C-infinity ring - Knowlpedia

Definition

A $C^\infty$ -ring is a set $A$ equipped, for every smooth map

f:\mathbb R^n\longrightarrow\mathbb R,

with an $n$ -ary operation $\Phi_f:A^n\to A$ . These operations satisfy

\Phi_{\pi_i}(a_1,\ldots,a_n)=a_i

for the coordinate projections and

\Phi_{g\circ(f_1,\ldots,f_m)} =\Phi_g\circ(\Phi_{f_1},\ldots,\Phi_{f_m})

for composites of smooth maps. Nullary operations encode real constants. Equivalently, a $C^\infty$ -ring is a product-preserving functor from the category of Euclidean spaces and smooth maps to sets.

Underlying ordinary algebra

Applying the definition to $x+y$ , $xy$ , $-x$ , and constant functions gives $A$ an underlying commutative unital $\mathbb R$ -algebra. The $C^\infty$ -structure contains more information: it also permits $\Phi_{\sin}$ , $\Phi_{\exp}$ , and every other smooth functional calculus. Consequently, an arbitrary homomorphism of the underlying $\mathbb R$ -algebras need not be a $C^\infty$ -ring morphism.

A morphism of $C^\infty$ -rings $u:A\to B$ commutes with every smooth operation:

u\bigl(\Phi_f(a_1,\ldots,a_n)\bigr) =\Phi_f\bigl(u(a_1),\ldots,u(a_n)\bigr).

Basic examples

For a smooth manifold $M$ , the algebra $C^\infty(M)$ is a $C^\infty$ -ring under pointwise smooth functional calculus:

\Phi_f(a_1,\ldots,a_n)(p) =f(a_1(p),\ldots,a_n(p)).

The quotient of a $C^\infty$ -ring by an ideal inherits a canonical $C^\infty$ -ring structure. Such quotients provide examples far beyond algebras of functions on manifolds.

References

Ieke Moerdijk and Gonzalo E. Reyes, Models for Smooth Infinitesimal Analysis, Springer, 1991. DOI record. Relevant: Chapter I, algebraic theories of smooth functions and $C^\infty$ -rings.
Dominic Joyce, “Algebraic Geometry over $C^\infty$ -rings,” Memoirs of the AMS 260 (2019). arXiv version. Relevant: §2, definitions and examples of $C^\infty$ -rings.