Tangent Space To A Functor

	Tangent Space To A Functor In algebraic geometry, the tangent space to a functor generalizes the classical construction of a tangent space such as the Zariski tangent space. The construction is based on the following observation. Let ''X'' be a scheme over a field ''k''. :To give a kepsilon(\epsilon)^2-point of ''X'' is the same thing as to give a ''k''-rational point ''p'' of ''X'' (i.e., the residue field of ''p'' is ''k'') together with an element of (\mathfrak_/\mathfrak_^2)^; i.e., a tangent vector at ''p''. (To see this, use the fact that any local homomorphism \mathcal_p \to kepsilon(\epsilon)^2 must be of the form :\delta_p^v: u \mapsto u(p) + \epsilon v(u), \quad v \in \mathcal_p^.) Let ''F'' be a functor from the category of ''k''-algebras to the category of sets. Then, for any ''k''-point p \in F(k), the fiber of \pi: F(kepsilon(\epsilon)^2) \to F(k) over ''p'' is called the tangent space to ''F'' at ''p''. If the functor ''F'' preserves fibered products (e.g. if it is a scheme), the tangent sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	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 :''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' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Rational Point In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point. Understanding rational points is a central goal of number theory and Diophantine geometry. For example, Fermat's Last Theorem may be restated as: for , the Fermat curve of equation x^n+y^n=1 has no other rational points than , , and, if is even, and . Definition Given a field ''k'', and an algebraically closed extension ''K'' of ''k'', an affine variety ''X'' over ''k'' is the set of common zeros in K^n of a collection of polynomials with coefficients in ''k'': :f_1(x_1,\ldots,x_n)=0,\ldots, f_r(x_1,\dots,x_n)=0. These common zeros are called the ''points'' of ''X''. A ''k''-rational point (or ''k''-point) of ''X'' is a point of ''X'' that belongs to ''k''''n'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Tangent Bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of manifold the tangent spaces and have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle , see tangent bundle#Examples, Examples section: all tangents to a circle lie in the plane of the circle. In order to make them disjoint it is necessary to align them in a plane perpendicular to the plane of the circle. of the tangent spaces of M . That is, : \begin TM &= \bigsqcup_ T_xM \\ &= \bigcup_ \left\ \times T_xM \\ &= \bigcup_ \left\ \\ &= \left\ \end where T_x M denotes the tangent space to M at the point x . So, an element of TM can be thought of as a ordered pair, pair (x,v), where x is a point in M and v is a tangent vector to M at x . There i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Springer Science+Business Media Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]