Editing Differential (mathematics) (section)

=== 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 element]]s. The simplest example is the ring of [[dual number]]s '''R'''[''ε''], where ''ε''<sup>2</sup> = 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''&nbsp;&minus; ''f''(''p'') belongs to the [[ideal (ring theory)|ideal]] ''I''<sub>''p''</sub> of functions on '''R''' which vanish at ''p''. If the derivative ''f'' vanishes at ''p'', then ''f''&nbsp;&minus; ''f''(''p'') belongs to the square ''I''<sub>''p''</sub><sup>2</sup> of this ideal. Hence the derivative of ''f'' at ''p'' may be captured by the equivalence class [''f''&nbsp;&minus; ''f''(''p'')] in the [[quotient space (linear algebra)|quotient space]]  ''I''<sub>''p''</sub>/''I''<sub>''p''</sub><sup>2</sup>, and the [[jet (mathematics)|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''<sub>''p''</sub><sup>2</sup>. 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''<sub>''p''</sub>) but '''R'''[''ε''] which is the quotient space of functions on '''R''' modulo ''I''<sub>''p''</sub><sup>2</sup>. Such a thickened point is a simple example of a [[Scheme (mathematics)|scheme]].<ref name="Harris1998" />

==== Algebraic geometry notions ====
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Differentials are also important in [[algebraic geometry]], and there are several important notions.
* [[Abelian differential]]s usually mean differential one-forms on an [[algebraic curve]] or [[Riemann surface]].
* [[Quadratic differential]]s (which behave like "squares" of abelian differentials) are also important in the theory of Riemann surfaces.
* [[Kähler differential]]s provide a general notion of differential in algebraic geometry.