In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point P on an algebraic variety V (and more generally). It does not use differential calculus, being based directly on abstract algebra, and in the most concrete cases just the theory of a system of linear equations.
Motivation For example, suppose C is a plane curve defined by a polynomial equation
F(X,Y) = 0 and take P to be the origin (0,0). Erasing terms of higher order than 1 would produce a 'linearised' equation reading
L(X,Y) = 0 in which all terms XaYb have been discarded if a + b > 1. We have two cases: L may be 0, or it may be the equation of a line. In the first case the (Zariski) tangent space to C at (0,0) is the whole plane, considered as a two-dimensional affine space. In the second case, the tangent space is that line, considered as affine space. (The question of the origin comes up, when we take P as a general point on C; it is better to say 'affine space' and then note that P is a natural origin, rather than insist directly that it is a vector space.) It is easy to see that over the real field we can obtain L in terms of the first partial derivatives of F. When those both are 0 at P, we have a singular point (double point, cusp or something more complicated). The general definition is that singular points of C are the cases when the tangent space has dimension 2.
Definition The cotangent space of a local ring R, with maximal ideal m {\displaystyle {\mathfrak {m}}} is defined to be
m / m 2 {\displaystyle {\mathfrak {m}}/{\mathfrak {m}}^{2}}
where m {\displaystyle {\mathfrak {m}}} 2 is given by the product of ideals. It is a vector space over the residue field k:= R/ m {\displaystyle {\mathfrak {m}}} . Its dual (as a k-vector space) is called tangent space of R. This definition is a generalization of the above example to higher dimensions: suppose given an affine algebraic variety V and a point v of V. Morally, modding out m {\displaystyle {\mathfrak {m}}} 2 corresponds to dropping the non-linear terms from the equations defining V inside some affine space, therefore giving a system of linear equations that define the tangent space. The tangent space T P ( X ) {\displaystyle T_{P}(X)} and cotangent space T P ∗ ( X ) {\displaystyle T_{P}^{*}(X)} to a scheme X at a point P is the (co)tangent space of O X , P {\displaystyle {\mathcal {O}}_{X,P}} . Due to the functoriality of Spec, the natural quotient map f : R → R / I {\displaystyle f:R\rightarrow R/I} induces a homomorphism g : O X , f − 1 ( P ) → O Y , P {\displaystyle g:{\mathcal {O}}_{X,f^{-1}(P)}\rightarrow {\mathcal {O}}_{Y,P}} for X=Spec(R), P a point in Y=Spec(R/I). This is used to embed T P ( Y ) {\displaystyle T_{P}(Y)} in T f − 1 P ( X ) {\displaystyle T_{f^{-1}P}(X)} . Since morphisms of fields are injective, the surjection of the residue fields induced by g is an isomorphism. Then a morphism k of the cotangent spaces is induced by g, given by
m P / m P 2 {\displaystyle {\mathfrak {m}}_{P}/{\mathfrak {m}}_{P}^{2}}
≅ ( m f − 1 P / I ) / ( ( m f − 1 P 2 + I ) / I ) {\displaystyle \cong ({\mathfrak {m}}_{f^{-1}P}/I)/(({\mathfrak {m}}_{f^{-1}P}^{2}+I)/I)}
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