}^{-1}$ is the identity transformation as desired. $\square$


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