Concept
Homological algebra
Concepts associés (54)
Objet projectif
En théorie des catégories, un objet projectif est une forme de généralisation des modules projectifs. Les objets projectifs dans les catégories abéliennes sont utilisés en algèbre homologique. La notion duale d'objet projectif est celle d'. Un objet dans une catégorie est dit projectif si pour tout épimorphisme et tout morphisme , il existe un morphisme tel que , c'est-à-dire que le diagramme suivant commute : 150px|center Autrement dit, tout morphisme se factorise par les épimorphismes .
Quasi-isomorphisme
En mathématiques, un quasi-isomorphisme est une application induisant un isomorphisme en homologie. Cette définition s'applique aux morphismes de complexes différentiels et notamment aux complexes de chaines ou de cochaines, mais aussi aux applications continues entre espaces topologiques via les différentes théories d'homologie. Toute équivalence d'homotopie est un quasi-isomorphisme mais la réciproque est fausse. En particulier, l'existence d'un quasi-isomorphisme entre deux espaces n'implique pas l'existence d'un quasi-isomorphisme réciproque.
Graded-commutative ring
In algebra, a graded-commutative ring (also called a skew-commutative ring) is a graded ring that is commutative in the graded sense; that is, homogeneous elements x, y satisfy where |x | and |y | denote the degrees of x and y. A commutative (non-graded) ring, with trivial grading, is a basic example. An exterior algebra is an example of a graded-commutative ring that is not commutative in the non-graded sense. A cup product on cohomology satisfies the skew-commutative relation; hence, a cohomology ring is graded-commutative.
Normal morphism
In 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. A monomorphism is normal if it is the of some morphism, and an epimorphism is conormal if it is the 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.
Subquotient
In the mathematical fields of and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, though this conflicts with in category theory. In the literature about sporadic groups wordings like " is involved in " can be found with the apparent meaning of " is a subquotient of ." A quotient of a subrepresentation of a representation (of, say, a group) might be called a subquotient representation; e.
Godement resolution
The Godement resolution of a sheaf is a construction in homological algebra that allows one to view global, cohomological information about the sheaf in terms of local information coming from its stalks. It is useful for computing sheaf cohomology. It was discovered by Roger Godement. Given a topological space X (more generally, a topos X with enough points), and a sheaf F on X, the Godement construction for F gives a sheaf constructed as follows. For each point , let denote the stalk of F at x.
Hyperhomology
In homological algebra, the hyperhomology or hypercohomology () is a generalization of (co)homology functors which takes as input not objects in an but instead chain complexes of objects, so objects in . It is a sort of cross between the derived functor cohomology of an object and the homology of a chain complex since hypercohomology corresponds to the derived global sections functor . Hyperhomology is no longer used much: since about 1970 it has been largely replaced by the roughly equivalent concept of a derived functor between derived categories.
Dimension homologique
En algèbre, la dimension homologique d'un anneau R diffère en général de sa dimension de Krull et se définit à partir des résolutions projectives ou injectives des R-modules. On définit également la dimension faible à partir des résolutions plates des R-modules. La dimension de Krull (respectivement homologique, faible) de R peut être vue comme une mesure de l'éloignement de cet anneau par rapport à la classe des anneaux artiniens (resp. semi-simples, ), cette dimension étant nulle si, et seulement si R est artinien (resp.
Algèbre d'opérateurs
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings. The results obtained in the study of operator algebras are often phrased in algebraic terms, while the techniques used are often highly analytic. Although the study of operator algebras is usually classified as a branch of functional analysis, it has direct applications to representation theory, differential geometry, quantum statistical mechanics, quantum information, and quantum field theory.
Coherent duality
In mathematics, coherent duality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the 'local' theory. The historical roots of the theory lie in the idea of the adjoint linear system of a linear system of divisors in classical algebraic geometry. This was re-expressed, with the advent of sheaf theory, in a way that made an analogy with Poincaré duality more apparent.