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Topic: Tor functors

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	Derived functor - Wikipedia, the free encyclopedia
		The functor which assigns to each such sheaf L the group L(X) of global sections is left exact, and the right derived functors are the sheaf cohomology functors, usually written as H
		This is a left-exact functor, and its right derived functors are the group cohomology functors, typically written as H
		becomes a functor from the functor category of all left exact functors from A to B to the full functor category of all functors from A to B.
	en.wikipedia.org /wiki/Derived_functor (1185 words)

	Tor functor - Wikipedia, the free encyclopedia
		Then T is a right exact functor from Mod-R to the category of abelian groups Ab (in case R is commutative, it is a right exact functor from Mod-R to Mod-R) and its left derived functors L
		In case R is commutative, we have additive functors from Mod-R × Mod-R to Mod-R.
		The Tor functors commute with arbitrary direct sums: there is a natural isomorphism
	en.wikipedia.org /wiki/Tor_functor (347 words)

	PlanetMath: derived functor (Site not responding. Last check: 2007-10-31)
		A completely analogous construction can be carried out for right-exact functors and for contravariant functors exact on either side, but it is traditional to only describe one case, as doing the others mostly consists of reversing arrows (and replacing “injective” with projective when appropriate), and the result is that of a left derived functor
		Étale cohomology arises as the right derived functors of the global sections functor on the category of étale sheaves; this example includes as special cases the previous two.
		This is version 17 of derived functor, born on 2003-02-10, modified 2006-05-15.
	planetmath.org /encyclopedia/DerivedFunctor.html (382 words)

	Derived functor - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-31)
		Suppose we are given a covariant left exact functor F : A → B between two abelian categories A and B.
		The right derived functors of the covariant left-exact functor F : A → B is then defined as follows.
		Second, suppose η : F → G is a natural transformation from the left exact functor F to the left exact functor G.
	www.encyclopedia-online.info /Derived_functor (1234 words)

	Homology (mathematics) - Wikipedia, the free encyclopedia
		In abstract algebra, one uses homology to define derived functors, for example the Tor functors.
		can be viewed as a covariant functor from the category of chain complexes to the category of abelian groups (or modules).
		are covariant functors from the category that X belongs to into the category of abelian groups (or modules).
	en.wikipedia.org /wiki/Homology_%28mathematics%29 (728 words)

	[No title]
		A functor F is said to be homotopy exact if whenever X is a co-Cartesian square, the diagram F X is co-Cartesian as well.
		If F is an n- exact functor and Sk is a sequence as given above, then there is a spectral seq* *uence with the following properties: (1) The spectral sequence is a second quadrant sequence of homological type (2) The E1 term is completely contained in the band -n p -k.
		The GAHSS for 2-exact Functors In this section, we review Baues' work with quadratic Z-modules and the qua- dratic tensor product, and use this to examine the E2 term of the GAHSS in the case where F is 2-exact and X is a Moore space M(A, n) with n 2.
	www.math.purdue.edu /research/atopology/Wodarz/ExactHomotopyFunctors.txt (7707 words)

	ozl(1) - The Manual Page Database (Site not responding. Last check: 2007-10-31)
		How- ever, deployment of an application gets harder in the presence of a large number of functors: (1) Installing the application requires cor- rect installation of a large number of functors, (2) Execution might be slow due to frequent file- or even network accesses.
		The commandline tool ozl eases deployment by creating a new functor that includes imported functors in a prelinked fashion: it is possible to collapse a hierarchy of functors into a single equivalent one.
		The model that should be kept in mind, is that the newly created functor employs an internal, private module manager that excutes the toplevel application functor together with all included functors.
	manpage.willempen.org /1/ozl (528 words)

	[No title]
		Tor and Ext in topology and algebra 26 8.
		Homotopy-preserving functors on spectra that do not preserve weak equivalences are transported to the stable category by first replacing their va* *riables by weakly equivalent CW spectra.
		The functor C* satisfies H(C(M)) ~=ss(M) and carri es the functors* ^HR and FHR to the functors R and Hom R. The inverse equivalence satisfies ss((X)) ~=H(X) and carries the functors R and Hom R to the functors ^HR and FHR.
	www.math.purdue.edu /research/atopology/Elmendorf-Kriz-May/modern_foundations.txt (12867 words)

	tor - Hutchinson encyclopedia article about tor (Site not responding. Last check: 2007-10-31)
		However, it is eroded by freeze-thaw weathering and by hydrolysis (a form of chemical erosion).
		The unweathered rock forms a tor, an upstanding, isolated mass of rock, like this one on Dartmoor.
		The barns are granite tors protruding from the upper slopes of Bynack More (1,090 m/3,576 ft), a mountain to the east of Cairn Gorm itself.
	encyclopedia.farlex.com /tor (314 words)

	Derived functor (Site not responding. Last check: 2007-10-31)
		The functor which assigns to each such sheaf L the groupL(X) of global sections is left exact, and the right derived functors are the sheaf cohomology functors,usually written as H
		A G-module M is an abeliangroup M together with a group action of G onM as a group of automorphisms.
		becomes a functor from the functor category of all left exact functors from A to B to the fullfunctor category of all functors from A to B.
	www.therfcc.org /derived-functor-113933.html (1107 words)

	Tor functor peee.org (Site not responding. Last check: 2007-10-31)
		In mathematics, the Tor functors of homological algebra are the derived functor s of the tensor product functor.
		Then T is a right exact functor from Mod-R to the category of abelian groups Ab (in case R is commutative, it is a right exact functor from Mod-R to Mod-R) and its derived functor s L
		In case R is commutative, we have additive functors from Mod-R × Mod-R to Mod-R. - As is true for every family of derived functors, every short exact sequence :
	tor.functor.en.peee.org (432 words)

	Derived functor (Site not responding. Last check: 2007-10-31)
		The functor which assigns to each sheaf L the group L (X) of global sections is left exact the right derived functors are the sheaf functors usually written as H
		A G -module M is an abelian group M together with a group action of G on M as a group of automorphisms.
		This is a left-exact functor and right derived functors are the group cohomology functors typically written as H
	www.freeglossary.com /Derived_functor (1418 words)

	The Rising Sea » Mathematics Notes
		Modules over a Scheme: (MOS) Ideals, special functors (extension by zero and coextension of scalars), locally free sheaves, sheaf Hom, extension of coherent sheaves.
		Derived Functors: (DF) (co)chain complexes in an abelian category, (co)homology, projective and injective resolutions, left and right derived functors of additive functors between abelian categories, long exact (co)homology sequences, long exact sequences of derived functors, dimension shifting and acyclic resolutions, change of base, homology and colimits, cohomology and limits, delta functors.
		Tor: (TOR) Tor on the left and right and balancing the two, dimension shifting, Tor and colimits, Tor for commutative rings and bimodules, criteria for flatness.
	therisingsea.org /?page_id=3 (1462 words)

	Samuel Eilenberg, September 30, 1913—January 30, 1998 \| By Hyman Bass, Henri Cartan, Peter Freyd, Alex Heller, and ... (Site not responding. Last check: 2007-10-31)
		This book used categories to show that they all could be described conceptually as presenting homology functors from the category of pairs of spaces to groups or to rings, satisfying suitable axioms such as "excision".
		All these various examples of the construction of new functors as "derived" functors of given ones were at hand for Eilenberg.
		Categories were defined in order to define functors, which in turn were defined in order to define natural transformations, which were defined finally in order to prove theorems that could not be proved before.
	stills.nap.edu /html/biomems/seilenberg.html (6599 words)

	Homological algebra (Site not responding. Last check: 2007-10-31)
		Central to homological algebra is the notion of exact sequence; these can be used to perform actual calculations.
		A classical tool of homological algebra is that of derived functor; the most basic examples are Ext and Tor.
		The computational sledgehammer par excellence is the spectral sequence; these are essential in the Cartan-Eilenberg and Tohoku approaches where they are needed, for instance, to compute the derived functors of a composition of two functors.
	www.tocatch.info /en/Homological_algebra.htm (279 words)

	PlanetMath: derived functor (Site not responding. Last check: 2007-10-31)
		Now, we define the classical right derived functors
		are the left derived functors of the tensor product.
		Cross-references: étale, category, group cohomology, sheaves, global section, sheaf cohomology, tensor product, Tor, Ext, imply, the following are equivalent, satisfy, necessary, short exact sequences, morphism, exact sequence, sequence, properties, projective resolution, arrows, side, groups, cohomology, right, independent, term, complex, injectives, functor, chain homotopy equivalence, injective resolution, object, enough injectives, abelian categories
	www.planetmath.org /encyclopedia/DerivedFunctor.html (382 words)

	[No title]
		This self-equivalence is not isomorphic to the identity functor unless R i* *s free of rank one.
		This uses that P is projective and so that A(P, -) is an e* xact functor* and both sides of the map (2.6)are right exact in X. Since P is a generator, every object can be written as the cokernel of a morphi* *sm between sums of copies of P.
		JP in A. Since the functor A(P, -) is exact, X is the image of th* *e cokernel of f.
	hopf.math.purdue.edu /Schwede/Morita.txt (4235 words)

	[No title]
		Actually, as applied to categories, "left exact" is a thrice-dead metaphor (twice-dead as applied to functors, since "exact sequence" is a dead metaphor for "exact differential", and "left exact" as applied to functors is a dead metaphor for "preserving the left- hand ends of exact sequences").
		functors to set preserving filtered colimits and connected limits v having finite limits and stable disjoint small coproducts with net image the coprimes.
		An intensive quantity type is a contravariant functor from a category of space which also takes coproducts to products ; but more: intensive quantities usually act linearly on extensive quantities, lending them (not only a linear but also a) multiplicative structure which is also preserved by the contravariant functorality.
	www.mta.ca /~cat-dist/catlist/1999/extensive (7570 words)

	commalg.org - the center for commutative algebra
		The second application is the Vanishing for Maps of Tor for log-terminal singularities: if $A\subset R$ is a Noether Normalization of a finitely generated $\mathbb C$-algebra $R$ and $S$ is a finitely generated $R$-algebra with log-terminal singularities, then the natural morphism $\operatorname{Tor}^A_i(M,R) \to \operatorname{Tor}^A_i(M,S)$ is zero, for every $A$-module $M$ and every $i\geq 1$.
		The final application is the Kawamata-Viehweg Vanishing Theorem for a connected projective variety $X$ of finite type over $\mathbb C$ whose affine cone has a log-terminal vertex (for some choice of polarization).
		Then the derived category $D^b(Sq)$ of $Sq$ has three duality functors which act on $D^b(Sq)$ just like three transpositions of the symmetric group $S_3$ (up to translation).
	www.commalg.org /preprints/2003_03.shtml (2821 words)

	Re: Quillen's Model Categories
		The category of chain complexes in an abelian category is an example of a model category, but Quillen wanted to go beyond this into more "nonabelian" realms, like the category of (nice) topological spaces.
		Lots of people have had mild misgivings about Quillen's precise definition of model category, so there are various mild modifications on the market, but you're talking about something more drastic.
		Thomas Derived categories for the working mathematician Abstract: It is becoming increasingly difficult for geometers and even physicists to avoid papers containing phrases like `triangulated category', not to mention derived functors.
	www.lns.cornell.edu /spr/2002-08/msg0043189.html (530 words)

	Amazon.ca: An Introduction to Homological Algebra: Books: Charles A. Weibel (Site not responding. Last check: 2007-10-31)
		The first half of the book takes as its subject the canonical topics in homological algebra: derived functors, Tor and Ext, projective dimensions and spectral sequences.
		Intermingled are less canonical topics, such as the derived inverse limit functor lim1, local cohomology, Galois cohomology, and affine Lie algebras.
		The last part of the book covers less traditional topics that are a vital part of the modern homological toolkit: simplicial methods, Hochschild and cyclic homology, derived categories and total derived functors.
	www.amazon.ca /exec/obidos/ASIN/0521559871 (287 words)

	Commutative Algebra: Homological and Birational Theory
		Although these topics have been studied extensively, it came as a major surprise recently that vanishing of Ext over complete intersections is symmetric in its arguments, and there are strong indications that similar results may hold over general Gorenstein rings.
		This active area of research, pursued by Avramov, Buchweitz, Huneke, Claudia Miller and their collaborators, is remarkable in that it exposes basic questions about the structure of ubiquitous homological functors.
		Some of these fundamental problems were raised a long time ago by Auslander and Reiten in connection with the so-called Generalized Nakayama Conjecture, which states that the vanishing of certain Ext modules forces a module to be projective.
	www.pims.math.ca /birs/workshops/2004/04w5027 (1330 words)

	Torsion Info - Bored Net - Boredom (Site not responding. Last check: 2007-10-31)
		An abelian group is called torsion-free if all its elements have infinite order.
		In the Tor functors of homological algebra, which arise because tensor product does not in general preserve exact sequences, the symbol Tor does stand for this kind of algebraic torsion, historically speaking anyway.
		These functors were introduced in order to make systematic the universal coefficient theorem of homology theory, in cases where the homology groups H
	www.borednet.com /e/n/encyclopedia/t/to/torsion.html (238 words)

	PlanetMath: Tor (Site not responding. Last check: 2007-10-31)
		This is version 6 of Tor, born on 2004-08-09, modified 2005-03-04.
		Object id is 6090, canonical name is Tor.
		(Associative rings and algebras :: Homological methods :: Homological functors on modules)
	www.planetmath.org /encyclopedia/Tor2.html (89 words)

	Geometry.Net - Pure_And_Applied_Math: Algebraic Topology
		These in turn have necessitated the development of more complex algebraic tools such as derived functors and spectral sequences; the machinery (mostly derived from homological algebra) is powerful if rather daunting.
		In all cases, the "naturality" of the construction implies that a map between spaces induces a map between the groups.
		The emphasis of the conference is on categorical decomposition techniques, especially calculus of functors and homology decompositions of classifying spaces.
	www.geometry.net /detail/pure_and_applied_math/algebraic_topology.html (2828 words)

	Cohomology group (Site not responding. Last check: 2007-10-31)
		By applying the functor F to this sequence, one obtains achain complex; the homology H
		of this complex depends only on F and X and is, bydefinition, the n-th derived functor of F, applied to X.
		are covariant functors from the category thatX belongs to into the category of abelian groups (or modules).
	www.therfcc.org /cohomology-group-220055.html (697 words)

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