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$C^\infty$-superrings and $C^\infty$-superschemes

Cristian Danilo Olarte, Pedro Rizzo, Alexander Torres-Gomez

TL;DR

The paper builds a rigorous algebro-geometric framework for $C^{\infty}$-superrings and their superschemes by defining $C^{\infty}$-superrings, studying split cases, and developing a spectrum theory via both prime-radical and $\mathbb{R}$-point perspectives. It establishes a fundamental equivalence between fair affine $C^{\infty}$-superschemes and fair $C^{\infty}$-superrings, extends the spectrum-Γ adjunction to the super context, and introduces notions of locality, localization, and $C^{\infty}$-radicals to support geometric constructions. The work also proves that the category of split affine $C^{\infty}$-superschemes admits fiber products and surveys fairfication as a functorial mechanism, setting the stage for a Batchelor-type expansion in future work. Overall, it provides the essential foundations for a geometric theory of smooth superspaces, with potential applications in derived differential geometry and supergeometry.

Abstract

This paper develops a theory of $C^\infty$-superrings and their associated $C^\infty$-superschemes. We prove a key equivalence between the category of fair affine $C^\infty$-superschemes and the category of fair $C^\infty$-superrings. We place special emphasis on split $C^\infty$-superrings, which generalize the function algebras of supermanifolds and serve as building blocks for more complex, non-split structures.

$C^\infty$-superrings and $C^\infty$-superschemes

TL;DR

The paper builds a rigorous algebro-geometric framework for -superrings and their superschemes by defining -superrings, studying split cases, and developing a spectrum theory via both prime-radical and -point perspectives. It establishes a fundamental equivalence between fair affine -superschemes and fair -superrings, extends the spectrum-Γ adjunction to the super context, and introduces notions of locality, localization, and -radicals to support geometric constructions. The work also proves that the category of split affine -superschemes admits fiber products and surveys fairfication as a functorial mechanism, setting the stage for a Batchelor-type expansion in future work. Overall, it provides the essential foundations for a geometric theory of smooth superspaces, with potential applications in derived differential geometry and supergeometry.

Abstract

This paper develops a theory of -superrings and their associated -superschemes. We prove a key equivalence between the category of fair affine -superschemes and the category of fair -superrings. We place special emphasis on split -superrings, which generalize the function algebras of supermanifolds and serve as building blocks for more complex, non-split structures.

Paper Structure

This paper contains 15 sections, 42 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $C^{\infty}$-rings and $C^{\infty}$-algebraic geometry: A short background
  4. Superrings: a quick overview
  5. $C^{\infty}$-Superrings
  6. Definition and examples
  7. Split $C^{\infty}$-superrings
  8. Categorical properties
  9. Local $C^{\infty}$-superrings
  10. Localization
  11. The spectrum of a $C^{\infty}$-Superring
  12. The prime spectrum
  13. The space of $\mathbb{R}$-points
  14. Fair $C^{\infty}$-superrings
  15. Spectrum of a $C^{\infty}$-superring as a locally ringed space.

Key Result

Proposition 2.14

Let $\mathfrak{C}$ be a $C^{\infty}$-ring and $\mathfrak{C}_x$ its localization at $x \in X_\mathfrak{C}$. Then the assignment for each open set $U\subseteq X_\mathfrak{C}$ defines a sheaf of $C^{\infty}$-rings on $X_\mathfrak{C}$, where $\pi_y:\mathfrak{C}\rightarrow\mathfrak{C}_y$ is the localization morphism at $y$.

Figures (1)

  • Figure :

Theorems & Definitions (177)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • Remark 2.9
  • Example 2.10
  • ...and 167 more