## Tag `0CLE`

Chapter 87: Morphisms of Algebraic Stacks > Section 87.38: Valuative criteria

Lemma 87.38.4. Assume given a $2$-commutative diagram $$ \xymatrix{ \mathop{\rm Spec}(K) \ar[r]_-{x'} \ar[d]_j & \mathcal{X}' \ar[d]^p \ar[r]_q & \mathcal{X} \ar[d]^f \\ \mathop{\rm Spec}(A) \ar[r]^-{y'} & \mathcal{Y}' \ar[r]^g & \mathcal{Y} } $$ 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.This lemma, although a bit of a brain teaser, is straightforward. (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 87.38.3; in fact this analogue may very well be wrong.) To prove the lemma we are allowed to replace $\mathcal{X}'$ by the $2$-fibre product $\mathcal{Y}' \times_\mathcal{Y} \mathcal{X}$ as described in Categories, Lemma 4.31.3. 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. $\square$

The code snippet corresponding to this tag is a part of the file `stacks-morphisms.tex` and is located in lines 8665–8686 (see updates for more information).

```
\begin{lemma}
\label{lemma-cat-dotted-arrows-base-change}
Assume given a $2$-commutative diagram
$$
\xymatrix{
\Spec(K) \ar[r]_-{x'} \ar[d]_j &
\mathcal{X}' \ar[d]^p \ar[r]_q &
\mathcal{X} \ar[d]^f \\
\Spec(A) \ar[r]^-{y'} &
\mathcal{Y}' \ar[r]^g &
\mathcal{Y}
}
$$
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 $\gamma$.
\end{lemma}
\begin{proof}
This lemma, although a bit of a brain teaser, is straightforward.
(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 \ref{lemma-cat-dotted-arrows-independent};
in fact this analogue may very well be wrong.)
To prove the lemma we are allowed to replace
$\mathcal{X}'$ by the $2$-fibre product
$\mathcal{Y}' \times_\mathcal{Y} \mathcal{X}$
as described in Categories, Lemma
\ref{categories-lemma-2-product-categories-over-C}.
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.
\end{proof}
```

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