Closed Immersion
In algebraic geometry, a closed immersion of schemes is a morphism of schemes f: Z \to X that identifies ''Z'' as a closed subset of ''X'' such that locally, regular functions on ''Z'' can be extended to ''X''. The latter condition can be formalized by saying that f^\#:\mathcal_X\rightarrow f_\ast\mathcal_Z is surjective. An example is the inclusion map \operatorname(R/I) \to \operatorname(R) induced by the canonical map R \to R/I. Other characterizations The following are equivalent: #f: Z \to X is a closed immersion. #For every open affine U = \operatorname(R) \subset X, there exists an ideal I \subset R such that f^(U) = \operatorname(R/I) as schemes over ''U''. #There exists an open affine covering X = \bigcup U_j, U_j = \operatorname R_j and for each ''j'' there exists an ideal I_j \subset R_j such that f^(U_j) = \operatorname (R_j / I_j) as schemes over U_j. #There is a quasicoherent sheaf of ideals \mathcal on ''X'' such that f_\ast\mathcal_Z\cong \mathcal_X/\mathcal and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Scheme (mathematics)
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem. Formally, a scheme is a topological space together with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Morphism Of Schemes
In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes. A morphism of algebraic stacks generalizes a morphism of schemes. Definition By definition, a morphism of schemes is just a morphism of locally ringed spaces. A scheme, by definition, has open affine charts and thus a morphism of schemes can also be described in terms of such charts (compare the definition of morphism of varieties). Let ƒ:''X''→''Y'' be a morphism of schemes. If ''x'' is a point of ''X'', since ƒ is continuous, there are open affine subsets ''U'' = Spec ''A'' of ''X'' containing ''x'' and ''V'' = Spec ''B'' of ''Y'' such that ƒ(''U'') ⊆ ''V''. Then ƒ: ''U'' → ''V'' is a morphism of affine schemes and thus is induced by some ring homomorphism ''B'' → ''A'' (cf. #Affine case.) In fact, one can use this description to "define" a morphism of schemes; o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Regular Function
In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular function. A regular map whose inverse is also regular is called biregular, and they are isomorphisms in the category of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no nonconstant regular functions on projective varieties – the weaker condition of a rational map and birational maps are frequently used as well. Definition If ''X'' and ''Y'' are closed subvarieties of \mathbb^n and \mathbb^m (so they are affine varieties), then a regular map f\colon X\to Y is the restriction of a polynomial map \mathbb^n\to \mathbb^m. Explicitly, it has the form: :f = (f_1, \dots, f_m) where the f_is are in the coordinate ring of ''X'': :k = k _1, \dots, x_nI, where ''I'' is the ideal defining ''X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Global Spec
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings \mathcal. Zariski topology For any ideal ''I'' of ''R'', define V_I to be the set of prime ideals containing ''I''. We can put a topology on \operatorname(R) by defining the collection of closed sets to be :\. This topology is called the Zariski topology. A basis for the Zariski topology can be constructed as follows. For ''f'' ∈ ''R'', define ''D''''f'' to be the set of prime ideals of ''R'' not containing ''f''. Then each ''D''''f'' is an open subset of \operatorname(R), and \ is a basis for the Zariski topology. \operatorname(R) is a compact space, but almost never Hausdorff: in fact, the maximal ideals in ''R'' are precisely the closed points in this topology. By the same reasoning, it is not, in general, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Finite Morphism
In algebraic geometry, a finite morphism between two affine varieties In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zerolocus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime idea ... X, Y is a dense Regular map (algebraic geometry), regular map which induces isomorphic inclusion k\left[Y\right]\hookrightarrow k\left[X\right] between their Coordinate ring, coordinate rings, such that k\left[X\right] is integral over k\left[Y\right]. This definition can be extended to the quasiprojective varieties, such that a Regular map (algebraic geometry), regular map f\colon X\to Y between quasiprojective varieties is finite if any point like y\in Y has an affine neighbourhood V such that U=f^(V) is affine and f\colon U\to V is a finite map (in view of the previous definition, because it is between affine varieties). Definition by Schemes A morphism ''f'': ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Radicial Morphism
In algebraic geometry, a morphism of schemes :''f'': ''X'' → ''Y'' is called radicial or universally injective, if, for every field ''K'' the induced map ''X''(''K'') → ''Y''(''K'') is injective. (EGA I, (3.5.4)) This is a generalization of the notion of a purely inseparable extension of fields (sometimes called a radicial extension, which should not be confused with a radical extension.) It suffices to check this for ''K'' algebraically closed. This is equivalent to the following condition: ''f'' is injective on the topological spaces and for every point ''x'' in ''X'', the extension of the residue fields :''k''(''f''(''x'')) ⊂ ''k''(''x'') is radicial, i.e. purely inseparable. It is also equivalent to every base change of ''f'' being injective on the underlying topological spaces. (Thus the term ''universally injective''.) Radicial morphisms are stable under composition, products and base change. If ''gf'' is radicial, so is ''f''. References * , section ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Separated Morphism
In algebraic geometry, given a morphism of schemes p: X \to S, the diagonal morphism :\delta: X \to X \times_S X is a morphism determined by the universal property of the fiber product X \times_S X of ''p'' and ''p'' applied to the identity 1_X : X \to X and the identity 1_X. It is a special case of a graph morphism: given a morphism f: X \to Y over ''S'', the graph morphism of it is X \to X \times_S Y induced by f and the identity 1_X. The diagonal embedding is the graph morphism of 1_X. By definition, ''X'' is a separated scheme over ''S'' (p: X \to S is a separated morphism) if the diagonal morphism is a closed immersion. Also, a morphism p: X \to S locally of finite presentation is an unramified morphism if and only if the diagonal embedding is an open immersion. Explanation As an example, consider an algebraic variety over an algebraically closed field ''k'' and p: X \to \operatorname(k) the structure map. Then, identifying ''X'' with the set of its ''k''rational points, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Segre Embedding
In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product (of sets) of two projective spaces as a projective variety. It is named after Corrado Segre. Definition The Segre map may be defined as the map :\sigma: P^n \times P^m \to P^\ taking a pair of points ( \in P^n \times P^m to their product :\sigma:( _0:X_1:\cdots:X_n _0:Y_1:\cdots:Y_m \mapsto _0Y_0: X_0Y_1: \cdots :X_iY_j: \cdots :X_nY_m (the ''XiYj'' are taken in lexicographical order). Here, P^n and P^m are projective vector spaces over some arbitrary field, and the notation : _0:X_1:\cdots:X_n is that of homogeneous coordinates on the space. The image of the map is a variety, called a Segre variety. It is sometimes written as \Sigma_. Discussion In the language of linear algebra, for given vector spaces ''U'' and ''V'' over the same field ''K'', there is a natural way to map their cartesian product to their tensor product. : \varphi: U\times V \to U\otimes V.\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Regular Embedding
In algebraic geometry, a closed immersion i: X \hookrightarrow Y of schemes is a regular embedding of codimension ''r'' if each point ''x'' in ''X'' has an open affine neighborhood ''U'' in ''Y'' such that the ideal of X \cap U is generated by a regular sequence of length ''r''. A regular embedding of codimension one is precisely an effective Cartier divisor. Examples and usage For example, if ''X'' and ''Y'' are smooth over a scheme ''S'' and if ''i'' is an ''S''morphism, then ''i'' is a regular embedding. In particular, every section of a smooth morphism is a regular embedding. If \operatornameB is regularly embedded into a regular scheme, then ''B'' is a complete intersection ring. The notion is used, for instance, in an essential way in Fulton's approach to intersection theory. The important fact is that when ''i'' is a regular embedding, if ''I'' is the ideal sheaf of ''X'' in ''Y'', then the normal sheaf, the dual of I/I^2, is locally free (thus a vector bundle) and the na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Stacks Project
The Stacks Project is an open source collaborative mathematics textbook writing project with the aim to cover "algebraic stacks and the algebraic geometry needed to define them". , the book consists of 115 chapters (excluding the license and index chapters) spreading over 7500 pages. The maintainer of the project, who reviews and accepts the changes, is Aise Johan de Jong. See alsoKerodona Stacks project inspired online textbook on categorical homotopy theory maintained by Jacob Lurie Jacob Alexander Lurie (born December 7, 1977) is an American mathematician who is a professor at the Institute for Advanced Study. Lurie is a 2014 MacArthur Fellow. Life When he was a student in the Science, Mathematics, and Computer Science ... References External linksProject website*Latest from the Stacks Project(as of 2013) (Accessed 20200401) Mathematics textbooks {{mathematicslitstub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 