The Stacks project

Lemma 99.4.1. The notion above does indeed define an equivalence relation on morphisms from spectra of fields into the algebraic stack $\mathcal{X}$.

Proof. It is clear that the relation is reflexive and symmetric. Hence we have to prove that it is transitive. This comes down to the following: Given a diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(\Omega ) \ar[r]_ b \ar[d]_ a & \mathop{\mathrm{Spec}}(L) \ar[d]^ q & \mathop{\mathrm{Spec}}(\Omega ') \ar[l]^{b'} \ar[d]^{a'} \\ \mathop{\mathrm{Spec}}(K) \ar[r]^ p & \mathcal{X} & \mathop{\mathrm{Spec}}(K') \ar[l]_{p'} } \]

with both squares $2$-commutative we have to show that $p$ is equivalent to $p'$. By the $2$-Yoneda lemma (see Algebraic Stacks, Section 93.5) the morphisms $p$, $p'$, and $q$ are given by objects $x$, $x'$, and $y$ in the fibre categories of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(K)$, $\mathop{\mathrm{Spec}}(K')$, and $\mathop{\mathrm{Spec}}(L)$. The $2$-commutativity of the squares means that there are isomorphisms $\alpha : a^*x \to b^*y$ and $\alpha ' : (a')^*x' \to (b')^*y$ in the fibre categories of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(\Omega )$ and $\mathop{\mathrm{Spec}}(\Omega ')$. Choose any field $\Omega ''$ and embeddings $\Omega \to \Omega ''$ and $\Omega ' \to \Omega ''$ agreeing on $L$. Then we can extend the diagram above to

\[ \xymatrix{ & \mathop{\mathrm{Spec}}(\Omega '') \ar[ld]_ c \ar[d]^{q'} \ar[rd]^{c'} \\ \mathop{\mathrm{Spec}}(\Omega ) \ar[r]_ b \ar[d]_ a & \mathop{\mathrm{Spec}}(L) \ar[d]^ q & \mathop{\mathrm{Spec}}(\Omega ') \ar[l]^{b'} \ar[d]^{a'} \\ \mathop{\mathrm{Spec}}(K) \ar[r]^ p & \mathcal{X} & \mathop{\mathrm{Spec}}(K') \ar[l]_{p'} } \]

with commutative triangles and

\[ (q')^*(\alpha ')^{-1} \circ (q')^*\alpha : (a \circ c)^*x \longrightarrow (a' \circ c')^*x' \]

is an isomorphism in the fibre category over $\mathop{\mathrm{Spec}}(\Omega '')$. Hence $p$ is equivalent to $p'$ as desired. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 99.4: Points of algebraic stacks

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Your email address will not be published.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 04XF. Beware of the difference between the letter 'O' and the digit '0'.