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Differential forms on $C^\infty$-ringed spaces

Eugene Lerman

TL;DR

This work develops a functorial complex of differential forms for local $C^ ablafty$-ringed spaces, unifying manifolds, Sikorski differential spaces, and $C^ ablafty$-schemes under a single de Rham-like framework. Central to the construction are the $C^ ablafty$-Kähler differentials $\\Omega^1_ ablafty$ with a universal derivation $d_ ablafty$, and the $C^ ablafty$-algebraic de Rham complex $(\Lambda^ullet \Omega^1_ ablafty, \wedge, d)$, which is functorial in the base ring and recovers the classical de Rham forms on manifolds with corners. The theory extends to differential forms on local $C^ ablafty$-ringed spaces via a sheaf $m{\Upomega}^ullet_ ablafty$, and supports pullbacks, integration over $k$-simplices, and a Stokes theorem for $C^ ablafty$-algebraic differential forms. In particular, on manifolds with corners the constructed forms agree with ordinary de Rham forms, and the formalism provides a robust extension of differential geometry to broader geometric contexts with potential applications in generalized spaces and synthetic differential geometry.

Abstract

We construct a complex of differential forms on a local $C^\infty$-ringed space. The two main classes of spaces we have in mind are differential spaces in the sense of Sikorski and $C^\infty$-schemes. Just as in the case of manifolds the construction is functorial. Consequently forms can be integrated over simplices and Stokes' theorem holds.

Differential forms on $C^\infty$-ringed spaces

TL;DR

This work develops a functorial complex of differential forms for local -ringed spaces, unifying manifolds, Sikorski differential spaces, and -schemes under a single de Rham-like framework. Central to the construction are the -Kähler differentials with a universal derivation , and the -algebraic de Rham complex , which is functorial in the base ring and recovers the classical de Rham forms on manifolds with corners. The theory extends to differential forms on local -ringed spaces via a sheaf , and supports pullbacks, integration over -simplices, and a Stokes theorem for -algebraic differential forms. In particular, on manifolds with corners the constructed forms agree with ordinary de Rham forms, and the formalism provides a robust extension of differential geometry to broader geometric contexts with potential applications in generalized spaces and synthetic differential geometry.

Abstract

We construct a complex of differential forms on a local -ringed space. The two main classes of spaces we have in mind are differential spaces in the sense of Sikorski and -schemes. Just as in the case of manifolds the construction is functorial. Consequently forms can be integrated over simplices and Stokes' theorem holds.
Paper Structure (13 sections, 30 theorems, 195 equations)

This paper contains 13 sections, 30 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Background
  3. $C^\infty$-rings
  4. Local $C^\infty$-ringed spaces
  5. Differential spaces
  6. $C^\infty$-schemes
  7. Graded objects
  8. Kähler differentials of $C^\infty$-rings
  9. $C^\infty$-algebraic de Rham complex
  10. Differential forms on $C^\infty$-ringed spaces
  11. Integration and Stokes' theorem
  12. Differential spaces and bump functions
  13. Differential spaces as local $C^\infty$-ringed spaces

Key Result

Lemma 2.12

The $C^\infty$-ring $C^\infty(\mathbb{R}^n)$ of smooth functions on $\mathbb{R}^n$ is a free ring on $n$ generators. The generators are the coordinate functions $x_1,\ldots, x_n:\mathbb{R}^n\to \mathbb{R}$.

Theorems & Definitions (168)

  • Remark 2.1
  • Definition 2.2: The category $\mathsf{Euc}$ of Euclidean spaces
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Example 2.8
  • Example 2.9
  • Example 2.10
  • Definition 2.11
  • ...and 158 more