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 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 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 T_P(X) and cotangent space T_P^*(X) to a scheme X at a point P is the (co)tangent space of. Due to the functoriality of Spec, the natural quotient map induces a homomorphism for X=Spec(R), P a point in Y=Spec(R/I). This is used to embed T_P(Y) in. 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 is an injection. (One often defines the tangent and cotangent spaces for a manifold in the analogous manner.)

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 = (F(X,Y)), and I+m2 = (L(X,Y))+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 at a point 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 K[t]/(t2), the dual numbers for K; in the parlance of schemes, morphisms from 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. In general, the dimension of the Zariski tangent space can be extremely large. For example, let be the ring of continuously differentiable real-valued functions on \mathbf{R}. Define to be the ring of germs of such functions at the origin. Then R is a local ring, and its maximal ideal m consists of all germs which vanish at the origin. The functions x^\alpha for define linearly independent vectors in the Zariski cotangent space, so the dimension of is at least the , the cardinality of the continuum. The dimension of the Zariski tangent space is therefore at least. On the other hand, the ring of germs of smooth functions at a point in an n-manifold has an n-dimensional Zariski cotangent space.

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