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. For example, suppose given a plane curve C 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. The cotangent space of a local ring R, with maximal ideal is defined to be where 2 is given by the product of ideals. It is a vector space over the residue field k:= R/. 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 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 and cotangent space to a scheme X at a point P is the (co)tangent space of .