The Stacks project

Remark 97.27.7. In Situation 97.27.1. 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 97.27.1 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 97.27.5 and 97.27.6, 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 $V$.

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