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Topic: Cochain complex

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Cohomology

Projective module

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Saunders Mac Lane

Samuel Eilenberg

September 30

Norman Steenrod

	Cohomology - Wikipedia, the free encyclopedia
		In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex.
		That is, cohomology is defined as the abstract study of cochains, cocycles and coboundaries.
		A cohomology theory is a family of contravariant functors from the category of pairs of topological spaces and continuous functions (or some subcategory thereof such as the category of CW complexes) to the category of Abelian groups and group homomorphisms that satisfies the Eilenberg-Steenrod axioms
	en.wikipedia.org /wiki/Cohomology (679 words)

	What is ... (Site not responding. Last check: 2007-10-20)
		The idea of Khovanov homology is to associate a cochain complex to a link diagram whose homology is a link invariant and whose graded Euler characteristic is the Jones polynomial.
		Here is my prefered way: a complex line bundle-with-connection consists of a familty of one dimensional vector spaces (fibres) parametrized by some space (the base space) together with a rule for "parallel transporting" fibres along paths in the base space.
		By this I mean a complex line bundle over the "space" of n-manifolds in some space X together with a rule for parallel transporting fibres over n+1-cobordisms in X. The rule for parallel transport should depend only on the homotopy class (mod boundary) of the cobordism.
	www.ma.hw.ac.uk /~paul/whatis.html (476 words)

	Cochain complex (Site not responding. Last check: 2007-10-20)
		A variant on the concept of chain complex is that of cochain complex.
		Suppose we are given a topological space X.
		The homology of this complex is the de Rham cohomology
	www.therfcc.org /cochain-complex-219789.html (202 words)

	sem8.html (Site not responding. Last check: 2007-10-20)
		G-algebras arose in M.Gerstenhaber’s work when he showed that the Hochschild cohomology of an associative algebra has the natural structure of a G-algebra, Later, F.Cohen showed that an algebra over the homology little discs operad is nothing but a G-algebra.
		Later, Deligne’s conjecture was proved: ‘the Hochschild cochain complex has the natural structure of an algebra over a chain operad of the little discs operad’, in other words, ‘it has a natural structure of homotopy G-algebra’.
		Drawing resemblance to Hochschild cohomology, we show that the disalgebra cohomology admits a G-algebra structure, which is induced by ahomotopy G-algebra structure on the cochain level.
	www.math.iisc.ernet.in /seminar/senov05.html (110 words)

	Chain complex (Site not responding. Last check: 2007-10-20)
		In homological algebra a chain complex $(A_\bullet d_\bullet) is a sequence of abelian groups or modules A$
		A variant on the concept of chain is that of cochain complex.
		A cochain complex $(A^\bullet d^\bullet) is a sequence of abelian groups or modules A$
	www.freeglossary.com /Chain_complex (417 words)

	Cohomologies? - Physics Help and Math Help - Physics Forums (Site not responding. Last check: 2007-10-20)
		A resolution of a module is an acyclic chain complex of projective modules, with the exception of the degree -1 term where it is the module you start with.
		What I gave isn't a paraphrase of the steenrod rod axioms, it is the definition of the cohomology groups of a cochain complex (when it makes sense to talk of these things).
		Remember that any chain complex is turned into a cochain complex by applying a contravariant functor.
	www.physicsforums.com /showthread.php?t=26363&goto=nextoldest (2798 words)

	Citations: The cohomology structure of an associative ring - Gerstenhaber (ResearchIndex)
		....1) cocycle m, the cochain m(m 1 1 1 m 1 1 1 m) 2 C 5; 2 (5.9) is again a cocycle, and that the cohomology class of (5.
		Hochschild cohomology of associative algebras was rst described in [17] and used to study the in nitesimal deformations of associative algebras in [9] 10] In the Gerstenhaber bracket was de ned on the space of cochains of an associative algebra, which equips the space of 1991 Mathematics Subject Classi cation.
		In particular, in ghost number one BRST cohomology we obtain a (noncentral) extension of the Lie algebra of vector fields in the plane by an....
	citeseer.ist.psu.edu /context/434250/0 (2229 words)

	[No title]
		Abstract.We apply the tools of stable homotopy theory to the study of mod- ules over the Steenrod algebra A; in particular, we study the (triangula ted) category Stable(A) of unbounded cochain* complexes of injective comodules over A, the dual of A, in which the morphisms are cochain* homotopy class* *es of maps.
		We let Stable(A) be the category whose objects are cochain complexes of injective left A-comodules, and whose morphisms are cochain homo- topy classes of maps.
		This material applies when is the dual of a group algebra, the dual of an enve* loping algebra, or the dual of the Steenrod algebra; in these cases, Ext(k; k) is the ordinary cohomology of with coefficients in k.
	hopf.math.purdue.edu /Palmieri/palmieri-steenrod.txt (9567 words)

	de rham cohomology (Site not responding. Last check: 2007-10-20)
		= 0 follows essentially from symmetry of second derivatives, so the vector spaces of k-forms along with the exterior derivative are a cochain complex, the de Rham complex:
		In the official differential geometry terminology, forms which are exterior derivatives are called exact and forms whose exterior derivatives are 0 are called closed (see closed and exact differential forms); d
		The cohomology of the de Rham complex, that is the vector spaces of closed forms modulo exact forms, are called the de Rham cohomology groups H
	www.yourencyclopedia.net /de_rham_cohomology.html (336 words)

	Non-Abelian Cohomology and Supermanifolds - Onishchik (ResearchIndex)
		Abstract: this paper, we apply the machinery of non-abelian cohomology to the problem of classification of complex analytic supermanifolds with a given retract.
		Namely, we associate to any holomorphic vector bundle E over a complex manifold M a non-abelian cochain complex whose 1-cohomology describes the set of all complex analytic supermanifolds such that the corresponding split supermanifold is (M; (Update)
		2 Homogeneous supermanifolds associated with the complex proje..
	citeseer.ist.psu.edu /onishchik98nonabelian.html (391 words)

	CCSD thèses-EN-ligne: Homologies d'algèbres Artin-Schelter régulières cubiques
		La propriété de Koszul généralisée nous permet d'écrire un quasi-isomorphisme explicite entre le complexe qui calcule la cohomologie de Hochschild de $A$ et le complexe qui calcule l'homologie de Hochschild de $A$, obtenant ainsi une dualité de Poincaré.
		The de Rham cohomology, cyclic and periodic cyclic homologies are deduced from the Hochschild homology using standard results.
		The Koszul property allows us to give an explicit quasi-isomorphism between the Hochschild cochain complex and the Hochschild chain complex.
	tel.ccsd.cnrs.fr /documents/archives0/00/00/77/63/index_fr.html (563 words)

	Combinatorial operad actions on cochains (Site not responding. Last check: 2007-10-20)
		The purpose of this article is to prove that the associative algebra structure on the normalized cochain complex of a simplicial set extends to the structure of an algebra over the Barratt-Eccles operad.
		We also prove that differential graded algebras over the Barratt-Eccles operad form a closed model category.
		More precisely, the Hochschild cochain complex is acted on by a suboperad of the Barratt-Eccles operad which is equivalent to the classical little squares operad.
	www-gat.univ-lille1.fr /~fresse/CochainModel.html (133 words)

	Derived Categories for Dummies, Part III \| The String Coffee Table
		is a cochain map between such cochains modulo quasi-isomorphisms.
		A chain complex of such functors is a commuting rectangle in
		A morphism of such chain complexes is a 3-dimensional commuting diagram in
	golem.ph.utexas.edu /string/archives/000536.html (949 words)

	Chain complex (Site not responding. Last check: 2007-10-20)
		In mathematics, in the field of homological algebra, a chain complex
		A bounded complex is one in which almost all the A
		An example is the complex defining the homology theory of a (finite) simplicial complex.
	www.omniknow.com /common/wiki.php?in=en&term=Cochain_complex (842 words)

	Derived Categories for Dummies, Part II \| The String Coffee Table
		, we can find its right Cartan-Eilenberg resolution, which is a (cochain) complex of cochain complexes, i.e.
		a double complex, with injective objects everywhere, being a resolution of the complex
		th hypercohomology of this (the cohomology of the total complex) is the
	golem.ph.utexas.edu /string/archives/000535.html (493 words)

	publications AY
		On Residue Complexes, Dualizing Sheaves and Local Cohomology Modules (with P. Sastry)
		Residue Complexes over Noncommutative Rings (with J.J. Zhang)
		Rigid Dualizing Complexes on Schemes (with J.J. Zhang)
	www.cs.bgu.ac.il /%7Eamyekut/publications/publications.html (194 words)

	publications AY
		The Residue Complex of a Noncommutative Graded Algebra
		Dualizing complexes, Morita equivalence and the derived Picard group of a ring
		The Continuous Hochschild Cochain Complex of a Scheme (Survey)
	www.math.bgu.ac.il /~amyekut/publications/publications.html (194 words)

	AMCA: Some geometric aspects of quandle (co)homology and cocycle invariants of knotted curves and surfaces by Masahico ... (Site not responding. Last check: 2007-10-20)
		A cohomology theory of quandles is defined and applications to knot theory will be given.
		Cocycle conditions are derived from Reidemeister moves, and generalized to a cochain complex.
		Quandle 2-cocycles are assigned to crossings of classical knot diagrams that are colored by elements of a finite quandle, and 3-cocycles are assigned to triple points of colored diagrams of knotted surfaces.
	at.yorku.ca /c/a/e/a/11.htm (222 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		/pub/Turner/opsseq Operations and Spectral Sequences I James M. Turner Department of Mathematics University of Virginia Charlottesville, VA 22903 Abstract: This is the first in a series of papers which examines a general type a chain complex (over F_2) whose homology supports a well-defined action of operations.
		We call such complexes Dold algebras, which include the singular cochain complex of a space and the singular chain complex of an infinite loop space, and we give conditions on filtrations of such objects so that there is a compatible action of operations on the associated spectral sequences.
		For applications, we recover W. Singer's result of the action of Steenrod operations on the Serre spectral sequence and we extend A. Bahri's action of Dyer-Lashof operations on the Eilenberg-Moore spectral sequence.
	claude.math.wesleyan.edu /~mhovey/archive/letter38 (567 words)

	CJM - Vol. 54, n6
		Effective Actions of the Unitary Group on Complex Manifolds
		Continued Fractions Associated with SL) and Units in Complex Cubic Fields
		The Continuous Hochschild Cochain Complex of a Scheme
	www.journals.cms.math.ca /CJM/v54n6/index.en.html (81 words)

	Math arXiv: Search results (Site not responding. Last check: 2007-10-20)
		math.AG/0111094 The Continuous Hochschild Cochain Complex of a Scheme.
		math.AG/0005127 Decomposition of the Hochschild Complex of a Scheme in Arbitrary Characteristic.
		math.RA/9810134 Dualizing Complexes, Morita Equivalence and the Derived Picard Group of a Ring.
	front.math.ucdavis.edu /author/Yekutieli-A* (183 words)

	[No title]
		James E. McClure and Jeffrey H. Smith mcclure@math.purdue.edu jhs@math.purdue.edu ABSTRACT: Deligne asked in 1993 whether the Hochschild cochain complex of an associative ring has a natural action by the singular chains of the little 2-cubes operad.
		In this paper we give an affirmative answer to this question.
		Generalizing the rational case proved by Sullivan, Anick proved that if $X$ is a finite $r$-connected CW-complex of dimension $\leq rp$ then the algebra of singular cochains $C^{*}(X;\mathbb{F}_p)$ can be replaced by a commutative differential graded algebra $A(X)$ with the same cohomology.
	claude.math.wesleyan.edu /~mhovey/archive/letter97 (1044 words)

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