The Stacks project

Lemma 89.13.3. Let $\mathcal{F}$ be a category cofibred in groupoids over $\mathcal{C}_\Lambda $ which has (S1). Let $B \to A$ be a surjection in $\mathcal{C}_\Lambda $ with kernel $I$ annihilated by $\mathfrak m_ B$. Let $x \in \mathcal{F}(A)$. The set of ideals

\[ \mathcal{J} = \{ J \subset I \mid \text{there exists an }y \to x\text{ lying over }B/J \to A\} \]

has a smallest element.

Proof. Note that $\mathcal{J}$ is nonempty as $I \in \mathcal{J}$. Also, if $J \in \mathcal{J}$ and $J \subset J' \subset I$ then $J' \in \mathcal{J}$ because we can pushforward the object $y$ to an object $y'$ over $B/J'$. Let $J$ and $K$ be elements of the displayed set. We claim that $J \cap K \in \mathcal{J}$ which will prove the lemma. Since $I$ is a $k$-vector space we can find an ideal $J \subset J' \subset I$ such that $J \cap K = J' \cap K$ and such that $J' + K = I$. By the above we may replace $J$ by $J'$ and assume that $J + K = I$. In this case

\[ A/(J \cap K) = A/J \times _{A/I} A/K. \]

Hence the existence of an element $z \in \mathcal{F}(A/(J \cap K))$ mapping to $x$ follows, via (S1), from the existence of the elements we have assumed exist over $A/J$ and $A/K$. $\square$

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