If $X$ is a manifold then the $mathbb R$-algebra $C^infty (X)$ of smooth functions $c:X
ightarrow mathbb R$ is a $C^infty $-ring. That is, for each smooth function $f:mathbb R^n
ightarrow mathbb R$ there is an $n$-fold operation $Phi _f:C^infty (X)^n
ightarrow C^infty (X)$ acting by $Phi _f:(c_1,ldots ,c_n)mapsto f(c_1,ldots ,c_n)$, and these operations $Phi _f$ satisfy many natural identities. Thus, $C^infty (X)$ actually has a far richer structure than the obvious $mathbb R$-algebra structure. The author explains the foundations of a version of algebraic geometry in which rings or...
If $X$ is a manifold then the $mathbb R$-algebra $C^infty (X)$ of smooth functions $c:X
ightarrow mathbb R$ is a $C^infty $-ring. That is, for each sm...
