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Topic: Injective resolution

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	Injective Resolution (Site not responding. Last check: 2007-11-05)
		If ''G'' is a finite group and ''k'' a field with characteristic 0, then one shows in the theory of group representation s that any subrepresentation of a given one is already a direct summand of the given one.
		It is an injective cogenerator in the category of abelian groups, which means that it is injective and any other module is contained in a suitably large product of copies of Q/'''Z'''.
		These injective resolutions are used to define the injective dimension of a module (the length of the shortest injective resolution ending in zeros, if such a finite resolution exists) as well as derived functor s.
	www.seattleluxury.com /encyclopedia/entry/Injective_resolution (928 words)

	Injective module - Wikipedia, the free encyclopedia
		Injective modules were introduced by Reinhold Baer in 1940.
		The injective dimension of a module M is the infimum of lengths of an injective resolution of M.
		In general, submodules, factor modules or infinite direct sums of injective modules need not be injective.
	en.wikipedia.org /wiki/Injective_dimension (1022 words)

	Springer Online Reference Works
		More generally, the notion of a resolution of an object can be defined in any Abelian category in a completely similar way, [a1].
		Resolutions are the main tool in the calculation of derived functors (cf.
		One such is the resolution afforded by the Koszul complex, which is something like an exterior algebra pulled apart.
	eom.springer.de /r/r081570.htm (275 words)

	Opposite Algebras
		Furthermore the dual of a projective OA-module is an injective A-module and the dual of a projective OA-resolution of a module M is an A-injective resolution of the dual of M. Subsections
		Injective hulls, and injective resolutions of a module are computed by taking the projective cover or projective resolution of the dual module over the opposite algebra and then again taking the dual to retrieve modules or complexes over the original algebra.
		The complex giving the minimal injective resolution of the module M together with the inclusion homomorphism from M into its injective hull.
	www.math.lsu.edu /magma/text858.htm (747 words)

	Derived functor (Site not responding. Last check: 2007-11-05)
		The crucial assumption we need to make about our abelian category A is that it have enough injectives, meaning that for every object A in A there exists a monomorphism A → I where I is an injective object in A.
		are all injective (this is known as an injective resolution of X).
		If X is a topological space, then the category of all sheaves of abelian groups on X is an abelian category with enough injectives (a result of Grothendieck).
	bopedia.com /en/wikipedia/d/de/derived_functor.html (937 words)

	Injective sheaf - Wikipedia, the free encyclopedia
		In mathematics, injective sheaves of abelian groups are used to construct the resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext.).
		An injective sheaf F is just a sheaf that is an injective element of the category of abelian sheaves; in other words, homomorphisms from A to F can always be lifted to any sheaf B containing A.
		Flasque resolutions, that is, resolutions by means of flasque sheaves, are one approach to defining sheaf cohomology.
	en.wikipedia.org /wiki/Injective_sheaf (671 words)

	Springer Online Reference Works
		The length of the shortest injective resolution is called the injective dimension of the module (cf.
		Injective modules are extensively used in the description of various classes of rings (cf.
		in its injective hull under endomorphisms of the latter.
	eom.springer.de /i/i051210.htm (450 words)

	[No title]
		INJECTIVE RESOLUTIONS OF UNSTABLE MODULES Errokh Mohamed Jmoui Chariya P. Peterson July 9, 1993 abstract We characterize unstable modules over the Steenrod algebra in terms of their in* jective resolutions*.
		The injective unstable modules are divided i* nto two classes: The reduced injective* modules and the nilpotent injective modules.* * A reduced injective module is a direct sum of direct factors of polynomial algebr* *as.
		Injective modules which are not of this type are nilpotent.
	www.math.purdue.edu /research/atopology/PetersonC-Jmoui/Inj_res_of_unst_mod.txt (4248 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		Some years ago, Friedlander and Suslin constructed an explicit injective resolution of twisted symmetric powers in the category of strict polynomial functors over a ground field of characteristic 2.
		The resolutions we construct are in fact "p-resolutions", that is, the differential does not vanish when composed twice, but only when composed p times.
		This result should unable us to constuct an injective resolution of any twisted functor if we know an injective resolution of the corresponding non twisted functor.
	www.math.purdue.edu /research/atopology/Troesch/troesch_Resolution_of_symmetric_powers.abstract (199 words)

	Resolution -- from Wolfram MathWorld
		Resolution is a widely used word with many different meanings.
		It can refer to resolution of equations, resolution of singularities (in algebraic geometry), resolution of modules or more sophisticated structures, etc. In a block design, a partition
		is known as resolution and is significant for automated theorem proving.
	mathworld.wolfram.com /Resolution.html (172 words)

	Abstracts (Representations of Finite Groups and Related Algebras) (Site not responding. Last check: 2007-11-05)
		We classify the injective modules over the Tate cohomology ring of a finite group, and we show how to construct kG-modules whose Tate cohomology is equal to a given injective.
		This is related to a conjectural way of calculating the variety of an infinite dimensional module by forming an injective resolution of its Tate cohomology.
		The Springer correspondence is an injective map from the set of unipotent classes of an algebraic group to the set of characters of its Weyl group.
	www.maths.bris.ac.uk /~majcr/durham/abstracts.html (1561 words)

	Math Forum Discussions
		Given injective resolutions X of f and Y of h.
		Injective objects in an arrow category of modules over a ring
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.org /kb/message.jspa?messageID=3730612&tstart=0 (300 words)

	resolution (Site not responding. Last check: 2007-11-05)
		The state or quality of being resolute; firm determination.
		The substitution of one metrical unit for another, especially the substitution of two short syllables for one long syllable in quantitative verse.
		The resolution was carried at the previous plenary session.
	dict.yqie.com /english/r/resolution.htm (358 words)

	Edgar E. Enochs - Page 2
		Injective covers and resolutions, (with Overtoun Jenda), Proceedings of the joint China-Japan Ring Theory Conference, Guilin, P. China, (1993), 42 -45.
		Resolutions by Gorenstein injective and projective modules and modules of finite injective dimension over Gorenstein rings, (with Overtoun Jenda), Communications in Algebra, 23, (1995), 869 -877.
		The Gorenstein injective envelope of the residue field of a local ring (with Richard Belshoff), to appear in Comm.
	www.ms.uky.edu /~enochs/info.html (2476 words)

	3 Extensions to [3] (Site not responding. Last check: 2007-11-05)
		A similar definition can be given with the property injective replaced by the property D-acyclic in the context of a left-exact functor C
		(L,G) is a quasi-isomorphism of complexes of injectives.
		C' be a left-exact functor and assume that C has enough injectives.
	www.imsc.res.in /~kapil/papers/spectral/spectralse3.html (545 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		You said what it was, too: you take your group G, you form a free resolution, you hom that resolution into A, then you take the cohomology of the resulting cochain complex to get the cohomology groups H^n(G,A).
		You may recall that G-bundles over X are secretly the same thing as maps from X to a space BG, called the "classifying space" of G. If you don't, well, take a peek at this: http://math.ucr.edu/home/baez/week151.html We can use this trick to classify the possible bundles at each stage of this Postnikov tower game.
		An injective resolution of the module M is an exact sequence 0 -> M -> E1 -> E2 ->...
	www.math.niu.edu /~rusin/known-math/00_incoming/postnikov (1497 words)

	Derived Categories for Dummies, Part II \| The String Coffee Table
		In order to understand hyper-derived functors it is finally necessary to first understand ordinary derived functors (to be distinguished from the ‘total’ derived functors that I started with).
		There are certain such resolutions which enjoy a property called injectivity.
		a double complex, with injective objects everywhere, being a resolution of the complex
	golem.ph.utexas.edu /string/archives/000535.html (493 words)

	Proceedings of the American Mathematical Society
		by use of the minimal injective resolution of
		--, Minimal pure injective resolutions of flat modules, J. of Algebra 105 (2) (1987), 351-364.
		Jinzhong Xu, Minimal injective and flat resolutions of modules over Gorenstein rings, J. of Algebra 175 (1995), 451-477.
	www.mathaware.org /proc/1997-125-04/S0002-9939-97-03662-9/home.html (296 words)

	Algebra- och geometriseminarier 02/03 (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-11-05)
		The Tate resolution of M gives information about the cohomology of this complex of sheaves.
		Especially, for a monomial ideal in the exterior algebra, which corresponds to a simplicial complex, we determine when the associated complex of sheaves reduces to a single coherent sheaf which is locally Cohen-Macaulay.
		I will in the talk describe free resolutions of $\mathbb{Z}$ over $\mathbb{Z}A$ for an Artin group $A$, (and indeed more generally for the case when $A$ is `locally Gaussian'), obtained by Squier, Salvetti, Dehornoy and others.
	www.math.su.se.cob-web.org:8888 /matematik/forskning/seminar/Alggeomsem2002-2003.html (3812 words)

	Injective Resolutions of Some Regular Rings - Ajitabh, Smith, Zhang (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		If your firewall is blocking outgoing connections to port 3125, you can use these links to download local copies.
		Consider a minimal injective resolution of A as an A-module 0 \Gamma!
		Ajitabh, S. Smith and J. Zhang, Injective resolutions of some regular rings, submitted.
	citeseer.ist.psu.edu /472912.html (425 words)

	Math 737, Fall 2005 (Site not responding. Last check: 2007-11-05)
		I will start by reviewing the basics of categories and functors, in particular I will focus on the important notion of representable functor.
		In this first part I will also introduce the notion of limit, additive and abelian category, the definition of a complex in an abelian category, and that of resolution (in particular injective resolution) and derived functor.
		The second part of the course will focus on the notion of sheaf on a topological space: its stalks, sections and support.
	www.math.jhu.edu /~kc/mat737/index.html (636 words)

	► » RHom (Site not responding. Last check: 2007-11-05)
		(3) the injective hull in Mod A of any object of T is in T. Using Hartshorne, Alg.
		This then is an injective module and the map M to I is injective.
		which is injective in Mod(A) as well and the previous argument works.
	www.science-chat.org /RHom-3620128.html (1072 words)

	[No title]
		Our method utilizes: computation of the Ext-groups between twisted divided and symmetric powers due to Franjou-Friedlander-Scorichenko-Suslin, resolutions of functors by divided and symmetric powers, interplay between functors and representations of the symmetric group.
		The simplicial model for the Hopf map is then the projection K x_{eta} S^2 -> S^2.
		http://hopf.math.purdue.edu/cgi-bin/generate?/Troesch/troesch_Resolution_of_symmetric_powers Title: A propos d'une question de Friedlander et Suslin I -- Une r'esolution injective des puissances sym'etriques twist'ees (in French) Author: Alain Troesch Address of Author: Institut de Mathematiques de Jussieu, Case 82 4 place Jussieu, F-75252 PARIS CEDEX 05 e-mail address: troesch@math.jussieu.fr Abstract.
	www.lehigh.edu /~dmd1/h1125.txt (1717 words)

	commalg.org - the center for commutative algebra (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-11-05)
		Singh: p-torsion elements in local cohomology modules; Associated primes of local cohomology modules, June 22, 2004
		Chan-Huang: Module structure of an injective resolution, June 22, 2004
		Holm-Jorgensen: Cohen-Macaulay injective, projective, and flat dimension; Semi-dualizing modules and related Gorenstein homological dimensions, May 28, 2004
	www.commalg.org.cob-web.org:8888 /preprints (581 words)

	commalg.org - the center for commutative algebra (Site not responding. Last check: 2007-11-05)
		Holm-Jorgensen: Cohen-Macaulay injective, projective, and flat dimension; Semi-dualizing modules and related Gorenstein homological dimensions
		Helm-Miller: Algorithms for graded injective resolutions and local cohomology over semigroup rings
		Simon-Strooker: Stiffness of finite free resolutions and the Canonical Element Conjecture
	www.commalg.org /preprints/archives.shtml (2774 words)

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