The Stacks project

Lemma 101.39.4. Assume given a $2$-commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_-{x'} \ar[d]_ j & \mathcal{X}' \ar[d]^ p \ar[r]_ q & \mathcal{X} \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[r]^-{y'} & \mathcal{Y}' \ar[r]^ g & \mathcal{Y} } \]

with the right square $2$-cartesian. Choose a $2$-arrow $\gamma ' : y' \circ j \to p \circ x'$. Set $x = q \circ x'$, $y = g \circ y'$ and let $\gamma : y \circ j \to f \circ x$ be the composition of $\gamma '$ with the $2$-arrow implicit in the $2$-commutativity of the right square. Then the category of dotted arrows for the left square and $\gamma '$ is equivalent to the category of dotted arrows for the outer rectangle and $\gamma $.

Proof. (We do not know how to prove the analogue of this lemma if instead of the category of dotted arrows we look at the set of isomorphism classes of morphisms producing two $2$-commutative triangles as in Lemma 101.39.3; in fact this analogue may very well be wrong.) First proof: this lemma is a special case of Categories, Lemma 4.44.2. Second proof: we are allowed to replace $\mathcal{X}'$ by the $2$-fibre product $\mathcal{Y}' \times _\mathcal {Y} \mathcal{X}$ as described in Categories, Lemma 4.32.3. Then the object $x'$ becomes the triple $(y' \circ j, x, \gamma )$. Then we can go from a dotted arrow $(a, \alpha , \beta )$ for the outer rectangle to a dotted arrow $(a', \alpha ', \beta ')$ for the left square by taking $a' = (y', a, \beta )$ and $\alpha ' = (\text{id}_{y' \circ j}, \alpha )$ and $\beta ' = \text{id}_{y'}$. Details omitted. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 101.39: Valuative criteria

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Your email address will not be published.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 0CLE. Beware of the difference between the letter 'O' and the digit '0'.