Differential (infinitesimal) - Algebraic Geometry

Algebraic Geometry

In algebraic geometry, differentials and other infinitesimal notions are handled in a very explicit way by accepting that the coordinate ring or structure sheaf of a space may contain nilpotent elements. The simplest example is the ring of dual numbers R, where ε2 = 0.

This can be motivated by the algebro-geometric point of view on the derivative of a function f from R to R at a point p. For this, note first that f − f(p)1 (where 1 is the identity function) belongs to the ideal I_p of functions on R which vanish at p. If the derivative f vanishes at p, then f − f(p)1 belongs to the square I_p2 of this ideal. Hence the derivative of f at p may be captured by the equivalence class in the quotient space I_p/I_p2, and the 1-jet of f (which encodes its value and its first derivative) is the equivalence class of f in the space of all functions modulo I_p2. Algebraic geometers regard this equivalence class as the restriction of f to a thickened version of the point p whose coordinate ring is not R (which is the quotient space of functions on R modulo I_p) but R which is the quotient space of functions on R modulo I_p2. Such a thickened point is a simple example of a scheme.