Zariski tangent space

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 given a plane curve "C" defined by a polynomial equation
and take "P" to be the origin (0,0). Erasing terms of higher order than 1 would produce a 'linearised' equation reading
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 formula_1 is defined to be
where formula_12 is given by the product of ideals. It is a vector space over the residue field "k := R/formula_1". 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 "formula_12" 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 formula_6 and cotangent space formula_7 to a scheme "X" at a point "P" is the (co)tangent space of formula_8. Due to the functoriality of Spec, the natural quotient map formula_9 induces a homomorphism formula_10 for "X"=Spec("R"), "P" a point in "Y"=Spec("R/I"). This is used to embed formula_11 in formula_12. 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
Since this is a surjection, the transpose formula_17 is an injection.
Analytic functions.
If "V" is a subvariety of an "n"-dimensional vector space, defined by an ideal "I", then "R = Fn/I", where "Fn" is the ring of smooth/analytic/holomorphic functions on this vector space. The Zariski tangent space at "x" is 
where "mn" is the maximal ideal consisting of those functions in "Fn" vanishing at "x".
In the planar example above, "I" = "", and "I+m2 = <L>+m2.
Properties.
If "R" is a Noetherian local ring, the dimension of the tangent space is at least the dimension of "R":
"R" is called regular if equality holds. In a more geometric parlance, when "R" is the local ring of a variety "V" in "v", one also says that "v" is a regular point. Otherwise it is called a singular point.
The tangent space has an interpretation in terms of homomorphisms to the dual numbers for "K",
in the parlance of schemes, morphisms "Spec K[t]/t2" to a scheme "X" over "K" correspond to a choice of a rational point "x ∈ X(k)" and an element of the tangent space at "x". Therefore, one also talks about tangent vectors. See also: tangent space to a functor.