Lemma 7.42.
A map \(f\) is \(!\)-able if and only if it factors as a proper map followed by an open immersion.
Proof
For a map \(A\to B\), equip \(B^{\triangleright}\) first with the analytic structure induced from \(A\). The map from \(A\) to this induced analytic ring is proper by definition, while the comparison from the induced analytic ring to \(B\) is an open immersion precisely when \(A\to B\) is \(!\)-able. Conversely, any factorization of this form satisfies the defining condition. See [Scholze 2025, Definition 9.16].
References
- Peter Scholze. Six-Functor Formalisms. 2025.