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Absolute Cohomological Purity Conjecture
In algebraic geometry, the theorem of absolute (cohomological) purity is an important theorem in the theory of étale cohomology. It states:A version of the theorem is stated at given *a regular scheme ''X'' over some base scheme, *i: Z \to X a closed immersion of a regular scheme of pure codimension ''r'', *an integer ''n'' that is invertible on the base scheme, *\mathcal a locally constant étale sheaf with finite stalks and values in \mathbb/n\mathbb, for each integer m \ge 0, the map :\operatorname^m(Z_; \mathcal) \to \operatorname^_Z(X_; \mathcal(r)) is bijective, where the map is induced by cup product with c_r(Z). The theorem was introduced in SGA 5 Exposé I, § 3.1.4. as an open problem. Later, Thomason proved it for large ''n'' and Gabber in general. See also *purity (algebraic geometry) In the mathematical field of algebraic geometry, purity is a theme covering a number of results and conjectures, which collectively address the question of proving that "when something ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
étale Cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type. History Étale cohomology was introduced by , using some suggestions by Jean-Pierre Serre, and was motivated by the attempt to construct a Weil cohomology theory in order to prove the Weil conjectures. The foundations were soon after worked out by Grothendieck together with Michael Artin, and published as and SGA 4. Grothendieck used étale cohomology to prove some of the Weil conjectures (Bernard Dwork had already managed to prove the rationality part of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Scheme
In algebraic geometry, a regular scheme is a locally Noetherian scheme whose local rings are regular everywhere. Every smooth scheme is regular, and every regular scheme of finite type over a perfect field is smooth.. For an example of a regular scheme that is not smooth, see Geometrically regular ring#Examples. See also *Étale morphism *Dimension of an algebraic variety *Glossary of scheme theory This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. ... * Smooth completion References Algebraic geometry Scheme theory {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Wayne Thomason
Robert Wayne Thomason (5 November 1952 Tulsa, Oklahoma, U.S. – 5 November 1995, Paris, France) was an American mathematician who worked on algebraic K-theory. His results include a proof that all infinite loop space machines are in some sense equivalent, and progress on the Quillen–Lichtenbaum conjecture. Thomason did his undergraduate studies at Michigan State University, graduating with a B.S. in mathematics in 1973. He completed his Ph.D. at Princeton University in 1977, under the supervision of John Moore. From 1977 to 1979 he was a C. L. E. Moore instructor at the Massachusetts Institute of Technology, and from 1979 to 1982 he was a Dickson Assistant Professor at the University of Chicago. After spending a year at the Institute for Advanced Study, he was appointed as faculty at Johns Hopkins University in 1983. Thomason suffered from diabetes Diabetes, also known as diabetes mellitus, is a group of metabolic disorders characterized by a high blood sugar level ( hype ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ofer Gabber
Ofer Gabber (עופר גאבר; born May 16, 1958) is a mathematician working in algebraic geometry. Life In 1978 Gabber received a Ph.D. from Harvard University for the thesis ''Some theorems on Azumaya algebras,'' written under the supervision of Barry Mazur. Gabber has been at the Institut des Hautes Études Scientifiques in Bures-sur-Yvette in Paris since 1984 as a CNRS senior researcher. He won the Erdős Prize in 1981 and the Prix Thérèse Gautier from the French Academy of Sciences in 2011. In 1981 Gabber with Victor Kac published a proof of a conjecture stated by Kac in 1968. Books * With Lorenzo Ramero: ''Almost Ring Theory'', Springer, Lecture Notes in Computer Science, vol 1800, 2003. * With Brian Conrad, Gopal Prasad: ''Pseudo-reductive Groups'', Cambridge University Press, 20102015, 2nd editionref> See also *almost ring theory *Theorem of absolute purity In algebraic geometry, the theorem of absolute (cohomological) purity is an important theorem in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Purity (algebraic Geometry)
In the mathematical field of algebraic geometry, purity is a theme covering a number of results and conjectures, which collectively address the question of proving that "when something happens, it happens in a particular codimension". Purity of the branch locus For example, ramification is a phenomenon of codimension 1 (in the geometry of complex manifolds, reflecting as for Riemann surfaces that ramify at single points that it happens in real codimension two). A classical result, Zariski–Nagata purity of Masayoshi Nagata and Oscar Zariski, called also purity of the branch locus, proves that on a non-singular algebraic variety a ''branch locus'', namely the set of points at which a morphism ramifies, must be made up purely of codimension 1 subvarieties (a Weil divisor). There have been numerous extensions of this result into theorems of commutative algebra and scheme theory, establishing purity of the branch locus in the sense of description of the restrictions on the possib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
