Topologization and Functional Analytification I: Intrinsic Morphisms of Commutative Algebras

TL;DR

This paper develops intrinsic definitions of key morphisms in algebraic geometry for various advanced ring structures, extending classical concepts to the realm of $ ext{∞}$-rings and stacks, and explores their interrelations.

Contribution

It introduces intrinsic definitions of étale, lisse, and non-ramified morphisms for a broad class of rings, including $ ext{∞}$-Banach and $ ext{∞}$-Borné rings, unifying classical and modern approaches.

Findings

Defined étale-like, lisse-like, and non-ramifié-like morphisms for $ ext{∞}$-rings.

Connected the generalizations to Huber's work in the noetherian case.

Explored the use of infinitesimal and crystalline stacks in this context.

Abstract

Eventually after Dieudonn\'e-Grothendieck, we give intrinsic definitions of \'etale, lisse and non-ramifi\'e morphisms for general adic rings and general locally convex rings. And we investigate the corresponding \'etale-like, lisse-like and non-ramifi\'e-like morphisms for general $\infty$-Banach, $\infty$-Born\'e and $\infty$-ind-Fr\'echet $\infty$-rings and $\infty$-functors into $\infty$-groupoid (as in the work of Bambozzi-Ben-Bassat-Kremnizer) in some intrinsic way by using the corresponding infinitesimal stacks and crystalline stacks. The two directions of generalization will intersect at Huber's book in the strongly noetherian situation.