The Stacks project

Lemma 97.27.4. In Situation 97.27.1 the rule $F$ that sends a locally Noetherian scheme $V$ over $S$ to the set of triples $(Z, u', \hat x)$ satisfying the compatibility condition and which sends a a morphism $\varphi : V_2 \to V_1$ of locally Noetherian schemes over $S$ to the map

\[ F(\varphi ) : F(V_1) \longrightarrow F(V_2) \]

sending an element $(Z_1, u'_1, \hat x_1)$ of $F(V_1)$ to $(Z_2, u'_2, \hat x_2)$ in $F(V_2)$ given by

  1. $Z_2 \subset V_2$ is the inverse image of $Z_1$ by $\varphi $,

  2. $u'_2$ is the composition of $u'_1$ and $\varphi |_{V_2 \setminus Z_2} : V_2 \setminus Z_2 \to V_1 \setminus Z_1$,

  3. $\hat x_2$ is the composition of $\hat x_1$ and $\varphi _{/Z_2} : V_{2, /Z_2} \to V_{1, /Z_1}$

is a contravariant functor.

Proof. To see the compatibility condition between $u'_2$ and $\hat x_2$, let $V'_1 \to V_1$, $\hat x'_1$, and $x'_1$ witness the compatibility between $u'_1$ and $\hat x_1$. Set $V'_2 = V_2 \times _{V_1} V'_1$, set $\hat x'_2$ equal to the composition of $\hat x'_1$ and $V'_{2, /Z_2} \to V'_{1, /Z_1}$, and set $x'_2$ equal to the composition of $x'_1$ and $V'_2 \to V'_1$. Then $V'_2 \to V_2$, $\hat x'_2$, and $x'_2$ witness the compatibility between $u'_2$ and $\hat x_2$. We omit the detailed verification. $\square$

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