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Hochschild Homology
In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by for algebras over a field, and extended to algebras over more general rings by . Definition of Hochschild homology of algebras Let ''k'' be a field, ''A'' an associative ''k''-algebra, and ''M'' an ''A''-bimodule. The enveloping algebra of ''A'' is the tensor product A^e=A\otimes A^o of ''A'' with its opposite algebra. Bimodules over ''A'' are essentially the same as modules over the enveloping algebra of ''A'', so in particular ''A'' and ''M'' can be considered as ''Ae''-modules. defined the Hochschild homology and cohomology group of ''A'' with coefficients in ''M'' in terms of the Tor functor and Ext functor by : HH_n(A,M) = \operatorname_n^(A, M) : HH^n(A,M) = \operatorname^n_(A, M) Hochschild complex Let ''k'' be a ring, ''A'' an associative ''k''-alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplex Category
In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving maps. It is used to define simplicial and cosimplicial objects. Formal definition The simplex category is usually denoted by \Delta. There are several equivalent descriptions of this category. \Delta can be described as the category of ''non-empty finite ordinals'' as objects, thought of as totally ordered sets, and ''(non-strictly) order-preserving functions'' as morphisms. The objects are commonly denoted = \ (so that is the ordinal n+1 ). The category is generated by coface and codegeneracy maps, which amount to inserting or deleting elements of the orderings. (See simplicial set for relations of these maps.) A simplicial object is a presheaf on \Delta, that is a contravariant functor from \Delta to another category. For instance, simplicial sets are contravariant with the codomain category being the catego ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wedge Product
A wedge is a triangular shaped tool, a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converting a force applied to its blunt end into forces perpendicular ( normal) to its inclined surfaces. The mechanical advantage of a wedge is given by the ratio of the length of its slope to its width..''McGraw-Hill Concise Encyclopedia of Science & Technology'', Third Ed., Sybil P. Parker, ed., McGraw-Hill, Inc., 1992, p. 2041. Although a short wedge with a wide angle may do a job faster, it requires more force than a long wedge with a narrow angle. The force is applied on a flat, broad surface. This energy is transported to the pointy, sharp end of the wedge, hence the force is transported. The wedge simply transports energy in the form of friction and collects it to the pointy end, consequently breaking the item. History Wedges have existed fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morphism (category Theory)
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Although many examples of morphisms are structure-preserving maps, morphisms need not to be maps, but they can be composed in a way that is similar to function composition. Morphisms and objects are constituents of a category. Morphisms, also called ''maps'' or ''arrows'', relate two objects called the ''source'' and the ''target'' of the morphism. There is a partial operation, called ''composition'', on the morphisms of a category that is defined if the target of the first morphism equals the source of the second morphism. The composition of morphisms behaves like function composition ( associativity of composition when it is defined, and existence of an identity morphism for every object). Morphisms and categories recur in much of co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skeleton (category Theory)
In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ..., a skeleton of a category (category theory), category is a subcategory that, roughly speaking, does not contain any extraneous isomorphisms. In a certain sense, the skeleton of a category is the "smallest" equivalence of categories, equivalent category, which captures all "categorical properties" of the original. In fact, two categories are equivalence of categories, equivalent iff, if and only if they have isomorphism of categories, isomorphic skeletons. A category is called skeletal if isomorphic objects are necessarily identical. Definition A skeleton of a category ''C'' is an Equivalence (category theory), equivalent category ''D'' in which isomorphic objects are equal. Typically, a skeleton is taken to be a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Circle
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, * a 0-dimensional simplex is a point, * a 1-dimensional simplex is a line segment, * a 2-dimensional simplex is a triangle, * a 3-dimensional simplex is a tetrahedron, and * a 4-dimensional simplex is a 5-cell. Specifically, a -simplex is a -dimensional polytope that is the convex hull of its vertices. More formally, suppose the points u_0, \dots, u_k are affinely independent, which means that the vectors u_1 - u_0,\dots, u_k-u_0 are linearly independent. Then, the simplex determined by them is the set of points C = \left\. A regular simplex is a simplex that is also a regular polytope. A regular -simplex may be constructed from a regular -simplex by connecting a new vertex to all original vertices by the common edge l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-isomorphism
In homological algebra, a branch of mathematics, a quasi-isomorphism or quism is a morphism ''A'' → ''B'' of chain complexes (respectively, cochain complexes) such that the induced morphisms :H_n(A_\bullet) \to H_n(B_\bullet)\ (\text H^n(A^\bullet) \to H^n(B^\bullet)) of homology groups (respectively, of cohomology groups) are isomorphisms for all ''n''. In the theory of model categories, quasi-isomorphisms are sometimes used as the class of weak equivalences when the objects of the category are chain or cochain complexes. This results in a homology-local theory, in the sense of Bousfield localization in homotopy theory. See also * Derived category In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction pr ... References *Gelfand, Sergei I., Manin, Yuri I. ''Methods of Homological Alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kähler Differential
In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods from calculus and geometry over the complex numbers to contexts where such methods are not available. Definition Let and be commutative rings and be a ring homomorphism. An important example is for a field and a unital algebra over (such as the coordinate ring of an affine variety). Kähler differentials formalize the observation that the derivatives of polynomials are again polynomial. In this sense, differentiation is a notion which can be expressed in purely algebraic terms. This observation can be turned into a definition of the module :\Omega_ of differentials in different, but equivalent ways. Definition using derivations An -linear '' derivation'' on is an - ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fiber 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 '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scheme (mathematics)
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations and define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise '' Éléments de géométrie algébrique'' (EGA); one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem. Schemes elaborate the fundamental idea that an a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Scheme
In algebraic geometry, a derived scheme is a homotopy-theoretic generalization of a scheme in which classical commutative rings are replaced with derived versions such as differential graded algebras, commutative simplicial rings, or commutative ring spectra. From the functor of points point-of-view, a derived scheme is a sheaf ''X'' on the category of simplicial commutative rings which admits an open affine covering \. From the locally ringed space point-of-view, a derived scheme is a pair (X, \mathcal) consisting of a topological space ''X'' and a sheaf \mathcal either of simplicial commutative rings or of commutative ring spectra on ''X'' such that (1) the pair (X, \pi_0 \mathcal) is a scheme and (2) \pi_k \mathcal is a quasi-coherent \pi_0 \mathcal- module. A derived stack is a stacky generalization of a derived scheme. Differential graded scheme Over a field of characteristic zero, the theory is closely related to that of a differential graded scheme. By definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
