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Comodule Over A Hopf Algebroid
In mathematics, at the intersection of algebraic topology and algebraic geometry, there is the notion of a Hopf algebroid which encodes the information of a presheaf of groupoids whose object sheaf and arrow sheaf are represented by algebras. Because any such presheaf will have an associated site, we can consider quasi-coherent sheaves on the site, giving a topos-theoretic notion of modules. Duallypg 2, comodules over a Hopf algebroid are the purely algebraic analogue of this construction, giving a purely algebraic description of quasi-coherent sheaves on a stack: this is one of the first motivations behind the theory. Definition Given a commutative Hopf-algebroid (A,\Gamma) a left comodule Mpg 302 is a left A-module M together with an A-linear map\psi: M \to \Gamma\otimes_AMwhich satisfies the following two properties # (counitary) (\varepsilon\otimes Id_M)\circ \psi = Id_M # (coassociative) (\Delta\otimes Id_M) \circ \psi = (Id_\Gamma \otimes \psi) \circ \psi A right comodule i ... [...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] |
Spectrum (topology)
In algebraic topology, a branch of mathematics, a spectrum is an object representable functor, representing a Cohomology#Generalized cohomology theories, generalized cohomology theory. Every such cohomology theory is representable, as follows from Brown's representability theorem. This means that, given a cohomology theory\mathcal^*:\text^ \to \text,there exist spaces E^k such that evaluating the cohomology theory in degree k on a space X is equivalent to computing the homotopy classes of maps to the space E^k, that is\mathcal^k(X) \cong \left[X, E^k\right].Note there are several different category (mathematics), categories of spectra leading to many technical difficulties, but they all determine the same homotopy category, known as the stable homotopy category. This is one of the key points for introducing spectra because they form a natural home for stable homotopy theory. The definition of a spectrum There are many variations of the definition: in general, a ''spectrum'' is any s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopical Algebra
In mathematics, homotopical algebra is a collection of concepts comprising the ''nonabelian'' aspects of homological algebra as well as possibly the abelian aspects as special cases. The ''homotopical'' nomenclature stems from the fact that a common approach to such generalizations is via abstract homotopy theory, as in nonabelian algebraic topology, and in particular the theory of closed model categories. This subject has received much attention in recent years due to new foundational work of Vladimir Voevodsky, Eric Friedlander, Andrei Suslin, and others resulting in the A1 homotopy theory for quasiprojective varieties over a field. Voevodsky has used this new algebraic homotopy theory to prove the Milnor conjecture (for which he was awarded the Fields Medal) and later, in collaboration with Markus Rost, the full Bloch–Kato conjecture. References * * * See also *Derived algebraic geometry * Derivator * Cotangent complex - one of the first objects discovered u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Algebras
Hopf is a German surname. Notable people with the surname include: *Eberhard Hopf (1902–1983), Austrian mathematician *Hans Hopf (1916–1993), German tenor *Heinz Hopf (1894–1971), German mathematician *Heinz Hopf (actor) (1934–2001), Swedish actor *Ludwig Hopf (1884–1939), German physicist *Maria Hopf Maria Hopf (13 September 1913 – 24 August 2008) was a pioneering archaeobotanist, based at the RGZM, Mainz. Career Hopf studied botany from 1941–44, receiving her doctorate in 1947 on the subject of soil microbes. She then worked in phyto ... (1914-2008), German botanist and archaeologist {{surname, Hopf German-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steenrod Algebra
In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology. For a given prime number p, the Steenrod algebra A_p is the graded Hopf algebra over the field \mathbb_p of order p, consisting of all stable cohomology operations for mod p cohomology. It is generated by the Steenrod squares introduced by for p=2, and by the Steenrod reduced pth powers introduced in and the Bockstein homomorphism for p>2. The term "Steenrod algebra" is also sometimes used for the algebra of cohomology operations of a generalized cohomology theory. Cohomology operations A cohomology operation is a natural transformation between cohomology functors. For example, if we take cohomology with coefficients in a ring R, the cup product squaring operation yields a family of cohomology operations: :H^n(X;R) \to H^(X;R) :x \mapsto x \smile x. Cohomology operations need not be homomorphisms of graded rings; see the Cartan formula below. Thes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adams Spectral Sequence
In mathematics, the Adams spectral sequence is a spectral sequence introduced by which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now called stable homotopy theory. It is a reformulation using homological algebra, and an extension, of a technique called 'killing homotopy groups' applied by the French school of Henri Cartan and Jean-Pierre Serre. Motivation For everything below, once and for all, we fix a prime ''p''. All spaces are assumed to be CW complexes. The ordinary cohomology groups H^*(X) are understood to mean H^*(X; \Z/p\Z). The primary goal of algebraic topology is to try to understand the collection of all maps, up to homotopy, between arbitrary spaces ''X'' and ''Y''. This is extraordinarily ambitious: in particular, when ''X'' is S^n, these maps form the ''n''th homotopy group of ''Y''. A more reasonable (but still very difficult!) goal is to understand ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adams-Novikov Spectral Sequence
In mathematics, the Adams spectral sequence is a spectral sequence introduced by which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now called stable homotopy theory. It is a reformulation using homological algebra, and an extension, of a technique called 'killing homotopy groups' applied by the French school of Henri Cartan and Jean-Pierre Serre. Motivation For everything below, once and for all, we fix a prime ''p''. All spaces are assumed to be CW complexes. The ordinary cohomology groups H^*(X) are understood to mean H^*(X; \Z/p\Z). The primary goal of algebraic topology is to try to understand the collection of all maps, up to homotopy, between arbitrary spaces ''X'' and ''Y''. This is extraordinarily ambitious: in particular, when ''X'' is S^n, these maps form the ''n''th homotopy group of ''Y''. A more reasonable (but still very difficult!) goal is to understand the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Localization (commutative Algebra)
In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module ''R'', so that it consists of fractions \frac, such that the denominator ''s'' belongs to a given subset ''S'' of ''R''. If ''S'' is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the field \Q of rational numbers from the ring \Z of integers. The technique has become fundamental, particularly in algebraic geometry, as it provides a natural link to sheaf theory. In fact, the term ''localization'' originated in algebraic geometry: if ''R'' is a ring of functions defined on some geometric object (algebraic variety) ''V'', and one wants to study this variety "locally" near a point ''p'', then one considers the set ''S'' of all functions that are not zero at ''p'' and localizes ''R'' wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Group Law
In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were introduced by . The term formal group sometimes means the same as formal group law, and sometimes means one of several generalizations. Formal groups are intermediate between Lie groups (or algebraic groups) and Lie algebras. They are used in algebraic number theory and algebraic topology. Definitions A one-dimensional formal group law over a commutative ring ''R'' is a power series ''F''(''x'',''y'') with coefficients in ''R'', such that # ''F''(''x'',''y'') = ''x'' + ''y'' + terms of higher degree # ''F''(''x'', ''F''(''y'',''z'')) = ''F''(''F''(''x'',''y''), ''z'') (associativity). The simplest example is the additive formal group law ''F''(''x'', ''y'') = ''x'' + ''y''. The idea of the definition is that ''F'' should be something like the formal power series expansion of the product of a Lie group, where we choose coordinates so that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brown–Peterson Cohomology
In mathematics, Brown–Peterson cohomology is a generalized cohomology theory introduced by , depending on a choice of prime ''p''. It is described in detail by . Its representing spectrum is denoted by BP. Complex cobordism and Quillen's idempotent Brown–Peterson cohomology BP is a summand of MU(''p''), which is complex cobordism MU localized at a prime ''p''. In fact MU''(p)'' is a wedge product of suspensions of BP. For each prime ''p'', Daniel Quillen showed there is a unique idempotent map of ring spectra ε from MUQ(''p'') to itself, with the property that ε( P''n'' is P''n''if ''n''+1 is a power of ''p'', and 0 otherwise. The spectrum BP is the image of this idempotent ε. Structure of BP The coefficient ring \pi_*(\text) is a polynomial algebra over \Z_ on generators v_n in degrees 2(p^n-1) for n\ge 1. \text_*(\text) is isomorphic to the polynomial ring \pi_*(\text) _1, t_2, \ldots/math> over \pi_*(\text) with generators t_i in \text_(\text) of degrees 2 (p^i-1) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches of algebraic topology Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy gro ... [...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] |
