Connected components of qcqs schemes and projective spaces

TL;DR

The paper proves that every quasi-component of a quasi-spectral space is connected, extending a classical compact-space result to a broader topological setting. This foundational result is then used to describe the connected components of qcqs schemes in terms of the global sections ring $R=\mathscr{O}_X(X)$ and the canonical map to $\operatorname{Spec}(R)$, yielding a canonical homeomorphism $\pi_0(X)\simeq\pi_0(\operatorname{Spec}(R))$ and clarifying the role of clopen sets via idempotents. It further analyzes connected components of projective spaces and Proj of graded rings, showing they decompose according to connected components of the base, with explicit descriptions of irreducible components. Finally, the authors establish equivalences that characterize when a space (or a ring) has finitely many connected components in terms of Boolean rings of clopen sets and idempotent decompositions, tying topological finiteness to algebraic structure.

Abstract

In this article, we first prove a general result in topology which asserts that every quasi-component of a quasi-spectral space is connected. As an important application, the structure of all connected components of every quasi-compact quasi-separated (qcqs) scheme $X$ is fully characterized. They are exactly of the form ...

Theorems & Definitions (77)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 3.1
• Example 3.2
• Theorem 3.3
• proof
• Corollary 3.4
• proof
• Corollary 3.5