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Motivic Sheaf
In mathematics, a motivic sheaf is a motivic-cohomology counterpart of an l-adic sheaf. It was first introduced by Morel and Voevodsky and was later developed by J. Ayoub,Joseph Ayoub, A guide to (étale) motivic sheaves, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 1101–1124 Deniz-Charles Cisinski, F. Déglise, F. Morel, and others. For Nori motive In algebraic geometry, a Nori motive is a mixed motive constructed by Madhav Nori. Today, it is known that Nori's 1-motive coincides with that of Ayoub. The construction is based on Nori's basic lemma and his tannakian theorem.§ 4.4.4., Bruno K ...s, the first construction is due to D. Arapura. In practice, a motivic sheaf is sometimes used instead of an l-adic sheaf because the former’s cycle-theoretic nature may be important. In the language of Ayoub,§ 1.4.1. of References Further reading * Adeel KhanMotivic sheaves on algebraic stacks* https://ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Motivic Cohomology
Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geometry and number theory are attempts to understand motivic cohomology. Motivic homology and cohomology Let ''X'' be a scheme of finite type over a field ''k''. A key goal of algebraic geometry is to compute the Chow groups of ''X'', because they give strong information about all subvarieties of ''X''. The Chow groups of ''X'' have some of the formal properties of Borel–Moore homology in topology, but some things are missing. For example, for a closed subscheme ''Z'' of ''X'', there is an exact sequence of Chow groups, the localization sequence :CH_i(Z) \rightarrow CH_i(X) \rightarrow CH_i(X-Z) \rightarrow 0, whereas in topology this would be part of a long exact sequence. This problem was resolved by generalizing Chow groups to a bigr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L-adic Sheaf
In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right. For example, comparing the expansion of the rational number \tfrac15 in base vs. the -adic expansion, \begin \tfrac15 &= 0.01210121\ldots \ (\text 3) &&= 0\cdot 3^0 + 0\cdot 3^ + 1\cdot 3^ + 2\cdot 3^ + \cdots \\ mu\tfrac15 &= \dots 121012102 \ \ (\text) &&= \cdots + 2\cdot 3^3 + 1 \cdot 3^2 + 0\cdot3^1 + 2 \cdot 3^0. \end Formally, given a prime number , a -adic number can be defined as a series s=\sum_^\infty a_i p^i = a_k p^k + a_ p^ + a_ p^ + \cdots where is an integer (possibly negative), and each a_i is an integer such that 0\le a_i < p. A -adic integer is a -adic number such that < ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Denis-Charles Cisinski
Denis-Charles Cisinski (born March 10, 1976) is a mathematician focussing on higher category theory, homotopy theory, K-theory and algebraic geometry. In 2001, Cisinski model structures on topoi were introduced and later named after him. Since 2016, Denis-Charles Cisinski works at the Universität Regensburg. Research Denis-Charles Cisinski obtained his PhD in 2002 at the Paris Diderot University with a thesis supervised by Georges Maltsiniotis and titled ''Les préfaisceaux comme modèles des types d'homotopie'' ( Presheaves as models for homotopy types). It was expanded and released as a book in 2006, further developing the theory from ''Pursuing Stacks'' by Alexander Grothendieck. In 2015, Denis-Charles Cisinski gave a talk at the Séminaire Nicolas Bourbaki summarizing the current state of research titled ''Catégories supérieures et théorie des topos'' (Higher categories and theory of toposes). '' Higher Categories and Homotopical Algebra'', a mathematical textbook about ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nori Motive
In algebraic geometry, a Nori motive is a mixed motive constructed by Madhav Nori. Today, it is known that Nori's 1-motive coincides with that of Ayoub. The construction is based on Nori's basic lemma and his tannakian theorem.§ 4.4.4., Bruno Kahn, Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry in Handbook of K-Theory. See also * motivic sheaf References * * * A. Bruguières. On a tannakian result due to Nori, preprint of the Département de Mathématiques de l'Université Montpellier II. * Fakhruddin, Najmuddin. Notes of Nori's lectures on mixed motives. T.I.F.R., Mumbai, 2000. Further reading Reference for Nori motives* Nori motive in nLab * Caramello, Olivia. "Motivic toposes." arXiv preprint arXiv:1507.06271 (2015). * L. Barbieri-Viale, O. Caramello & L. Lafforgue: Syntactic categories for Nori motives, arXiv:1506.06113v1 ath.AG(2015). See alsAn example of incorrect behaviour of a senior mathematician towards a young mathematicianfor some background ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |