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Cohomological Descent
In algebraic geometry, a cohomological descent is, roughly, a "derived" version of a fully faithful descent in the classical descent theory. This point is made precise by the below: the following are equivalent: in an appropriate setting, given a map ''a'' from a simplicial space ''X'' to a space ''S'', *a^*: D^+(S) \to D^+(X) is fully faithful. *The natural transformation \operatorname_ \to Ra_* \circ a^* is an isomorphism. The map ''a'' is then said to be a morphism of cohomological descent. The treatment in SGA uses a lot of topos theory. Conrad's notes gives a more down-to-earth exposition. See also *hypercovering In mathematics, and in particular homotopy theory, a hypercovering (or hypercover) is a simplicial object that generalises the Čech nerve of a cover. For the Čech nerve of an open cover one can show that if the space X is compact and if every in ..., of which a cohomological descent is a generalization References *SGA4 Vbis [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 proceeds on the basis that the objects of ''D''(''A'') should be chain complexes in ''A'', with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated spectral sequences. The development of the derived category, by Alexander Grothendieck and his student Jean-Louis Verdier shortly after 1960, now appears as one terminal point in the explosive development of homological algebra in the 1950s, a decade in which it had made remarkab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descent Theory
In mathematics, the idea of descent extends the intuitive idea of 'gluing' in topology. Since the topologists' glue is the use of equivalence relations on topological spaces, the theory starts with some ideas on identification. Descent of vector bundles The case of the construction of vector bundles from data on a disjoint union of topological spaces is a straightforward place to start. Suppose ''X'' is a topological space covered by open sets ''Xi''. Let ''Y'' be the disjoint union of the ''Xi'', so that there is a natural mapping :p: Y \rightarrow X. We think of ''Y'' as 'above' ''X'', with the ''Xi'' projection 'down' onto ''X''. With this language, ''descent'' implies a vector bundle on ''Y ''(so, a bundle given on each ''Xi''), and our concern is to 'glue' those bundles ''Vi'', to make a single bundle ''V'' on X. What we mean is that ''V'' should, when restricted to ''Xi'', give back ''Vi'', up to a bundle isomorphism. The data needed is then this: on each overlap :X_, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topos Theory
In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion of localization; they are a direct generalization of point-set topology. The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic. The mathematical field that studies topoi is called topos theory. Grothendieck topos (topos in geometry) Since the introduction of sheaves into mathematics in the 1940s, a major theme has been to study a space by studying sheaves on a space. This idea was expounded by Alexander Grothendieck by introducing the notion of a "topos". The main utility of this notion is in the abundance of situations in mathematics where topological heuristics are very effective, but an honest topological space is lacking; it is sometimes possible to find a topos formaliz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypercovering
In mathematics, and in particular homotopy theory, a hypercovering (or hypercover) is a simplicial object that generalises the Čech nerve of a cover. For the Čech nerve of an open cover one can show that if the space X is compact and if every intersection of open sets in the cover is contractible, then one can contract these sets and get a simplicial set that is weakly equivalent to X in a natural way. For the étale topology and other sites, these conditions fail. The idea of a hypercover is to instead of only working with n-fold intersections of the sets of the given open cover \mathcal U, to allow the pairwise intersections of the sets in \mathcal U=\mathcal U_0 to be covered by an open cover \mathcal U_1, and to let the triple intersections of this cover to be covered by yet another open cover \mathcal U_2, and so on, iteratively. Hypercoverings have a central role in étale homotopy and other areas where homotopy theory is applied to algebraic geometry, such as motivic homoto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
