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Mixed Motive (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 ... [...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] |
Betti Cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory. From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rigid Category
In category theory, a branch of mathematics, a rigid category is a monoidal category where every object is rigid, that is, has a dual ''X''* (the internal Hom 'X'', 1 and a morphism 1 → ''X'' ⊗ ''X''* satisfying natural conditions. The category is called right rigid or left rigid according to whether it has right duals or left duals. They were first defined (following Alexander Grothendieck) by Neantro Saavedra Rivano in his thesis on Tannakian categories. Definition There are at least two equivalent definitions of a rigidity. *An object ''X'' of a monoidal category is called left rigid if there is an object ''Y'' and morphisms \eta_X : \mathbf \to X \otimes Y and \epsilon_X : Y \otimes X \to \mathbf such that both compositions X ~ \xrightarrow ~ (X \otimes Y) \otimes X ~ \xrightarrow ~ X \otimes (Y \otimes X) ~ \xrightarrow ~ X Y ~ \xrightarrow ~ Y \otimes (X \otimes Y ) ~ \xrightarrow ~ (Y \otimes X) \otimes Y ~ \xrightarrow ~ Y are identities. A right rigid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tate Motive
In number theory, a cyclotomic character is a character of a Galois group giving the Galois action on a group of roots of unity. As a one-dimensional representation over a ring , its representation space is generally denoted by (that is, it is a representation ). ''p''-adic cyclotomic character Fix a prime, and let denote the absolute Galois group of the rational numbers. The roots of unity \mu_ = \left\ form a cyclic group of order p^n, generated by any choice of a primitive th root of unity . Since all of the primitive roots in \mu_ are Galois conjugate, the Galois group G_\mathbf acts on \mu_ by automorphisms. After fixing a primitive root of unity \zeta_ generating \mu_, any element of \mu_ can be written as a power of \zeta_, where the exponent is a unique element in (\mathbf/p^n\mathbf)^\times. One can thus write \sigma.\zeta := \sigma(\zeta) = \zeta_^ where a(\sigma,n) \in (\mathbf/p^n \mathbf)^\times is the unique element as above, depending on both \sigma and p. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lefschetz Motive
Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations. Life He was born in Moscow, the son of Alexander Lefschetz and his wife Sarah or Vera Lifschitz, Jewish traders who used to travel around Europe and the Middle East (they held Ottoman Empire, Ottoman passports). Shortly thereafter, the family moved to Paris. He was educated there in engineering at the École Centrale Paris, but emigrated to the US in 1905. He was badly injured in an industrial accident in 1907, losing both hands. He moved towards mathematics, receiving a Doctor of Philosophy, Ph.D. in algebraic geometry from Clark University in Worcester, Massachusetts in 1911. He then took positions in University of Nebraska and University of Kansas, moving to Princeton University in 1924, where he was soo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorical Adjunction
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Categorical may refer to: * Categorical imperative, a concept in philosophy developed by Immanuel Kant * Categorical theory, in mathematical logic * Morley's categoricity theorem, a mathematical theorem in model theory * Categorical data analysis * Categorical distribution, a probability distribution * Categorical logic, a branch of category theory within mathematics with notable connections to theoretical computer science * Categorical syllogism, a kind of logical argument * Categorical proposition, a part of deductive reasoning * Categorization * Categorical perception * Category theory in mathematics ** Categorical set theory * Recursive categorical syntax in linguistics See also *Category (other) Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) * Categories (Peirce) * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-abelian Category
In mathematics, specifically in category theory, a pseudo-abelian category is a category (mathematics), category that is preadditive category, preadditive and is such that every idempotent has a kernel (category theory), kernel. Recall that an idempotent morphism p is an endomorphism of an object with the property that p\circ p = p. Elementary considerations show that every idempotent then has a cokernel.Lars Brünjes, Forms of Fermat equations and their zeta functions, Appendix A The pseudo-abelian condition is stronger than preadditivity, but it is weaker than the requirement that every morphism have a kernel and cokernel, as is true for abelian categories. Synonyms in the literature for pseudo-abelian include pseudoabelian and Karoubian. Examples Any abelian category, in particular the category Ab of abelian groups, is pseudo-abelian. Indeed, in an abelian category, ''every'' morphism has a kernel. The category of associative rng (algebra), rngs (not ring (mathematics), rings ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karoubi Envelope
In mathematics the Karoubi envelope (or Cauchy completion or idempotent completion) of a category C is a classification of the idempotents of C, by means of an auxiliary category. Taking the Karoubi envelope of a preadditive category gives a pseudo-abelian category, hence the construction is sometimes called the pseudo-abelian completion. It is named for the French mathematician Max Karoubi. Given a category C, an idempotent of C is an endomorphism :e: A \rightarrow A with :e\circ e = e. An idempotent ''e'': ''A'' → ''A'' is said to split if there is an object ''B'' and morphisms ''f'': ''A'' → ''B'', ''g'' : ''B'' → ''A'' such that ''e'' = ''g'' ''f'' and 1''B'' = ''f'' ''g''. The Karoubi envelope of C, sometimes written Split(C), is the category whose objects are pairs of the form (''A'', ''e'') where ''A'' is an object of C and e : A \rightarrow A is an idempotent of C, and whose morphisms are the triples : (e, f, e^): (A, e) \rightarrow (A^, e^) where f: A \righta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Preadditive Category
In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-category C is a category such that every hom-set Hom(''A'',''B'') in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation. In formulas: f\circ (g + h) = (f\circ g) + (f\circ h) and (f + g)\circ h = (f\circ h) + (g\circ h), where + is the group operation. Some authors have used the term ''additive category'' for preadditive categories, but here we follow the current trend of reserving this term for certain special preadditive categories (see below). Examples The most obvious example of a preadditive category is the category Ab itself. More precisely, Ab is a closed monoidal category. Note that commutativity is crucial here; it ensures that the sum of two group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoidal Category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant diagrams commute. The ordinary tensor product makes vector spaces, abelian groups, ''R''-modules, or ''R''-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples. Every (small) monoidal category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category's objects and whose binary operation is given by the category's tensor product. A rather different application, of which monoidal categories can be considered an abstractio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chow Ring
In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups (compare Poincaré duality) and have a multiplication called the intersection product. The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general. Rational equivalence and Chow groups For what follows, define a variety over a field k to be an integral scheme of finite type over k. For any scheme X of finite type over k, an algebraic cycle on X means a finite linear combination of subvarieties of X with integer coefficients. (Here and below, subvarieties are understood to b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Correspondence (algebraic Geometry)
In algebraic geometry, a correspondence between algebraic varieties ''V'' and ''W'' is a subset ''R'' of ''V''×''W'', that is closed in the Zariski topology. In set theory, a subset of a Cartesian product of two sets is called a binary relation or correspondence; thus, a correspondence here is a relation that is defined by algebraic equations. There are some important examples, even when ''V'' and ''W'' are algebraic curves: for example the Hecke operators of modular form theory may be considered as correspondences of modular curves. However, the definition of a correspondence in algebraic geometry is not completely standard. For instance, Fulton, in his book on intersection theory, uses the definition above. In literature, however, a correspondence from a variety ''X'' to a variety ''Y'' is often taken to be a subset ''Z'' of ''X''×''Y'' such that ''Z'' is finite and surjective over each component of ''X''. Note the asymmetry in this latter definition; which talks about a correspo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
