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Sheaf Of Algebras
In algebraic geometry, a sheaf of algebras on a ringed space ''X'' is a sheaf of commutative rings on ''X'' that is also a sheaf of \mathcal_X-modules. It is quasi-coherent if it is so as a module. When ''X'' is a scheme, just like a ring, one can take the global Spec of a quasi-coherent sheaf of algebras: this results in the contravariant functor \operatorname_X from the category of quasi-coherent (sheaves of) \mathcal_X-algebras on ''X'' to the category of schemes that are affine over ''X'' (defined below). Moreover, it is an equivalence: the quasi-inverse is given by sending an affine morphism f: Y \to X to f_* \mathcal_Y. Affine morphism A morphism of schemes f: X \to Y is called affine if Y has an open affine cover U_i's such that f^(U_i) are affine. For example, a finite morphism is affine. An affine morphism is quasi-compact and separated; in particular, the direct image of a quasi-coherent sheaf along an affine morphism is quasi-coherent. The base change of an affine m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ringed Space
In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of rings called a structure sheaf. It is an abstraction of the concept of the rings of continuous (scalar-valued) functions on open subsets. Among ringed spaces, especially important and prominent is a locally ringed space: a ringed space in which the analogy between the stalk at a point and the ring of germs of functions at a point is valid. Ringed spaces appear in analysis as well as complex algebraic geometry and the scheme theory of algebraic geometry. Note: In the definition of a ringed space, most expositions tend to restrict the rings to be commutative rings, including Hartshorne and Wikipedia. "Éléments de géométrie algébrique", on the other hand, does not impose the commutativity assumption, although the book mostly consi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheaf Of Commutative Rings
In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of rings called a structure sheaf. It is an abstraction of the concept of the rings of continuous (scalar-valued) functions on open subsets. Among ringed spaces, especially important and prominent is a locally ringed space: a ringed space in which the analogy between the stalk at a point and the ring of germs of functions at a point is valid. Ringed spaces appear in analysis as well as complex algebraic geometry and the scheme theory of algebraic geometry. Note: In the definition of a ringed space, most expositions tend to restrict the rings to be commutative rings, including Hartshorne and Wikipedia. "Éléments de géométrie algébrique", on the other hand, does not impose the commutativity assumption, although the book mostly consid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheaf Of Modules
In mathematics, a sheaf of ''O''-modules or simply an ''O''-module over a ringed space (''X'', ''O'') is a sheaf (mathematics), sheaf ''F'' such that, for any open subset ''U'' of ''X'', ''F''(''U'') is an ''O''(''U'')-module and the restriction maps ''F''(''U'') → ''F''(''V'') are compatible with the restriction maps ''O''(''U'') → ''O''(''V''): the restriction of ''fs'' is the restriction of ''f'' times that of ''s'' for any ''f'' in ''O''(''U'') and ''s'' in ''F''(''U''). The standard case is when ''X'' is a scheme (mathematics), scheme and ''O'' its structure sheaf. If ''O'' is the constant sheaf \underline, then a sheaf of ''O''-modules is the same as a sheaf of abelian groups (i.e., an abelian sheaf). If ''X'' is the prime spectrum of a ring ''R'', then any ''R''-module defines an ''O''''X''-module (called an associated sheaf) in a natural way. Similarly, if ''R'' is a graded ring and ''X'' is the Proj construction, Proj of ''R'', then any graded module ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-coherent Sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free sheaves of infinite rank. Coherent sheaf cohomology is a powerful technique, in particular for studying the sections of a given coherent sheaf. Definitions A quasi-coherent sheaf on a ringed space (X, \mathcal O_X) is a sheaf \mathcal F of \mathcal O_X-modules which has a local presentation, that is, every point in X has an open neighborhood U in which there is an ex ... [...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] |
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] |
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] |
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 zero-locus 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 quasi-projective 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] |
Quasi-compact Morphism
In algebraic geometry, a morphism f: X \to Y between schemes is said to be quasi-compact if ''Y'' can be covered by open affine subschemes V_i such that the pre-images f^(V_i) are quasi-compact (as topological space). If ''f'' is quasi-compact, then the pre-image of a quasi-compact open subscheme (e.g., open affine subscheme) under ''f'' is quasi-compact. It is not enough that ''Y'' admits a covering by quasi-compact open subschemes whose pre-images are quasi-compact. To give an example, let ''A'' be a ring that does not satisfy the ascending chain conditions on radical ideals, and put X = \operatorname A. ''X'' contains an open subset ''U'' that is not quasi-compact. Let ''Y'' be the scheme obtained by gluing two ''Xs along ''U''. ''X'', ''Y'' are both quasi-compact. If f: X \to Y is the inclusion of one of the copies of ''X'', then the pre-image of the other ''X'', open affine in ''Y'', is ''U'', not quasi-compact. Hence, ''f'' is not quasi-compact. A morphism from a quasi-compac ... [...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] |
Locally Ringed Space
In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of rings called a structure sheaf. It is an abstraction of the concept of the rings of continuous (scalar-valued) functions on open subsets. Among ringed spaces, especially important and prominent is a locally ringed space: a ringed space in which the analogy between the stalk at a point and the ring of germs of functions at a point is valid. Ringed spaces appear in analysis as well as complex algebraic geometry and the scheme theory of algebraic geometry. Note: In the definition of a ringed space, most expositions tend to restrict the rings to be commutative rings, including Hartshorne and Wikipedia. "Éléments de géométrie algébrique", on the other hand, does not impose the commutativity assumption, although the book mostly considers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cone (algebraic Geometry)
In algebraic geometry, a cone is a generalization of a vector bundle. Specifically, given a scheme ''X'', the relative Spec :C = \operatorname_X R of a quasi-coherent graded ''O''''X''-algebra ''R'' is called the cone or affine cone of ''R''. Similarly, the relative Proj :\mathbb(C) = \operatorname_X R is called the projective cone of ''C'' or ''R''. Note: The cone comes with the \mathbb_m-action due to the grading of ''R''; this action is a part of the data of a cone (whence the terminology). Examples *If ''X'' = Spec ''k'' is a point and ''R'' is a homogeneous coordinate ring, then the affine cone of ''R'' is the (usual) affine cone over the projective variety corresponding to ''R''. *If R = \bigoplus_0^\infty I^n/I^ for some ideal sheaf ''I'', then \operatorname_X R is the normal cone to the closed scheme determined by ''I''. *If R = \bigoplus_0^\infty L^ for some line bundle ''L'', then \operatorname_X R is the total space of the dual of ''L''. *More generally, given a vecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |