Editing Dual number (section)

== Algebraic geometry ==
In modern [[algebraic geometry]], the dual numbers over a field <math>k</math> (by which we mean the ring <math>k[\varepsilon]/(\varepsilon^2)</math>) may be used to define the [[tangent vectors]] to the points of a <math>k</math>-[[scheme (mathematics)|scheme]].<ref name=":0" /> Since the field <math>k</math> can be chosen intrinsically, it is possible to speak simply of the tangent vectors to a scheme. This allows notions from [[differential geometry]] to be imported into algebraic geometry.

In detail: The ring of dual numbers may be thought of as the ring of functions on the "first-order neighborhood of a point" – namely, the <math> k</math>-[[scheme (mathematics)|scheme]] <math> \operatorname{Spec} (k[\varepsilon]/(\varepsilon^2))</math>.<ref name=":0">{{Citation |last=Shafarevich |first=Igor R. |title=Schemes |date=2013 |url=http://dx.doi.org/10.1007/978-3-642-38010-5_1 |work=Basic Algebraic Geometry 2 |pages=35–38 |access-date=2023-12-27 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |doi=10.1007/978-3-642-38010-5_1 |isbn=978-3-642-38009-9}}</ref> Then, given a <math> k</math>-scheme <math> X</math>, <math> k</math>-points of the scheme are in 1-1 correspondence with maps <math> \operatorname{Spec} k \to X </math>, while tangent vectors are in 1-1 correspondence with maps <math> \operatorname{Spec} (k[\varepsilon]/(\varepsilon^2)) \to X  </math>.

The field <math>k</math> above can be chosen intrinsically to be a [[residue field]]. To wit: Given a point <math>x</math> on a scheme <math>S</math>, consider the [[stalk (mathematics)|stalk]] <math>S_x</math>. Observe that <math>S_x</math> is a [[local ring]] with a unique [[maximal ideal]], which is denoted <math>\mathfrak m_x</math>. Then simply let <math>k = S_x / \mathfrak m_x</math>.