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Cartan–Eilenberg Resolution
In homological algebra, the Cartan–Eilenberg resolution is in a sense, a resolution of a chain complex. It can be used to construct hyper-derived functors. It is named in honor of Henri Cartan and Samuel Eilenberg. Definition Let \mathcal be an Abelian category with enough projectives, and let A_ be a chain complex with objects in \mathcal. Then a Cartan–Eilenberg resolution of A_ is an upper half-plane double complex P_ (i.e., P_ = 0 for q < 0) consisting of projective objects of and an "augmentation" chain map such that * If then the ''p''-th column is zero, i.e. for all ''q''. * For any fixed column , ** The complex of boundaries obtained by applying the horizontal differential to (the st column of ) forms a projective resolution |
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] |
Resolution (algebra)
In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category), which is used to define invariants characterizing the structure of a specific module or object of this category. When, as usually, arrows are oriented to the right, the sequence is supposed to be infinite to the left for (left) resolutions, and to the right for right resolutions. However, a finite resolution is one where only finitely many of the objects in the sequence are non-zero; it is usually represented by a finite exact sequence in which the leftmost object (for resolutions) or the rightmost object (for coresolutions) is the zero-object. Generally, the objects in the sequence are restricted to have some property ''P'' (for example to be free). Thus one speaks of a ''P resolution''. In particular, every module has free resolutions, projective resol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chain Complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathematics), image of each homomorphism is included in the kernel (algebra)#Group homomorphisms, kernel of the next. Associated to a chain complex is its Homology (mathematics), homology, which describes how the images are included in the kernels. A cochain complex is similar to a chain complex, except that its homomorphisms are in the opposite direction. The homology of a cochain complex is called its cohomology. In algebraic topology, the singular chain complex of a topological space X is constructed using continuous function#continuous functions between topological spaces, continuous maps from a simplex to X, and the homomorphisms of the chain complex capture how these maps restrict to the boundary of the simplex. The homology of this chain co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henri Cartan
Henri Paul Cartan (; 8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology. He was the son of the mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of composer , physicist and mathematician , and the son-in-law of physicist Pierre Weiss. Life According to his own words, Henri Cartan was interested in mathematics at a very young age, without being influenced by his family. He moved to Paris with his family after his father's appointment at Sorbonne in 1909 and he attended secondary school at Lycée Hoche in Versailles. available also at In 1923 he started studying mathematics at École Normale Supérieure, receiving an agrégation in 1926 and a doctorate in 1928. His PhD thesis, entitled ''Sur les systèmes de fonctions holomorphes a variétés linéaires lacunaires et leurs applications'', was supervised by Paul Montel. Cartan taught at Lycée Malherbe in Caen from 1928 to 1929, at Un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Samuel Eilenberg
Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to a Jewish family. He spent much of his career as a professor at Columbia University. He earned his Ph.D. from University of Warsaw in 1936, with thesis ''On the Topological Applications of Maps onto a Circle''; his thesis advisors were Kazimierz Kuratowski and Karol Borsuk. He died in New York City in January 1998. Career Eilenberg's main body of work was in algebraic topology. He worked on the axiomatic treatment of homology theory with Norman Steenrod (and the Eilenberg–Steenrod axioms are named for the pair), and on homological algebra with Saunders Mac Lane. In the process, Eilenberg and Mac Lane created category theory. Eilenberg was a member of Bourbaki and, with Henri Cartan, wrote the 1956 book ''Homological Algebra''. Later ... [...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] |
Enough Projectives
In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in Abelian category, abelian Category (mathematics), categories are used in homological algebra. The Duality (mathematics), dual notion of a projective object is that of an injective object. Definition An Object (category theory), object P in a category \mathcal is ''projective'' if for any epimorphism e:E\twoheadrightarrow X and morphism f:P\to X, there is a morphism \overline:P\to E such that e\circ \overline=f, i.e. the following diagram Commutative diagram, commutes: That is, every morphism P\to X mathematical jargon#factor through, factors through every epimorphism E\twoheadrightarrow X. If ''C'' is Category_(mathematics)#Small_and_large_categories, locally small, i.e., in particular \operatorname_C(P, X) is a Set (mathematics), set for any object ''X'' in ''C'', this definition is equivalent to the condition that the hom functor (also known as representab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Complex
In mathematics, specifically Homological algebra, a double complex is a generalization of a chain complex where instead of having a \mathbb-grading, the objects in the bicomplex have a \mathbb\times\mathbb-grading. The most general definition of a double complex, or a bicomplex, is given with objects in an additive category \mathcal. A bicomplex is a sequence of objects C_ \in \text(\mathcal) with two differentials, the horizontal differentiald^h: C_ \to C_and the vertical differentiald^v:C_ \to C_which have the compatibility relationd_h\circ d_v = d_v\circ d_hHence a double complex is a commutative diagram of the form\begin & & \vdots & & \vdots & & \\ & & \uparrow & & \uparrow & & \\ \cdots & \to & C_ & \to & C_ & \to & \cdots \\ & & \uparrow & & \uparrow & & \\ \cdots & \to & C_ & \to & C_ & \to & \cdots \\ & & \uparrow & & \uparrow & & \\ & & \vdots & & \vdots & & \\ \endwhere the rows and columns form chain complexes. Some authors instead require that the squares anticom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horseshoe Lemma
In homological algebra, the horseshoe lemma, also called the simultaneous resolution theorem, is a statement relating resolutions of two objects A' and A'' to resolutions of extensions of A' by A''. It says that if an object A is an extension of A' by A'', then a resolution of A can be built up inductively with the ''n''th item in the resolution equal to the coproduct of the ''n''th items in the resolutions of A' and A''. The name of the lemma comes from the shape of the diagram illustrating the lemma's hypothesis. Formal statement Let \mathcal be an abelian category with enough projectives. If is a diagram in \mathcal such that the column is exact and the rows are projective resolutions of A' and A'' respectively, then it can be completed to a commutative diagram where all columns are exact, the middle row is a projective resolution of A, and P_n=P'_n\oplus P''_n for all ''n''. If \mathcal is an abelian category with enough injectives, the dual statement also holds. The ... [...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] |
Hyperhomology
In homological algebra, the hyperhomology or hypercohomology (\mathbb_*(-), \mathbb^*(-)) is a generalization of (co)homology functors which takes as input not objects in an abelian category \mathcal but instead chain complexes of objects, so objects in \text(\mathcal). 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 \mathbf^*\Gamma(-). 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. Motivation One of the motivations for hypercohomology comes from the fact that there isn't an obvious generalization of cohomological long exact sequences associated to short exact sequences0 \to M' \to M \to M'' \to 0i.e. there is an associated long exact sequence0 \to H^0(M') \to H^0(M) \to H^0(M'')\to H^1(M') \to \cdots It turns out hypercohomol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
