The Stacks project

Lemma 59.40.3. Let $X$, $Y$ be schemes. Any two morphisms $a, b : X \to Y$ of schemes for which there exists a $2$-isomorphism $(a_{small}, a_{small}^\sharp ) \cong (b_{small}, b_{small}^\sharp )$ in the $2$-category of ringed topoi are equal.

Proof. Let us argue this carefully 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

\[ t : a_{small}^{-1}h_ V = h_{a^{-1}(V)} \to b_{small}^{-1}h_ V = h_{b^{-1}(V)} \]

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

\[ \xymatrix{ h_{b^{-1}(V)} \ar@{=}[r] & b_{small}^{-1}h_ V \ar[d]^{b_{small}^{-1}(f)} & & a_{small}^{-1}h_ V \ar[d]^{a_{small}^{-1}(f)} \ar[ll]^ t & h_{a^{-1}(V)} \ar@{=}[l] \\ & b_{small}^{-1}\mathcal{O}_ Y \ar[rd]_{b^\sharp } & & a_{small}^{-1}\mathcal{O}_ Y \ar[ll]^ t \ar[ld]^{a^\sharp } \\ & & \mathcal{O}_ X } \]

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. $\square$


Comments (0)


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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04LW. Beware of the difference between the letter 'O' and the digit '0'.