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Topic: Ext functors

	Ext functor - Wikipedia, the free encyclopedia
		In mathematics, the Ext functors of homological algebra are derived functors of Hom functors.
		Ext functors take their name from their relationship to extensions.
		Given R-modules A and B, there is a bijective correspondence between equivalence classes of extensions of A by B and elements of
	en.wikipedia.org /wiki/Ext_functor (142 words)

	NationMaster - Encyclopedia: Derived functor
		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 Derived functors and the long exact sequences are "natural" in several technical senses.
		Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps.
		Functors are often defined by universal properties; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits.
	www.nationmaster.com /encyclopedia/Derived-functor (1358 words)

	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 (1215 words)

	PlanetMath: derived functor
		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.
	www.planetmath.org /encyclopedia/DerivedFunctor.html (382 words)

	NationMaster - Encyclopedia: Ext functor
		In mathematics, the Ext functors of homological algebra are derived functors of
		Note that a completely analogous construction can be done 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).
		This is version 13 of derived functor, born on 2003-02-10, modified 2004-12-31.
	www.nationmaster.com /encyclopedia/Ext-functor (214 words)

	PlanetMath: Ext
		functors was given by S. Mac Lane, namely that there is a correspondence between
		This is version 6 of Ext, born on 2003-08-14, modified 2007-01-11.
		Object id is 4588, canonical name is Ext.
	planetmath.org /encyclopedia/Ext.html (97 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		This notion exploits a strong interplay between representations of the general linear group and the symmetric group coming from the action of these groups on the tensor power of a space, and may be thought of as a generalization of a classical notion of symmetrization of a representation.
		This functor known (unfortunately) also as the Schur functor is an important tool used to com- pare representations of the general group and symmetric group (see eg.
		Namely, our formula says that the functor F and the d-module S(F) are obtained by applying the same d-functor to the right d-structure of: V d in the first case, and k[ d] in the second.
	www.math.purdue.edu /research/atopology/Chalupnik/extpol.txt (7587 words)

	Derived functor - the free encyclopedia (Site not responding. Last check: 2007-10-20)
		functors may be derived to obtain other functors closely related to the original ones.
		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
		natural transformation from the left exact functor F to the left exact functor G.
	www.the-free-web-encyclopedia.com /default.asp?t=Derived_functor (1098 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		A diagram in S is a functor F from a small category C to S. A cone from a diagram F to an object X is a natural transformation from F to the constant diagram at X.
		Is it true that P l Ext k (M; N) consists of those extensions which pullback to zero under any map from a module of pure projective dimension at most l 1 to M? The relation between phantom maps and pure extensions.
		Ext (M; N) is an isomorphism, and the projective and pure projective dimensions of M agree.
	jdc.math.uwo.ca /papers/ideals.txt (14824 words)

	De
		'''Ext functors.''' If ''R'' is a ring (mathematics)ring, then the category of all left module (mathematics)''R''-modules is an abelian category with enough injectives.
		Naturality Derived functors and the long exact sequences are "natural" in several technical senses.
		Furthermore, this functor is compatible with the long exact sequences in the following sense: if :image:short_exact_sequence_ABC.png is a short exact sequence, then a commutative diagram image:two_long_exact_sequences2.png is induced.
	www.gateserver.net /Topicdetails.aspx?Topicid=39836&name=&catid=111&topicname=Derived_functor (1299 words)

	Group cohomology - One Language (Site not responding. Last check: 2007-10-20)
		A general paradigm in group theory is that a group G should be studied via its group representations.
		We may therefore form its derived functors; their values are abelian groups and they are denoted by H
		These Ext groups can also be computed via a projective resolution of Z, the advantage being that such a resolution only depends on G and not on M.
	www.onelang.com /encyclopedia/index.php/Group_cohomology (739 words)

	libstdc++-v3 Source: functional Source File
		The @c unary_compose 00099 * functor is constructed from two functions/functors, @c f and @c g.
		When a 00180 * functor is required but no operations are desired, this can be used as a 00181 * pass-through.
		They can be used (especially with 00191 * the composition functors) to "strip" data from a sequence before 00192 * performing the remainder of an algorithm.
	cs.brown.edu /~jwicks/libstdc++/html_user/ext_2functional-source.html (1176 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		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)

	13.1. Subfunctors
		These differences within one functor are expressed by various prepositional phrases, by using different cases or conjuctions.
		Subfunctors are not assigned to all PDT functors that would need a more detailed specification of their meaning but only to a limited group of functors.
		Subfunctors were assigned on the basis of the surface form of the modification with the particular functor.
	ufal.mff.cuni.cz /pdt2.0/doc/manuals/en/t-layer/html/ch07s13s01.html (4352 words)

	Cohomology operation - Biocrawler (Site not responding. Last check: 2007-10-20)
		In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if F is a functor defining a cohomology theory, then a cohomology operation should be a natural transformation from F to itself.
		The general theory of the Steenrod algebra of operations has been brought into close relation with that of the symmetric group.
		In the Adams spectral sequence the bicommutant aspect is implicit in the use of Ext functors, the derived functors of Hom-functors; if there is a bicommutant aspect, taken over the Steenrod algebra acting, it is only at a derived level.
	www.biocrawler.com /encyclopedia/Cohomology_operation (257 words)

	Derived functor (Site not responding. Last check: 2007-10-20)
		In mathematics, certain functor s may be derived to obtain other functors closely related to the original ones.
		A G -module M is an abelian group M together with a group action of G on M as a group of automorphisms.
		First, given a commutative diagram of the form : (where the rows are exact), the two resulting long exact sequences are related by commuting squares: Second, suppose η : F → G is a natural transformation from the left exact functor F to the left exact functor G.
	www.randomnugget.com /resource-Derived_functor.html (1231 words)

	Citations: New York: Springer Verlag - MacLane, Berlin (ResearchIndex) (Site not responding. Last check: 2007-10-20)
		, 42] For p 1, the elements in Ext G (A; B) are represented by p extensions of A by B i.e.
		, 46] For p 1, the elements in Ext p G (A; B) are represented by p extensions of A by B i.e.
		, 42] For p 1, the elements in Ext p G (A; B) are represented by p extensions of A by B i.e.
	citeseer.ist.psu.edu /context/22491/0 (917 words)

	PolyExt -- Ext groups between a holonomic module and a polynomial ring
		PolyExt -- Ext groups between a holonomic module and a polynomial ring
		groups between a D-module M and the polynomial ring are the derived functors of
		Does not yet compute explicit representations of Ext groups such as Yoneda representation.
	www.ima.umn.edu /~leykin/Dmodules/doc/___Poly__Ext.html (188 words)

	NSDL Metadata Record -- Ext
		For a ring R, and R-module A, we have a covariant functor...
		Ext gets its name from the existence of a natural bijection between elements of...
		A more modern interpretation of the Ext functors was given by S. Mac Lane, namely that there is a correspondence between...
	nsdl.org /mr/1034318 (88 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		The E_2-term involves the derived functors of direct sum in the category of "L-complete" E_*-modules.
		There are (n-1) such derived functors (n if we try to compute filtered homotopy colimits).
		Thus, Morava E-theory is "n derived functors away from being a homology theory".
	hopf.math.purdue.edu /Hovey/morava-E-SS.abstract (220 words)

	2003 Chevrolet Trailblazer Ext (Site not responding. Last check: 2007-10-20)
		EXT is a mod for the PC game Doom 3.
		According to the http://fragu.d3files.com/ext/index.html EXT website: EXT is a sci-fi shooter, placing you in a war against vicious aliens from another planet.
		To retake the resource rich planet, and keep the human race alive, the UN sent out a fleet to wage war with the hostile aliens, and take over the planet.
	www.awonky.com /pages2/0/2003-chevrolet-trailblazer-ext.html (458 words)

	Homological algebra
		A classical tool of homological algebra is that of derived functor; the most basic examples are Ext functorsExt and Tor functorsTor.
		With a diverse set of applications in mind, it was natural to try to put the whole subject on a uniform basis.
		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.territoriopc.com /eng/homological_algebra.php (292 words)

	[No title]
		\medskip Another means of studying resolutions is through the derived functors, Ext and Tor, which are again defined sequentially, and the sequences are often infinite.
		Generally, one is interested in whether or not the derived functors Ext and Tor are zero, a condition which we refer to as vanishing.
		Hence this symmetry in vanishing of Ext result recovers part of a theorem of Avramov and Buchweitz for complete intersections, this time without using the machinery of support varieties.
	dreadnought.uta.edu /~dave/research.dir/research.html (1786 words)

	Derived functor - The Jiggies Reference Guide (Site not responding. Last check: 2007-10-20)
		Suppose we are given a covariant left exact functor F : A → B between two abelian categories A and B.
		If A is a fixed left R-module, then the functor Hom(A,-) is left exact, and its right derived functors are the Ext functor Ext
		Second, suppose η : F → G is a natural transformation from the left exact functor F to the left exact functor G.
	www.jiggies.com /reference/Derived_functor (1301 words)

	ScriptedFunctor -- the class of all scripted functors (Site not responding. Last check: 2007-10-20)
		A scripted functor accepts a subscript or a superscript: the primary example is HH.
		Each object of class ScriptedFunctor is called a scripted functor.
		Each scripted functor is also a member of class MutableHashTable.
	www.msri.org /about/computing/docs/macaulay/2-0.9/0222.html (78 words)

	Ext (Site not responding. Last check: 2007-10-20)
		The 2006 Chevrolet TrailBlazer EXT is a carryover from 2005...
		The Cadillac Escalade EXT, a $53,000 chrome-trimmed luxury pickup with leather bucket seats, a seven-speaker stereo system with satellite radio and a global tracking system, is the vehicle most targeted by thieves, a study by the insurance...
		It's the second year in a row that an Escalade EXT has been at the top of the most-stolen list, which was...
	picofthe2007cadillacescalade.ionsescalade.com /ext (853 words)

	Amazon.com: "left derived functors": Key Phrase page (Site not responding. Last check: 2007-10-20)
		6.2 Derived Functors Left derived functors are manufactured in the same wav as Ext and Tor.
		a fixed module B, the functor X H Hom (X, B) is right exact, and it gives rise to its left derived functors.
		Also, the right derived functors of a left exact contravariant functor (use projective resolu-...
	www.amazon.com /phrase/left-derived-functors (520 words)

	Properties Group (Site not responding. Last check: 2007-10-20)
		TFuncObjProperty - Read/write property that uses functors to get/set class instance.
		TFuncObjPropertyVarPut - Read/write property that uses functors to set class instance, and reference to variable to get it..
		TFuncObjPropertyIdx - Read/write property that uses functors with index to get/set class type.
	owlnext.sourceforge.net /useowl_cls15.html (476 words)

	LMS Proceedings Abstract, paper PLMS 1352 (Site not responding. Last check: 2007-10-20)
		If $M$ has finite Gorenstein dimension $\mbox{G-dim}_RM$ in the sense of Auslander and Bridger, then it determines two other cohomology theories besides the one given by the absolute cohomology functors ${\rm Ext}^n_R(M,\)$.
		Relative cohomology functors ${\rm Ext}^n_{\mathcal G}(M,\)$ are defined for all non-negative integers $n$; they treat the modules of Gorenstein dimension $0$ as projectives and vanish for $n > \mbox{G-dim}_RM$.
		Tate cohomology functors $\widehat{\rm Ext}^n_R(M,\)$ are defined for all integers $n$; all groups $\widehat{\rm Ext}^n_R(M,N)$ vanish if $M$ or $N$ has finite projective dimension.
	www.lms.ac.uk /publications/proceedings/abstracts/p1352a.html (205 words)

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