|
Presheaf With Transfers
In algebraic geometry, a presheaf with transfers is, roughly, a presheaf that, like cohomology theory, comes with pushforwards, “transfer” maps. Precisely, it is, by definition, a contravariant additive functor from the category of finite correspondences (defined below) to the category of abelian groups (in category theory, “presheaf” is another term for a contravariant functor). When a presheaf ''F'' with transfers is restricted to the subcategory of smooth separated schemes, it can be viewed as a presheaf on the category with ''extra'' maps F(Y) \to F(X), not coming from morphisms of schemes but also from finite correspondences from ''X'' to ''Y'' A presheaf ''F'' with transfers is said to be \mathbb^1-homotopy invariant if F(X) \simeq F(X \times \mathbb^1) for every ''X''. For example, Chow groups as well as motivic cohomology groups form presheaves with transfers. Finite correspondence Let X, Y be algebraic schemes (i.e., separated and of finite type over a fie ... [...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 topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Monoidal Category
In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" \otimes is defined) such that the tensor product is symmetric (i.e. A\otimes B is, in a certain strict sense, naturally isomorphic to B\otimes A for all objects A and B of the category). One of the prototypical examples of a symmetric monoidal category is the category of vector spaces over some fixed field ''k,'' using the ordinary tensor product of vector spaces. Definition A symmetric monoidal category is a monoidal category (''C'', ⊗, ''I'') such that, for every pair ''A'', ''B'' of objects in ''C'', there is an isomorphism s_: A \otimes B \to B \otimes A that is natural in both ''A'' and ''B'' and such that the following diagrams commute: *The unit coherence: *: *The associativity coherence: *: *The inverse law: *: In the diagrams above, ''a'', ''l'' , ''r'' are the associativity isomorphism, the left unit isomorphism, and the right un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopical Algebra
In mathematics, homotopical algebra is a collection of concepts comprising the ''nonabelian'' aspects of homological algebra as well as possibly the abelian aspects as special cases. The ''homotopical'' nomenclature stems from the fact that a common approach to such generalizations is via abstract homotopy theory, as in nonabelian algebraic topology, and in particular the theory of closed model categories. This subject has received much attention in recent years due to new foundational work of Vladimir Voevodsky, Eric Friedlander, Andrei Suslin, and others resulting in the A1 homotopy theory for quasiprojective varieties over a field. Voevodsky has used this new algebraic homotopy theory to prove the Milnor conjecture (for which he was awarded the Fields Medal) and later, in collaboration with Markus Rost, the full Bloch–Kato conjecture. References * * * See also *Derived algebraic geometry * Derivator * Cotangent complex - one of the first objects discovered u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clay Mathematics Monographs
Clay Mathematics Monographs is a series of expositions in mathematics co-published by AMS and Clay Mathematics Institute The Clay Mathematics Institute (CMI) is a private, non-profit foundation (nonprofit), foundation dedicated to increasing and disseminating mathematics, mathematical knowledge. Formerly based in Peterborough, New Hampshire, the corporate address i .... Each volume in the series offers an exposition of an active area of current research, provided by a group of mathematicians. List of books * * * * * {{cite book , last=Hori , first=Kentaro , first2=Sheldon , last2=Katz , author2-link=Sheldon Katz , first3=Albrecht , last3=Klemm , first4=Rahul , last4=Pandharipande , author4-link=Rahul Pandharipande , first5=Richard , last5=Thomas , author5-link=Richard Thomas (mathematician) , first6=Cumrun , last6=Vafa , first7=Ravi , last7=Vakil , author7-link=Ravi Vakil , first8=Eric , last8=Zaslow , author8-link=Eric Zaslow , year=2003 , title=Mirror Symmetry , isbn= 978 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...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) po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Étale Topology
In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic. The étale topology was originally introduced by Grothendieck to define étale cohomology, and this is still the étale topology's most well-known use. Definitions For any scheme ''X'', let Ét(''X'') be the category of all étale morphisms from a scheme to ''X''. This is the analog of the category of open subsets of ''X'' (that is, the category whose objects are varieties and whose morphisms are open immersions). Its objects can be informally thought of as étale open subsets of ''X''. The intersection of two objects corresponds to their fiber product over ''X''. Ét(''X'') is a large category, meaning that its objects do not form a set. An étale presheaf on ''X'' is a contravariant functor from Ét(''X'') to the category of sets. A presheaf '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixed Motives (math)
In algebraic geometry, motives (or sometimes motifs, following French usage) is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically, a "motif" is the "cohomology essence" of a variety. In the formulation of Grothendieck for smooth projective varieties, a motive is a triple (X, p, m), where ''X'' is a smooth projective variety, p: X \vdash X is an idempotent correspondence, and ''m'' an integer, however, such a triple contains almost no information outside the context of Grothendieck's category of pure motives, where a morphism from (X, p, m) to (Y, q, n) is given by a correspondence of degree n-m. A more object-focused approach is taken by Pierre Deligne in ''Le Groupe Fondamental de la Droite Projective Moins Trois Points''. In that article, a motive is a "system of realisations" – that is, a t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relative Cycle
In algebraic geometry, a relative cycle is a type of algebraic cycle on a scheme. In particular, let X be a scheme of finite type over a Noetherian scheme S, so that X \rightarrow S. Then a relative cycle is a cycle on X which lies over the generic points of S, such that the cycle has a well-defined specialization to any fiber of the projection X \rightarrow S. The notion was introduced by Andrei Suslin and 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 var ... in 2000; the authors were motivated to overcome some of the deficiencies of sheaves with transfers. References * * *Appendix 1A of {{algebraic-geometry-stub Algebraic geometry ... [...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 bigrad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smash Product
In topology, a branch of mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) (''X,'' ''x''0) and (''Y'', ''y''0) is the quotient of the product space ''X'' × ''Y'' under the identifications (''x'', ''y''0) ∼ (''x''0, ''y'') for all ''x'' in ''X'' and ''y'' in ''Y''. The smash product is itself a pointed space, with basepoint being the equivalence class of (''x''0, ''y''0). The smash product is usually denoted ''X'' ∧ ''Y'' or ''X'' ⨳ ''Y''. The smash product depends on the choice of basepoints (unless both ''X'' and ''Y'' are homogeneous). One can think of ''X'' and ''Y'' as sitting inside ''X'' × ''Y'' as the subspaces ''X'' × and × ''Y''. These subspaces intersect at a single point: (''x''0, ''y''0), the basepoint of ''X'' × ''Y''. So the union of these subspaces can be identified with the wedge sum ''X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Of Schemes
In mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction. It has many interpretations and special cases. For example, the fiber product describes how an algebraic variety over one field determines a variety over a bigger field, or the pullback of a family of varieties, or a fiber of a family of varieties. Base change is a closely related notion. Definition The category of schemes is a broad setting for algebraic geometry. A fruitful philosophy (known as Grothendieck's relative point of view) is that much of algebraic geometry should be developed for a morphism of schemes ''X'' → ''Y'' (called a scheme ''X'' over ''Y''), rather than for a single scheme ''X''. For example, rather than simply studying algebraic curves, one can study families of curves over any base scheme ''Y''. Indeed, the two approaches enrich each other. In particular, a scheme over a commutative ring ''R'' means a scheme ''X'' together with a morphism ''X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
