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

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

(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 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. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)