The Stacks project

Remark 96.27.7. In Situation 96.27.1. Let $V$ be a locally Noetherian scheme over $S$. Let $(Z_ i, u'_ i, \hat x_ i) \in F(V)$ for $i = 1, 2$. Let $V'_ i \to V$, $\hat x'_ i$ and $x'_ i$ witness the compatibility between $u'_ i$ and $\hat x_ i$ for $i = 1, 2$.

Set $V' = V'_1 \times _ V V'_2$. Let $E' \to V'$ denote the equalizer of the morphisms

\[ V' \to V'_1 \xrightarrow {x'_1} X' \quad \text{and}\quad V' \to V'_2 \xrightarrow {x'_2} X' \]

Set $Z = Z_1 \cap Z_2$. Let $E_ W \to V_{/Z}$ be the equalizer of the morphisms

\[ V_{/Z} \to V_{/Z_1} \xrightarrow {\hat x_1} W \quad \text{and}\quad V_{/Z} \to V_{/Z_2} \xrightarrow {\hat x_2} W \]

Observe that $E' \to V$ is separated and locally of finite type and that $E_ W$ is a locally Noetherian formal algebraic space separated over $V$. The compatibilities between the various morphisms involved show that

  1. $\mathop{\mathrm{Im}}(E' \to V) \cap (Z_1 \cup Z_2)$ is contained in $Z = Z_1 \cap Z_2$,

  2. the morphism $E' \times _ V (V \setminus Z) \to V \setminus Z$ is a monomorphism and is equal to the equalizer of the restrictions of $u'_1$ and $u'_2$ to $V \setminus (Z_1 \cup Z_2)$,

  3. the morphism $E'_{/Z} \to V_{/Z}$ factors through $E_ W$ and the diagram

    \[ \xymatrix{ E'_{/Z} \ar[r] \ar[d] & X'_{/T'} \ar[d]^ g \\ E_ W \ar[r] & W } \]

    is cartesian. In particular, the morphism $E'_{/Z} \to E_ W$ is a formal modification as the base change of $g$,

  4. $E'$, $(E' \to V)^{-1}Z$, and $E'_{/Z} \to E_ W$ is a triple as in Situation 96.27.1 with base scheme the locally Noetherian scheme $V$,

  5. given a morphism $\varphi : A \to V$ of locally Noetherian schemes, the following are equivalent

    1. $(Z_1, u'_1, \hat x_1)$ and $(Z_2, u'_2, \hat x_2)$ restrict to the same element of $F(A)$,

    2. $A \setminus \varphi ^{-1}(Z) \to V \setminus Z$ factors through $E' \times _ V (V \setminus Z)$ and $A_{/\varphi ^{-1}(Z)} \to V_{/Z}$ factors through $E_ W$.

We conclude, using Lemmas 96.27.5 and 96.27.6, that if there is a solution $E \to V$ for the triple in (4), then $E$ represents $F \times _{\Delta , F \times F} V$ on the category of locally Noetherian schemes over $V$.


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Your email address will not be published.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