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 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.
In the first case the (Zariski) tangent space to C at (0,0) is the whole plane, considered as a two-dimensional 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
It is a vector space over the residue field k:= R/
Its dual (as a k-vector space) is called tangent space of R.[1] 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.
to a scheme X at a point P is the (co)tangent space of
Due to the functoriality of Spec, the natural quotient map
[2] 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
(One often defines the tangent and cotangent spaces for a manifold in the analogous manner.)
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.
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.
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.
In general, the dimension of the Zariski tangent space can be extremely large.
be the ring of continuously differentiable real-valued functions on
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.
define linearly independent vectors in the Zariski cotangent space
The dimension of the Zariski tangent space
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.