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List Of Homological Algebra Topics
{{Short description, none This is a list of homological algebra topics, by Wikipedia page. Basic techniques * Cokernel * Exact sequence * Chain complex * Differential module *Five lemma *Short five lemma * Snake lemma *Nine lemma *Extension (algebra) ** Central extension **Splitting lemma *Projective module * Injective module * Projective resolution * Injective resolution * Koszul complex *Exact functor * Derived functor * Ext functor *Tor functor * Filtration (abstract algebra) *Spectral sequence * Abelian category * Triangulated category * Derived category Applications *Group cohomology * Galois cohomology *Lie algebra cohomology *Sheaf cohomology * Whitehead problem * Homological conjectures in commutative algebra Homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homological Algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of module (mathematics), modules and Syzygy (mathematics), syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through both their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariant (mathematics), invariants of ring (mathematics), rings, modules, topological spaces, and other 'tan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Functor
In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much of the work in homological algebra is designed to cope with functors that ''fail'' to be exact, but in ways that can still be controlled. Definitions Let P and Q be abelian categories, and let be a covariant additive functor (so that, in particular, ''F''(0) = 0). We say that ''F'' is an exact functor if whenever :0 \to A\ \stackrel \ B\ \stackrel \ C \to 0 is a short exact sequence in P then :0 \to F(A) \ \stackrel \ F(B)\ \stackrel \ F(C) \to 0 is a short exact sequence in Q. (The maps are often omitted and implied, and one says: "if 0→''A''→''B''→''C''→0 is exact, then 0→''F''(''A'')→''F''(''B'')→''F''(''C'')→0 is also exact".) Further, we say that ''F'' is *left-exact if whenever 0→''A''→''B''→' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead Problem
In group theory, a branch of abstract algebra, the Whitehead problem is the following question: Saharon Shelah proved that Whitehead's problem is independent of ZFC, the standard axioms of set theory. Refinement Assume that ''A'' is an abelian group such that every short exact sequence :0\rightarrow\mathbb\rightarrow B\rightarrow A\rightarrow 0 must split if ''B'' is also abelian. The Whitehead problem then asks: must ''A'' be free? This splitting requirement is equivalent to the condition Ext1(''A'', Z) = 0. Abelian groups ''A'' satisfying this condition are sometimes called Whitehead groups, so Whitehead's problem asks: is every Whitehead group free? It should be mentioned that if this condition is strengthened by requiring that the exact sequence :0\rightarrow C\rightarrow B\rightarrow A\rightarrow 0 must split for any abelian group ''C'', then it is well known that this is equivalent to ''A'' being free. ''Caution'': The converse of Whitehead's problem, namely that ever ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheaf Cohomology
In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. The central work for the study of sheaf cohomology is Grothendieck's 1957 Tôhoku paper. Sheaves, sheaf cohomology, and spectral sequences were introduced by Jean Leray at the prisoner-of-war camp Oflag XVII-A in Austria. From 1940 to 1945, Leray and other prisoners organized a "université en captivité" in the camp. Leray's definitions were simplified and clarified in the 1950s. It became clear that sheaf cohomology was not only a new approach to cohomology in algebraic topology, but also a powerful method in complex analytic geometry and algebraic geometry. These subjects often involve constructing global functions with specified local properties, and sheaf cohomology is ideally suited to such problems. Man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra Cohomology
In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to properties of the Lie algebra. It was later extended by to coefficients in an arbitrary Lie module. Motivation If G is a compact simply connected Lie group, then it is determined by its Lie algebra, so it should be possible to calculate its cohomology from the Lie algebra. This can be done as follows. Its cohomology is the de Rham cohomology of the complex of differential forms on G. Using an averaging process, this complex can be replaced by the complex of left-invariant differential forms. The left-invariant forms, meanwhile, are determined by their values at the identity, so that the space of left-invariant differential forms can be identified with the exterior algebra of the Lie algebra, with a suitable differential. The construction of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Cohomology
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a natural way on some abelian groups, for example those constructed directly from ''L'', but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor. History The current theory of Galois cohomology came together around 1950, when it was realised that the Galois cohomology of ideal class groups in algebraic number theory was one way to formulate class field theory, at the time it was in the process of ridding itself of connections to L-functions. Galois cohomology makes no assumption that Galois groups are abelian groups, so this was a non-abelian theory. It was formulated abstractly as a theory of class ... [...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] |
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] |
Triangulated Category
In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology. Much of homological algebra is clarified and extended by the language of triangulated categories, an important example being the theory of sheaf cohomology. In the 1960s, a typical use of triangulated categories was to extend properties of sheaves on a space ''X'' to complexes of sheaves, viewed as objects of the derived category of sheaves on ''X''. More recently, triangulated categories have become objects of interest in their own right. Many equivalences between triangulated categories of different origins have been proved or conjectured. For example, the homological mirror symmetry conj ... [...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] |
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] |
Filtration (abstract Algebra)
In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory. A filtered algebra over the field k is an algebra (A,\cdot) over k that has an increasing sequence \ \subseteq F_0 \subseteq F_1 \subseteq \cdots \subseteq F_i \subseteq \cdots \subseteq A of subspaces of A such that :A=\bigcup_ F_ and that is compatible with the multiplication in the following sense: : \forall m,n \in \mathbb,\quad F_m\cdot F_n\subseteq F_. Associated graded algebra In general there is the following construction that produces a graded algebra out of a filtered algebra. If A is a filtered algebra then the ''associated graded algebra'' \mathcal(A) is defined as follows: The multiplication is well-defined and endows \mathcal(A) with the structure of a graded algebra, with gradation \_. Furthermore if A is associative then so is \mathcal(A). Also if A is uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
