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Regularly Embedded
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 n ... [...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 topo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Free 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 a ... [...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 internationall ... [...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] |
Regular Submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions. Formal definition In the following we assume all manifolds are differentiable manifolds of class ''C''''r'' for a fixed , and all morphisms are differentiable of class ''C''''r''. Immersed submanifolds An immersed submanifold of a manifold ''M'' is the image ''S'' of an immersion map ; in general this image will not be a submanifold as a subset, and an immersion map need not even be injective (one-to-one) – it can have self-intersections. More narrowly, one can require that the map be an injection (one-to-one), in which we call it an injective immersion, and define an immersed submanifold to be the image subset ''S'' together with a topology and differenti ... [...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 2020-04-01) Mathematics textbooks {{mathematics-lit-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acyclic Complex
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through both their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences. It has play ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
