Formal Superschemes over Fields: Basic Theory

TL;DR

This work develops a foundational theory of formal superschemes over a field by extending the classical formal-scheme framework to the super context. It constructs and analyzes formal super $\Bbbk$-functors ($\mathrm{SSpf}$, $\mathrm{PSSpf}$) and their dual coalgebraic viewpoints, proving a faithful flat descent theorem and a fiber-dimension type inequality for locally algebraic formal superschemes. The paper establishes a robust dictionary between super-coalgebras and formal superschemes via $\mathrm{SSp}^*$ and $\mathrm{SSpf}$, introduces a cofree coalgebra framework, and defines local superdimension, enabling a Krull-dimension-style measure in the super setting. Together, these results lay a comprehensive, field-based platform for morphisms, base change, and dimension theory in formal superschemes, with clear paths to further geometric and differential-geometric developments in supergeometry.

Abstract

This paper develops the basic theory of formal schemes over fields in the supersymmetric setting. We introduce the notion of a formal superscheme and investigate some of its fundamental properties. Particular emphasis is placed on the study of morphisms between formal superschemes, for which we establish a faithfully flat descent theorem and a fiber dimension-type theorem.

Theorems & Definitions (102)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• Remark 2.4
• Proposition 2.5