nLab differential geometry and algebraic geometry

differential geometry and algebraic geometry, cf. GAGA

The following table illustrates some parallels between algebraic geometry and differential geometry. There is also a third column for (holomorphic) complex geometry. These parallels are formalized using the notion of a Fermat theory.

algebraic geometry	differential geometry
commutative algebra	C^∞-ring
affine scheme	smooth locus
scheme	C^∞-scheme
derivation	C^∞-derivation
Kähler differential	Kähler C^∞-differential
Serre–Swan theorem	smooth Serre–Swan theorem
smooth morphism	submersion
unramified morphism	immersion