|
Cotangent Complex
In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric spaces such as manifolds or schemes. If f: X \to Y is a morphism of geometric or algebraic objects, the corresponding cotangent complex \mathbf_^\bullet can be thought of as a universal "linearization" of it, which serves to control the deformation theory of f. It is constructed as an object in a certain derived category of sheaves on X using the methods of homotopical algebra. Restricted versions of cotangent complexes were first defined in various cases by a number of authors in the early 1960s. In the late 1960s, Michel André and Daniel Quillen independently came up with the correct definition for a morphism of commutative rings, using simplicial methods to make precise the idea of the cotangent complex as given by taking the (non-abelian) left derived functor of Kähler differentials. Luc Illusie then globalized this def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Luc Illusie
Luc Illusie (; born 1940) is a French mathematician, specializing in algebraic geometry. His most important work concerns the theory of the cotangent complex and deformations, crystalline cohomology and the De Rham–Witt complex, and logarithmic geometry. In 2012, he was awarded the Émile Picard Medal of the French Academy of Sciences. Biography Luc Illusie entered the École Normale Supérieure in 1959. At first a student of the mathematician Henri Cartan, he participated in the Cartan–Schwartz seminar of 1963–1964. In 1964, following Cartan's advice, he began to work with Alexandre Grothendieck, collaborating with him on two volumes of the latter's Séminaire de Géométrie Algébrique du Bois Marie. In 1970, Illusie introduced the concept of the cotangent complex. A researcher in the Centre national de la recherche scientifique from 1964 to 1976, Illusie then became a professor at the University of Paris-Sud, retiring as emeritus professor in 2005. Between 1984 and 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conormal Exact Sequence
In category theory and its applications to mathematics, a normal monomorphism or conormal epimorphism is a particularly well-behaved type of morphism. A normal category is a category in which every monomorphism is normal. A conormal category is one in which every epimorphism is conormal. Definition A monomorphism is normal if it is the kernel of some morphism, and an epimorphism is conormal if it is the cokernel of some morphism. A category C is binormal if it's both normal and conormal. But note that some authors will use the word "normal" only to indicate that C is binormal. Examples In the category of groups, a monomorphism ''f'' from ''H'' to ''G'' is normal if and only if its image is a normal subgroup of ''G''. In particular, if ''H'' is a subgroup of ''G'', then the inclusion map ''i'' from ''H'' to ''G'' is a monomorphism, and will be normal if and only if ''H'' is a normal subgroup of ''G''. In fact, this is the origin of the term "normal" for monomorphisms. On the other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connecting Homomorphism
The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology. Homomorphisms constructed with its help are generally called ''connecting homomorphisms''. Statement In an abelian category (such as the category of abelian groups or the category of vector spaces over a given field), consider a commutative diagram: : where the rows are exact sequences and 0 is the zero object. Then there is an exact sequence relating the kernels and cokernels of ''a'', ''b'', and ''c'': :\ker a ~~ \ker b ~~ \ker c ~\overset~ \operatornamea ~~ \operatornameb ~~ \operatornamec where ''d'' is a homomorphism, known as the ''connecting homomorphism''. Furthermore, if the morphism ''f'' is a monomorphism, then so is the morphism \ker a ~~ \ker b, and if ''g is an epimorphism, then so is \operatorna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Functor
In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in various quite different settings that a short exact sequence often gives rise to a "long exact sequence". The concept of derived functors explains and clarifies many of these observations. Suppose we are given a covariant left exact functor ''F'' : A → B between two abelian categories A and B. If 0 → ''A'' → ''B'' → ''C'' → 0 is a short exact sequence in A, then applying ''F'' yields the exact sequence 0 → ''F''(''A'') → ''F''(''B'') → ''F''(''C'') and one could ask how to continue this sequence to the right to form a long exact sequence. Strictly speaking, this question is ill-posed, since there are always numerous different ways to continue a given exact sequence to the right. But it turns out that (if A is "nice" enough) the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deformation Theory
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesimal conditions are the result of applying the approach of differential calculus to solving a problem with constraints. The name is an analogy to non-rigid structures that deform slightly to accommodate external forces. Some characteristic phenomena are: the derivation of first-order equations by treating the ε quantities as having negligible squares; the possibility of ''isolated solutions'', in that varying a solution may not be possible, ''or'' does not bring anything new; and the question of whether the infinitesimal constraints actually 'integrate', so that their solution does provide small variations. In some form these considerations have a history of centuries in mathematics, but also in physics and engineering. For example, in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imperfection Module
The imperfect, or ''past imperfective'', is a verb form in linguistics. Imperfect or imperfection or imperfectionist may also refer to: Film and TV * ''Imperfect'' (2012 film), a 2012 Singaporean film * ''Imperfect'' (2019 film), a 2019 Indonesian film * '' An Imperfection'', a 2015 Sri Lankan Canadian film * "Imperfection" (''Star Trek: Voyager''), a TV episode * ''The Imperfects'', an 2022 TV series Music *'' Imperfectly'', a 1992 album by singer-songwriter Ani DiFranco *''Imperfection'', 2004 album by Real Life *"Imperfect", song by Joi Cardwell *"Imperfection", 2008 song by Glenn Hughes from ''First Underground Nuclear Kitchen'' * "Imperfection" (Evanescence song) from ''Synthesis'' *"Imperfection", 2015 song by Tinchy Stryder featuring vocals by Fuse ODG *"Imperfection", song by Skillet from '' Collide'' *"Imperfection", song by Gentleman from Aloe Blacc discography * "Imperfections" (song), song by Celine Dion from ''Courage'' *"Imperfections", song by Josh Osho from '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abelian groups, Ab. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very ''stable'' categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories are na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right Exact Functor
In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much of the work in homological algebra is designed to cope with functors that ''fail'' to be exact, but in ways that can still be controlled. Definitions Let P and Q be abelian categories, and let be a covariant additive functor (so that, in particular, ''F''(0) = 0). We say that ''F'' is an exact functor if whenever :0 \to A\ \stackrel \ B\ \stackrel \ C \to 0 is a short exact sequence in P then :0 \to F(A) \ \stackrel \ F(B)\ \stackrel \ F(C) \to 0 is a short exact sequence in Q. (The maps are often omitted and implied, and one says: "if 0→''A''→''B''→''C''→0 is exact, then 0→''F''(''A'')→''F''(''B'')→''F''(''C'')→0 is also exact".) Further, we say that ''F'' is *left-exact if whenever 0→''A''→''B''→' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Space
In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topology, while algebraic spaces are given by gluing together affine schemes using the finer étale topology. Alternatively one can think of schemes as being locally isomorphic to affine schemes in the Zariski topology, while algebraic spaces are locally isomorphic to affine schemes in the étale topology. The resulting category of algebraic spaces extends the category of schemes and allows one to carry out several natural constructions that are used in the construction of moduli spaces but are not always possible in the smaller category of schemes, such as taking the quotient of a free action by a finite group (cf. the Keel–Mori theorem). Definition There are two common ways to define algebraic spaces: they can be defined as either quotients of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
