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Five-term Exact Sequence
In mathematics, five-term exact sequence or exact sequence of low-degree terms is a sequence of terms related to the first step of a spectral sequence. More precisely, let :E_2^ \Rightarrow H^n(A) be a first quadrant spectral sequence, meaning that E_2^ vanishes except when ''p'' and ''q'' are both non-negative. Then there is an exact sequence :0 → ''E''21,0 → ''H'' 1(''A'') → ''E''20,1 → ''E''22,0 → ''H'' 2(''A''). Here, the map E_2^ \to E_2^ is the differential of the E_2-term of the spectral sequence. Example *The inflation-restriction exact sequence ::0 → ''H'' 1(''G''/''N'', ''A''''N'') → ''H'' 1(''G'', ''A'') → ''H'' 1(''N'', ''A'')''G''/''N'' → ''H'' 2(''G''/''N'', ''A''''N'') →''H'' 2(''G'', ''A'') :in group cohomology arises as the five-term exact sequence associated to the Lyndon–Hochschild–Serre spectral sequence ::''H'' ''p''(''G''/''N'', ''H'' ''q''(''N'', ''A'')) ⇒ ''H'' ''p+q''(''G, ''A'') : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context of group theory, a sequence :G_0\;\xrightarrow\; G_1 \;\xrightarrow\; G_2 \;\xrightarrow\; \cdots \;\xrightarrow\; G_n of groups and group homomorphisms is said to be exact at G_i if \operatorname(f_i)=\ker(f_). The sequence is called exact if it is exact at each G_i for all 1\leq i |
Spectral Sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra. Discovery and motivation Motivated by problems in algebraic topology, Jean Leray introduced the notion of a sheaf (mathematics), sheaf and found himself faced with the problem of computing sheaf cohomology. To compute sheaf cohomology, Leray introduced a computational technique now known as the Leray spectral sequence. This gave a relation between cohomology groups of a sheaf and cohomology groups of the direct image of a sheaf, pushforward of the sheaf. The relation involved an infinite process. Leray found that the cohomology groups of the pushforward formed a natural chain complex, so that he could take the cohomolo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inflation-restriction Exact Sequence
In mathematics, the inflation-restriction exact sequence is an exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context o ... occurring in group cohomology and is a special case of the five-term exact sequence arising from the study of spectral sequences. Specifically, let ''G'' be a group (mathematics), group, ''N'' a normal subgroup, and ''A'' an abelian group which is equipped with an action of ''G'', i.e., a homomorphism from ''G'' to the automorphism, automorphism group of ''A''. The quotient group ''G''/''N'' acts on ::''A''''N'' = . : Then the inflation-restriction exact sequence is: ::0 → ''H'' 1(''G''/''N'', ''A''''N'') → ''H'' 1(''G'', ''A'') → ''H'' 1(''N'', ''A'')''G''/''N'' → ''H'' 2(''G''/''N'', ''A''''N'') →''H'' 2(''G'', ''A'') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Cohomology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group ''G'' in an associated ''G''-module ''M'' to elucidate the properties of the group. By treating the ''G''-module as a kind of topological space with elements of G^n representing ''n''-simplices, topological properties of the space may be computed, such as the set of cohomology groups H^n(G,M). The cohomology groups in turn provide insight into the structure of the group ''G'' and ''G''-module ''M'' themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyndon–Hochschild–Serre Spectral Sequence
In mathematics, especially in the fields of group cohomology, homological algebra and number theory, the Lyndon spectral sequence or Hochschild–Serre spectral sequence is a spectral sequence relating the group cohomology of a normal subgroup ''N'' and the quotient group ''G''/''N'' to the cohomology of the total group ''G''. The spectral sequence is named after Roger Lyndon, Gerhard Hochschild, and Jean-Pierre Serre. Statement Let G be a group and N be a normal subgroup. The latter ensures that the quotient G/N is a group, as well. Finally, let A be a G-module. Then there is a spectral sequence of cohomological type :H^p(G/N,H^q(N,A)) \Longrightarrow H^(G,A) and there is a spectral sequence of homological type :H_p(G/N,H_q(N,A)) \Longrightarrow H_(G,A), where the arrow '\Longrightarrow' means convergence of spectral sequences. The same statement holds if G is a profinite group, N is a ''closed'' normal subgroup and H^* denotes the continuous cohomology. Examples Homo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Profinite Group
In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. Properties of the profinite group are generally speaking uniform properties of the system. For example, the profinite group is finitely generated (as a topological group) if and only if there exists d\in\N such that every group in the system can be generated by d elements. Many theorems about finite groups can be readily generalised to profinite groups; examples are Lagrange's theorem and the Sylow theorems. To construct a profinite group one needs a system of finite groups and group homomorphisms between them. Without loss of generality, these homomorphisms can be assumed to be surjective, in which case the finite groups will appear as quotient groups of the resulting profinite group; in a sense, these quotients approximate the profini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Subgroup
In mathematics, topological groups are logically the combination of Group (mathematics), groups and Topological space, topological spaces, i.e. they are groups and topological spaces at the same time, such that the Continuous function, continuity condition for the group operations connects these two structures together and consequently they are not independent from each other. Topological groups have been studied extensively in the period of 1925 to 1940. Alfréd Haar, Haar and André Weil, Weil (respectively in 1933 and 1940) showed that the Integral, integrals and Fourier series are special cases of a very wide class of topological groups. Topological groups, along with continuous group actions, are used to study continuous symmetry, symmetries, which have many applications, for example, Symmetry (physics), in physics. In functional analysis, every topological vector space is an additive topological group with the additional property that scalar multiplication is continuous; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G is normal in G if and only if gng^ \in N for all g \in G and n \in N. The usual notation for this relation is N \triangleleft G. Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of G are precisely the kernels of group homomorphisms with domain G, which means that they can be used to internally classify those homomorphisms. Évariste Galois was the first to realize the importance of the existence of normal subgroups. Definitions A subgroup N of a group G is called a normal subgroup of G if it is invariant under conjugation; that is, the conjugation of an element of N by an element of G is always in N. The usual notation for this re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Module
In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous. Examples A topological vector space is a topological module over a topological field. An abelian topological group can be considered as a topological module over \Z, where \Z is the ring of integers with the discrete topology. A topological ring is a topological module over each of its subrings. A more complicated example is the I-adic topology on a ring and its modules. Let I be an ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ... of a ring R. The sets of the form x + I^n for all x \in R and all positive integers n, form a base for a topology on R that makes R into a topological ring. Then for any left R-module M, the sets of the form x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
G-module
In mathematics, given a group ''G'', a ''G''-module is an abelian group ''M'' on which ''G'' acts compatibly with the abelian group structure on ''M''. This widely applicable notion generalizes that of a representation of ''G''. Group (co)homology provides an important set of tools for studying general ''G''-modules. The term ''G''-module is also used for the more general notion of an ''R''-module on which ''G'' acts linearly (i.e. as a group of ''R''-module automorphisms). Definition and basics Let G be a group. A left G-module consists of an abelian group M together with a left group action \rho:G\times M\to M such that :''g''·(''a''1 + ''a''2) = ''g''·''a''1 + ''g''·''a''2 and :(''g''2 ''x'' ''g''1)·''a'' = ''g''2·(''g''1·''a'') where ''g''·''a'' denotes ρ(''g'',''a'') and ''x'' denotes the binary operation inside the group ''G''. A right ''G''-module is defined similarly. Given a left ''G''-module ''M'', it can be turned into a right ''G''-module by defining '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Long Exact Sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context of group theory, a sequence :G_0\;\xrightarrow\; G_1 \;\xrightarrow\; G_2 \;\xrightarrow\; \cdots \;\xrightarrow\; G_n of groups and group homomorphisms is said to be exact at G_i if \operatorname(f_i)=\ker(f_). The sequence is called exact if it is exact at each G_i for all 1\leq i |
