The Stacks project

Lemma 58.40.2. Let $X$, $Y$ be schemes. Let $f : X \to Y$ be a morphism of schemes. Let $t$ be a $2$-morphism from $(f_{small}, f_{small}^\sharp )$ to itself, see Modules on Sites, Definition 18.8.1. Then $t = \text{id}$.

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

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

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 58.36.2 the map $h_{i'}$ is equal to

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

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

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

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 58.21.2 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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