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H Topology
In algebraic geometry, the ''h'' topology is a Grothendieck topology introduced by Vladimir Voevodsky to study the homology of schemes. It combines several good properties possessed by its related "sub"topologies, such as the ''qfh'' and ''cdh'' topologies. It has subsequently been used by Beilinson to study p-adic Hodge theory, in Bhatt and Scholze's work on projectivity of the affine Grassmanian, Huber and Jörder's study of differential forms, etc. Definition Voevodsky defined the ''h'' topology to be the topology associated to finite families \ of morphisms of finite type such that \amalg U_i \to X is a universal topological epimorphism (i.e., a set of points in the target is an open subset if and only if its preimage is open, and any base change also has this property). Voevodsky worked with this topology exclusively on categories Sch^_ of schemes of finite type over a ''Noetherian'' base scheme S. Bhatt-Scholze define the ''h'' topology on the category Sch^_ of schemes ... [...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] |
Grothendieck Topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site. Grothendieck topologies axiomatize the notion of an open cover. Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology. This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme. It has been used to define other cohomology theories since then, such as ℓ-adic cohomology, flat cohomology, and crystalline cohomology. While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as to John Tate's theory of rigid analytic geometry. There is a natural way to associate a site to an ordi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Voevodsky
Vladimir Alexandrovich Voevodsky (, russian: Влади́мир Алекса́ндрович Воево́дский; 4 June 1966 – 30 September 2017) was a Russian-American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal in 2002. He is also known for the proof of the Milnor conjecture and motivic Bloch–Kato conjectures and for the univalent foundations of mathematics and homotopy type theory. Early life and education Vladimir Voevodsky's father, Aleksander Voevodsky, was head of the Laboratory of High Energy Leptons in the Institute for Nuclear Research at the Russian Academy of Sciences. His mother Tatyana was a chemist. Voevodsky attended Moscow State University for a while, but was forced to leave without a diploma for refusing to attend classes and failing academically. He received his Ph.D. in mathematics from Harvard University in 1992 after being recommended with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homology Theory
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry. The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes. For instance, a circle is not a disk because the circle has a hole through it while the disk is solid, and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. However, because a hole is "not there", it is not immediately obvious how to define a hole or how to distinguish different kinds of holes. Homology was originally a rigorous mathematical method for ... [...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] |
V-topology
In mathematics, especially in algebraic geometry, the v-topology (also known as the universally subtrusive topology) is a Grothendieck topology whose covers are characterized by lifting maps from valuation rings. This topology was introduced by and studied further by , who introduced the name ''v''-topology, where ''v'' stands for valuation. Definition A universally subtrusive map is a map ''f'': ''X'' → ''Y'' of quasi-compact, quasi-separated schemes such that for any map ''v'': Spec (''V'') → ''Y'', where ''V'' is a valuation ring, there is an extension (of valuation rings) V \subset W and a map Spec ''W'' → ''X'' lifting ''v''. Examples Examples of ''v''-covers include faithfully flat maps, proper surjective maps. In particular, any Zariski covering is a ''v''-covering. Moreover, universal homeomorphisms, such as X_ \to X, the normalisation of the cusp, and the Frobenius in positive characteristic are ''v''-coverings. In fact, the perfection X_ \to X of a sch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Seminormal Ring
In algebra, a seminormal ring is a commutative reduced ring in which, whenever ''x'', ''y'' satisfy x^3 = y^2, there is ''s'' with s^2 = x and s^3 = y. This definition was given by as a simplification of the original definition of . A basic example is an integrally closed domain, i.e., a normal ring. For an example which is not normal, one can consider the non-integral ring \mathbb, y The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...xy, or the ring of a nodal curve. In general, a reduced scheme X can be said to be seminormal if every morphism Y \to X which induces a homeomorphism of topological spaces, and an isomorphism on all residue fields, is an isomorphism of schemes. A semigroup is said to be seminormal if its semigroup algebra is seminormal. References * * * * Char ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Nullstellensatz
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert, who proved the Nullstellensatz in his second major paper on invariant theory in 1893 (following his seminal 1890 paper in which he proved Hilbert's basis theorem). Formulation Let ''k'' be a field (such as the rational numbers) and ''K'' be an algebraically closed field extension (such as the complex numbers). Consider the polynomial ring k _1, \ldots, X_n/math> and let ''I'' be an ideal in this ring. The algebraic set V(''I'') defined by this ideal consists of all ''n''-tuples x = (''x''1,...,''x''''n'') in ''Kn'' such that ''f''(x) = 0 for all ''f'' in ''I''. Hilbert's Nullstellens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nisnevich Topology
In algebraic geometry, the Nisnevich topology, sometimes called the completely decomposed topology, is a Grothendieck topology on the category of schemes which has been used in algebraic K-theory, A¹ homotopy theory, and the theory of motives. It was originally introduced by Yevsey Nisnevich, who was motivated by the theory of adeles. Definition A morphism of schemes f:Y \to X is called a Nisnevich morphism if it is an étale morphism such that for every (possibly non-closed) point ''x'' ∈ ''X'', there exists a point ''y'' ∈ ''Y'' in the fiber such that the induced map of residue fields ''k''(''x'') → ''k''(''y'') is an isomorphism. Equivalently, ''f'' must be flat, unramified, locally of finite presentation, and for every point ''x'' ∈ ''X'', there must exist a point ''y'' in the fiber such that ''k''(''x'') → ''k''(''y'') is an isomorphism. A family of morphisms is a Nisnevich cover if each morphism in the family is étale and for every (possibly non-closed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
V-topology
In mathematics, especially in algebraic geometry, the v-topology (also known as the universally subtrusive topology) is a Grothendieck topology whose covers are characterized by lifting maps from valuation rings. This topology was introduced by and studied further by , who introduced the name ''v''-topology, where ''v'' stands for valuation. Definition A universally subtrusive map is a map ''f'': ''X'' → ''Y'' of quasi-compact, quasi-separated schemes such that for any map ''v'': Spec (''V'') → ''Y'', where ''V'' is a valuation ring, there is an extension (of valuation rings) V \subset W and a map Spec ''W'' → ''X'' lifting ''v''. Examples Examples of ''v''-covers include faithfully flat maps, proper surjective maps. In particular, any Zariski covering is a ''v''-covering. Moreover, universal homeomorphisms, such as X_ \to X, the normalisation of the cusp, and the Frobenius in positive characteristic are ''v''-coverings. In fact, the perfection X_ \to X of a sch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrei Suslin
Andrei Suslin (russian: Андре́й Алекса́ндрович Су́слин, sometimes transliterated Souslin) was a Russian mathematician who contributed to algebraic K-theory and its connections with algebraic geometry. He was a Trustee Chair and Professor of mathematics at Northwestern University. He was born on 27 December 1950 in St. Petersburg, Russia. As a youth, he was an "all Leningrad" gymnast. He received his PhD from Leningrad University in 1974; his thesis was titled ''Projective modules over polynomial rings''. In 1976 he and Daniel Quillen independently proved Serre's conjecture about the triviality of algebraic vector bundles on affine space. In 1982 he and Alexander Merkurjev proved the Merkurjev–Suslin theorem on the norm residue homomorphism in Milnor K2-theory, with applications to the Brauer group. Suslin was an invited speaker at the International Congress of Mathematicians in 1978 and 1994, and he gave a plenary invited address at th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
