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Motive (algebraic Geometry)
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 tuple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations called ''addition'' and ''multiplication'', which obey the same basic laws as addition and multiplication of integers, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. A ''ring'' may be defined as a set that is endowed with two binary operations called ''addition'' and ''multiplication'' such that the ring is an abelian group with respect to the addition operator, and the multiplication operator is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors apply the term ''ring'' to a further generalization, often called a '' rng'', that omits the requirement for a multiplicative identity, and instead call the structure defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-abelian Category
In mathematics, specifically in category theory, a pseudo-abelian category is a category that is preadditive and is such that every idempotent has a 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 rngs (not rings!) together with multiplicative morphisms is pseudo-abelian. A more complicated example is the category of Chow mot ... [...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 for additive categories, 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 ... [...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 this page reserves that 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 homomorphisms is again ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoidal Category
In mathematics, a monoidal category (or tensor category) is a category (mathematics), category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an Object (category theory), object ''I'' that is both a left identity, 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 diagram (category theory), diagrams commutative diagram, commute. The ordinary tensor product makes vector spaces, abelian groups, module (mathematics), ''R''-modules, or algebra (ring theory), ''R''-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples. Every (small category, 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 ... [...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 ... [...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] |
Adequate Equivalence Relation
In algebraic geometry, a branch of mathematics, an adequate equivalence relation is an equivalence relation on algebraic cycles of smooth projective varieties used to obtain a well-working theory of such cycles, and in particular, well-defined intersection products. Pierre Samuel formalized the concept of an adequate equivalence relation in 1958. Since then it has become central to theory of motives. For every adequate equivalence relation, one may define the category of pure motives with respect to that relation. Possible (and useful) adequate equivalence relations include ''rational'', ''algebraic'', ''homological'' and ''numerical equivalence''. They are called "adequate" because dividing out by the equivalence relation is functorial, i.e. push-forward (with change of codimension) and pull-back of cycles is well-defined. Codimension 1 cycles modulo rational equivalence form the classical group of divisors modulo linear equivalence. All cycles modulo rational equivalence form the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CW-complex
In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generalizes both manifolds and simplicial complexes and has particular significance for algebraic topology. It was initially introduced by J. H. C. Whitehead to meet the needs of homotopy theory. (open access) CW complexes have better categorical properties than simplicial complexes, but still retain a combinatorial nature that allows for computation (often with a much smaller complex). The C in CW stands for "closure-finite", and the W for "weak" topology. Definition CW complex A CW complex is constructed by taking the union of a sequence of topological spaces \emptyset = X_ \subset X_0 \subset X_1 \subset \cdots such that each X_k is obtained from X_ by gluing copies of k-cells (e^k_\alpha)_\alpha, each homeomorphic to the open k-ball B^ ... [...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 par ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Betti Cohomology
Betti may refer to: People * Betti (given name) * Betti (surname) Other uses * Betti number in topology, named for Enrico Betti * Betti's theorem in engineering theory, named for Enrico Betti * Betti reaction, a chemical addition reaction See also * Beti (other) * Betty (other) {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
