C∞-Rings and Smooth Functional Calculus

• C∞-rings are algebraic structures that encode the full smooth functional calculus by assigning operations corresponding to every smooth map on ℝⁿ.
• They generalize the algebra of smooth functions on manifolds, providing the foundation for C∞-schemes and synthetic differential geometry.
• C∞-rings support constructions such as quotients, localizations, and tensor products, mirroring classical commutative algebra in a smooth context.

A $C^\infty$-ring is an algebraic structure that encodes the full smooth functional calculus; it provides an abstraction of the algebra of smooth functions on manifolds. Formally, a $C^\infty$-ring consists of a set $A$ together with operations $f_A: A^n \rightarrow A$ for each smooth map $f: \mathbb{R}^n \rightarrow \mathbb{R}$, subject to projection and composition axioms mirroring the properties of smooth functions. The theory of $C^\infty$-rings forms a foundation for differential geometry in an algebraic (and categorical) manner, leading to concepts such as $C^\infty$-schemes and stacks, and serving as the algebraic underpinning for synthetic differential geometry and derived differential geometry (Joyce, 2009, Stel, 2013).

1. Definition and Fundamental Properties

Let $E$ denote the category whose objects are Cartesian powers $\mathbb{R}^n$ for $n\geq 0$, with morphisms all smooth maps. A $C^\infty$-ring $A$ is a product-preserving functor $A: E \to \mathrm{Set}$, equivalently a set $A$ equipped with a family of operations

$$\forall n \geq 0,\ \forall \varphi \in C^\infty(\mathbb{R}^n,\mathbb{R}),\quad \varphi_A: A^n \to A$$

satisfying:

• Projection Axioms: For each projection $\pi_i: \mathbb{R}^n \to \mathbb{R}$, $(\pi_i)_A$ is the projection $A^n \to A$.
• Composition Axioms: For $\varphi \in C^\infty(\mathbb{R}^n)$, $\psi_j \in C^\infty(\mathbb{R}^m)$, $j=1,\dots, n$, and $x \in A^m$,

$$\varphi_A\left( \psi_{1,A}(x), \dots, \psi_{n,A}(x) \right) = (\varphi \circ (\psi_1, \dots, \psi_n))_A(x).$$

This structure induces an underlying commutative $\mathbb{R}$-algebra on $A$ via the assignment: $$+ = \Phi_{(x,y)\mapsto x+y},\quad \cdot = \Phi_{(x,y)\mapsto x y},\quad 0 = \Phi_{0},\quad 1 = \Phi_{1}.$$ But the $C^\infty$-operations capture far more, allowing functional calculus for arbitrary smooth operations (Stel, 2013, Joyce, 2009).

2. Examples and Algebraic Machinery

Standard constructions and examples include:

• Coordinate $C^\infty$-rings: $C^\infty(\mathbb{R}^n)$, the algebra of smooth functions $\mathbb{R}^n \to \mathbb{R}$, is the free $C^\infty$-ring on $n$ generators.
• Smooth functions on manifolds: For any smooth manifold $M$, $C^\infty(M)$ is a $C^\infty$-ring via $$f_{C^\infty(M)}(h_1,\dots,h_n)(x) = f(h_1(x), \dots, h_n(x))$$ for $f: \mathbb{R}^n \to \mathbb{R}$.
• Quotients and Congruences: Every quotient $C^\infty(\mathbb{R}^n)/I$ for an ideal $I$ gives a $C^\infty$-ring, as do more general quotients by $C^\infty$-congruences. The lattice of $C^\infty$-congruences is isomorphic to the lattice of ideals (Berni et al., 2019).

General constructions in the category $\mathbf{C^\infty\text{-}Rng}$ mirror those in ordinary commutative algebra:

• Limits and filtered colimits exist and are computed on underlying sets.
• Coproducts become $C^\infty$-tensor products, e.g., $$C^\infty(\mathbb{R}^m) \otimes^\infty C^\infty(\mathbb{R}^n) \cong C^\infty(\mathbb{R}^{m+n})$$.
• Free $C^\infty$-rings on sets exist: the free $C^\infty$-ring on a set $E$ is the colimit $$\varinjlim_{E' \subset_{\mathrm{fin}} E} C^\infty(\mathbb{R}^{E'})$$.
• Localization at a subset $S \subseteq A$: $A[S^{-1}]$ exists via universal property, analogous to localization in commutative algebra (Joyce, 2009, Berni et al., 2019).

3. $C^\infty$-Ringed Spaces, Schemes, and Stacks

The geometric theory built on $C^\infty$-rings is $C^\infty$-algebraic geometry (Joyce, 2009, Joyce, 2011, Francis-Staite et al., 2019):

• An affine $C^\infty$-scheme is the locally $C^\infty$-ringed space $\mathrm{Spec}^\infty(A)$, where points are $\mathbb{R}$-valued $C^\infty$-ring homomorphisms $A \to \mathbb{R}$, with basic open sets $D(a) = \{x:\ x(a)\neq 0\}$.
• $C^\infty$-schemes are local $C^\infty$-ringed spaces locally modeled on affine $C^\infty$-schemes.
• Morphisms of $C^\infty$-rings induce morphisms of (affine) $C^\infty$-schemes, dual to the algebraic category.
• Stacks: The machinery extends to $C^\infty$-stacks and Deligne-Mumford $C^\infty$-stacks, serving as smooth analogues of orbifolds (Joyce, 2011).
• Cotangent modules/sheaves: There is a theory of Kähler differentials and cotangent modules for $C^\infty$-rings and an associated cotangent sheaf for $C^\infty$-schemes (Francis-Staite et al., 2019).

Manifolds embed fully faithfully as $C^\infty$-schemes via $M \mapsto \mathrm{Spec}^\infty(C^\infty(M))$, and the entire apparatus generalizes classical algebraic geometry with a smooth base.

4. Specialized Classes: Reduced, Local, von Neumann Regular, and $C^\infty$-Fields

Several important subclasses of $C^\infty$-rings are central to the structure theory:

• $C^\infty$-reduced: A $C^\infty$-ring is reduced if it has no nontrivial smooth nilpotents, i.e., the smooth radical of zero is zero (Berni et al., 2020).
• Local $C^\infty$-rings: Possess a unique maximal $C^\infty$-ideal; for example, stalks of structure sheaves at points in $C^\infty$-schemes.
• von Neumann regular $C^\infty$-rings (vNR): $A$ is vNR if for each $a \in A$ there exists $x \in A$ such that $a = a^2 x$ (or, equivalently, all principal ideals are generated by idempotents). The spectrum of such a ring is a Boolean space, and the subcategory of vNR $C^\infty$-rings is reflective (Berni et al., 2019, Berni et al., 2024). There is an anti-equivalence (up to conjugation) between the category of vNR $C^\infty$-rings and Boolean algebras.
• $C^\infty$-fields: A $C^\infty$-ring whose underlying commutative ring is a field. All $C^\infty$-fields are real closed, and in the finitely generated case are isomorphic to $(\mathbb{R}, \Phi)$ (Berni et al., 2020).

Key results relating spectra and order structures include the identification of the real spectrum (space of orderings) and the Boolean spectrum for vNR $C^\infty$-rings.

5. Cosimplicial $C^\infty$-Rings and de Rham Complexes

Cosimplicial and simplicial methods arise naturally in the study of forms, cohomology, and higher structures:

• The de Rham complex $\Omega^*(\mathbb{R}^m)$ assembles into a cosimplicial graded-commutative algebra, with cosimplicial structure inherited from the simplicial structure on Euclidean spaces via smoothly defined face and degeneracy maps (Stel, 2013).
• Each $\Omega^*(\mathbb{R}^p)$ admits a natural induced $C^\infty$-ring structure via pointwise functional calculus: for any $\varphi \in C^\infty(\mathbb{R}^n)$ and forms $\omega_1, \dots, \omega_n$, $\varphi_\Omega(\omega_1, \dots, \omega_n)$ is pointwise application on 0-forms and extends via Taylor expansion to higher-degree forms.
• The face and degeneracy maps commute with the $C^\infty$-operations, making $\{\Omega^*(\mathbb{R}^p)\}_{p\geq 0}$ a cosimplicial $C^\infty$-ring.
• Quillen’s tangent category and modules: For any (co)simplicial $C^\infty$-ring $R^\bullet$, Quillen modules over $R^\bullet$ align with (co)simplicial modules over the underlying commutative rings (Stel, 2013).

6. Universal Algebra, Presentations, and Spectra

Key developments in universal algebra situate $C^\infty$-rings as varieties in the sense of Lawvere theories (Berni et al., 2019, Berni et al., 2018):

• The Lawvere theory corresponding to $C^\infty$-rings has objects $\mathbb{R}^n$ and morphisms all smooth maps.
• Finitely generated $C^\infty$-rings have the form $C^\infty(\mathbb{R}^n)/I$, finitely presented if $I$ is finitely generated.
• For the theory of finite presentation, congruences correspond bijectively to ideals.
• There is an adjunction between $C^\infty$-rings and commutative rings via the forgetful functor $U$ and the free $C^\infty$-ring functor, with significant implications for classifying toposes.

The spectrum $\mathrm{Spec}^\infty(A)$ with its smooth Zariski topology encodes the geometric content of a $C^\infty$-ring, and the real spectrum $\mathrm{Sper}^\infty(A)$ provides the order-theoretic refinement pertinent to real algebraic geometry (Berni et al., 2020).

7. Applications and Connections

$C^\infty$-rings and their categories serve as the foundational algebra for:

• Synthetic Differential Geometry (SDG): Models involving well-adapted toposes with $C^\infty$-ring objects, enabling rigorous manipulation of nilpotent infinitesimals and supporting the Kock–Lawvere axiom (Berni et al., 2018).
• $C^\infty$-Algebraic Geometry: The base structure for $C^\infty$-schemes and $C^\infty$-stacks, leading to the study of geometric objects such as d-manifolds and C^\infty-orbifolds (Joyce, 2009, Joyce, 2011).
• Order Theory and Real Algebraic Geometry: Development of smooth real spectra, orderings, and their connection to real closed fields, allowing the extension of tools from real algebraic geometry to the $C^\infty$-context (Berni et al., 2020).
• Stone Duality and Boolean Spaces: The anti-equivalence between vNR $C^\infty$-rings and Boolean algebras generalizes classical Stone duality within the smooth category (Berni et al., 2024).
• C^\infty-Algebraic Geometry with Corners: Extension to $C^\infty$-rings with corners, broadening the reach of $C^\infty$-algebraic geometry to singular and stratified spaces (Francis-Staite et al., 2019).

The theory is essential for contemporary developments in derived differential geometry, condensed mathematics, and synthetic approaches to geometric and topological problems.

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Key References:

(Joyce, 2009, Joyce, 2011, Berni et al., 2019, Francis-Staite et al., 2019, Stel, 2013, Berni et al., 2019, Berni et al., 2018, Berni et al., 2020, Berni et al., 2024)