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Topic: Derived categories

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	[No title]
		DERIVED CATEGORIES AND PROJECTIVE CLASSES J. Abstract.An important example of a model category is the category of unb* ounded chain com- plexes of R-modules, which has as its homotopy category* the derived cate* gory of the ring R. This example shows that traditional homological algebra is encompassed by Qui *llen's homotopical alge- bra.
		An important corollary of the fact that a derived category D is the homotopy* * category of a model category is that D(X, Y) is a set (as opposed to a proper class) for any two c* *hain complexes X and Y.
		A cofibrantly generated model category is a model category M fo* r which there exist sets I and J of morphisms with domains which are small relative to I-cofand J-c of, respectively, such that I-cofis the category* of cofibrations and J-cofis the category of acyc* *lic cofibrations.
	www.math.purdue.edu /research/atopology/Christensen/derived.txt (6614 words)

	PlanetMath: derived functor
		This, along with a couple of other properties determine the derived functors completely, giving an axiomatic definition, though the construction used above is usually necessary to show existence.
		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)

	PlanetMath: derived category
		Derived categories seem somewhat obscure, but in fact, many mathematicians believe they are the appropriate place to do homological algebra.
		Cross-references: derived functors, global sections, fixed, right, functors, algebra, place, localization, inverse, complexes, homology groups, isomorphism, maps, classes, chain homotopy, morphisms, chain complexes, category, abelian category
		This is version 5 of derived category, born on 2003-02-10, modified 2005-02-16.
	planetmath.org /encyclopedia/DerivedCategory.html (182 words)

	Derived Categories: Arbeitsgemeinschaft, 2005-2006 (Site not responding. Last check: 2007-09-06)
		Derived categories originated as attempts by Verdier and Grothendieck to create a firmer foundation for homological algebra, extending Grothendieck's famous Tohoku paper.
		In particular, the derived category was needed for generalizing duality theorems to relative settings.
		The first purely algebro-geometric (and not trivial) derived equivalence was discovered by Mukai: an abelian variety and its dual have equivalent derived categories of coherent sheaves.
	www.math.ucsb.edu /~mckernan/Seminar/derived.html (507 words)

	[No title] (Site not responding. Last check: 2007-09-06)
		The machinery needed to define a derived category in full generality tends to obscure the simplicity of the phenomena.
		Thus the category of projective objects or the category of filtered objects of an abelian category are no longer abelian in general.
		Derived functors are constructed using a `generalized calculus of fractions'.
	www1.elsevier.com /homepage/saj/523281/h19.htm (346 words)

	Derived Categories for Dummies, Part IV \| The String Coffee Table
		For instance this derived categroy of coherent sheaves is equivalent to what is called a triangulated Fukaya categroy and also to (at least for a large number of cases) the derived category of representations of some quiver (which I mentioned already in part III).
		For instance the relation between triangulated Fukaya categories and derived categories of coherent sheaves is related to mirror symmetry.
		The derivation of this could be summarized by a couple of sub-steps, which are however not strictly necessary in order to move on to step 4.
	golem.ph.utexas.edu /string/archives/000538.html (1550 words)

	Derived category - Wikipedia, the free encyclopedia
		In mathematics, the derived category D(C) of a category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C (which therefore should already be an abelian category).
		The development of the derived category, by Alexander Grothendieck and his student Jean-Louis Verdier shortly after 1960, now appears as one terminal point in the explosive development of homological algebra in the 1950s, a decade in which it had made remarkable strides and became close to appearing a universal approach in mathematics.
		Despite the level of abstraction, the derived category methodology established itself over the following decades; and perhaps began to impose itself with the formulation of the Riemann-Hilbert correspondence in dimensions greater than 1 in derived terms, around 1980.
	en.wikipedia.org /wiki/Derived_category (538 words)

	ImpulseAdventure.com - IMatch Manage Version Script 2.3
		Extra categories are defined as categories that exist in a derived file that are not attached to the original file, or as categories that exist in the original file that are not attached to the derived versions.
		NOTE: This mode might copy categories from derived versions to originals requiring a second run of the script to ensure that these new categories in the original are then applied to all derived versions.
		Extra properties are defined as properties that have a non-null value in an derived version file that are null in the original file, or as properties that have a non-null value in the original file that are null in the derived versions.
	www.impulseadventure.com /assets/ManageVersions.htm (4190 words)

	The Rising Sea » Mathematics Notes
		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.
		Derived Categories Part I: (DTC) Homotopy categories, derived categories (extending functors, introduction to hearts, bounded derived categories), homotopy resolutions, homotopy direct limits, bousfield subcategories, existence of resolutions.
		Linearised Categories: (LC) Generalise the group ring construction to the linearisation of any small category with respect to a sheaf of rings, the graded version of this construction.
	therisingsea.org /?page_id=3 (1462 words)

	Negative K-theory of Derived Categories, by Marco Schlichting (Site not responding. Last check: 2007-09-06)
		We define negative K-groups for exact categories and for ``derived categories'' in the framework of Frobenius pairs, generalizing definitions of Bass, Karoubi, Carter, Pedersen and Thomason.
		We show that the first negative K-group of an abelian category vanishes, and that, in general, negative K-groups of a noetherian abelian category vanish.
		In the appendix, we replace Waldhausen's cylinder functor by a slightly weaker form of non-functorial factorization which is still sufficient to prove his approximation and fibration theorems.
	www.math.uiuc.edu /K-theory/0636 (156 words)

	Derived functor - Wikipedia, the free encyclopedia
		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).
		Derived functors and the long exact sequences are "natural" in several technical senses.
		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 (1209 words)

	List of Publications: Mikhail Kapranov
		On the derived category of coherent sheaves on Grassmann varieties, Mathematics of USSR - Izvestija, 24 (1985), 183--192.
		The derived category of coherent sheaves on a quadric, Funct.
		On the derived category and K-functor of coherent sheaves on intersections of quadrics, Math.
	www.math.toronto.edu /~kapranov/publ.html (774 words)

	Derived Categories Home Page (Site not responding. Last check: 2007-09-06)
		Originally, derived categories were used in algebraic geometry, but in the recent past their application has spread to many branches of mathematics, including non-commutative algebra, representation theory and algebraic topology.
		The main purpose of this conference on Derived Categories is to present to the uninitiated some of these different approaches, and to give experts from the various fields the opportunity to meet and discuss their work.
		To encourage research students and novices in derived categories to participate, in the mornings there will be several series of introductory lectures by international experts on the basics of derived categories, and on the use of derived categories in various areas of application as above.
	www.ma.hw.ac.uk /icms/meetings/2003/dercat/index.html (247 words)

	The Rising Sea
		Derived Categories of Quasi-coherent Sheaves (DCOQS) Added a section on “invertible complexes” which are complexes in the derived category of sheaves that are units under the derived tensor.
		Derived Categories of Quasi-Coherent Sheaves (DCOQS) On the level of sheaves of modules one defines derived inverse image and tensor using hoflat complexes, and to prove there are enough of these one uses the “extension by zero” construction.
		New notes: Derived Categories of Quasi-coherent Sheaves (DCOQS) In algebraic geometry it is the derived categories of quasi-coherent (or coherent) sheaves that are usually of interest.
	www.therisingsea.org (1949 words)

	AGBC Program: Derived Categories (Site not responding. Last check: 2007-09-06)
		In recent years derived categories of sheaves have come to be regarded as interesting geometric objects in their own right, and even as useful potential substitutes for varieties, spurred in part by developments in string theory (D-branes and homological mirror symmetry), noncommutative geometry and representation theory, and birational geometry.
		Another main topic could be the description of derived categories as categories of modules for algebras.
		Reconstruction of a variety from the derived category and groups of autoequivalences.
	www.math.utexas.edu /~benzvi/AGBC.html (280 words)

	preadditive categories are all Abelian
		Apparently the category of divisable groups (abelian groups in which you can divide by integers) is not an Abelian category: Q --> Q/Z is a monic, but is not a kernel.
		the category of chain complexes with morphisms that are split in each degree however is not abelian (the splitting maps need not commute with the differentials) but is exact (in particular additive).
		I read something like this: the category of vector bundles is equivalent to the category of finitely generated locally free OX-modules, where OX is the sheaf of functions (this looks very similar to Swan's theorem).
	www.physicsforums.com /showthread.php?p=833015 (1570 words)

	DAN CHRISTENSEN - Derived categories, projective classes and phantom maps
		An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R.
		The motivation for the work is the construction of the ``pure derived category'' of a ring R.
		Pure homological algebra has applications to phantom maps in the stable homotopy category and the (usual) derived category of a ring, and these connections will be described.
	www.cms.math.ca /Events/Winter97/w97-abs/node16.f (191 words)

	Compact course for PhD-students
		Simon Goodwin : An introduction to derived categories in representation theory.
		The highlights of the course (time permitting) will be Rickard's Morita theory for derived categories and Happel's Theorem, which concerns tilting in derived categories.
		In the first half of the course we will define the derived category of an abelian category and prove that it is a triangulated category.
	www.imf.au.dk /conferences/FMOA05 (475 words)

	To other homepages of (Site not responding. Last check: 2007-09-06)
		The notion of derived category has originated in homological algebra and algebraic topology in an attempt to formalize the theory of cohomologies.
		By now, derived categories have become an essential tool for mathematicians working in algebra, algebraic geometry, representation theory and algebraic topology.
		The course is intended for graduate students working in various areas of mathematics who need to use or want to learn the machinery of derived categories.
	www.maths.warwick.ac.uk /~rumynin/na/s1.html (230 words)

	Triangulated Categories in the Representation of Finite Dimensional Algebras - Cambridge University Press
		In recent years representation theory has been an area of intense research and the author shows that derived categories of finite-dimensional algebras are a useful tool in studying tilting processes.
		Results on the structure of derived categories of hereditary algebras are used to investigate Dynkin algebras and interated tilted algebras.
		The study of trivial extension algebras and repetitive algebras is then developed using the triangulated structure on the stable category of the algebra's module category.
	www.cambridge.org /catalogue/catalogue.asp?isbn=0521339227 (189 words)

	Derived categories for dummies, Part I \| The String Coffee Table
		An abelian category is a category equipped with the necessary structure so as to make it possible to have sequences of morphisms in that category which form a complex.
		Instead, its left derived functor is. This is usually denoted by an otimes with an L on top of it.
		A category is said to have a Grothendieck topology when its objects and morphisms sort of behave as if they were open sets and inclusions of open sets.
	golem.ph.utexas.edu /string/archives/000528.html (3457 words)

	Algebraic Geometry School 2004
		A large fraction of the material should be accessible to beginning graduate students with the appropriate motivation.
		Plan of Caldararu's lectures: I will present a mild introduction to derived categories, attempting to avoid the technical details of their inner workings.
		Plan of Reid's lectures: Nontechnical introduction to derived categories and comparison with motives, cohomology theories.
	www.mimuw.edu.pl /~jarekw/EAGER/announce04.html (291 words)

	Workshop on Triangulated Categories Leeds 2006 (Site not responding. Last check: 2007-09-06)
		Over the past decades, triangulated categories have made their way into many different parts of mathematics, to the extent that today, they can be viewed as a unifying theory underlying major parts of modern mathematics.
		The aim of this workshop is to bring together researchers from many parts of mathematics who all use triangulated methods.
		The event will promote cross fertilization leading to new applications of triangulated categories.
	www.maths.leeds.ac.uk /Pure/algebra/TriCat06.html (146 words)

	AMCA: Auto-equivalences of derived categories acting on group cohomology by Alexander Zimmermann
		AMCA: Auto-equivalences of derived categories acting on group cohomology by Alexander Zimmermann
		The derived category of a group ring attained quite some interest in recent years.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/f/e/02.htm (428 words)

	Math arXiv: Search results (Site not responding. Last check: 2007-09-06)
		math.AG/0605095 DG deformation theory of objects in homotopy and derived categories I. Valery A. Lunts, Dmitri Orlov.
		math.AG/0512620 Derived categories of coherent sheaves and motives.
		math.AG/0503632 Derived categories of coherent sheaves and triangulated categories of singularities.
	front.math.ucdavis.edu /author/Orlov-D* (214 words)

	Table of contents for Library of Congress control number 2002190826 (Site not responding. Last check: 2007-09-06)
		57 II.1 The Language of Categories and Functors.................
		57 II.2 Categories and Structures, Equivalence of Categories........
		153 III.4 Derived Category as the Localization of Homotopic Category 159 III.5 The Structure of the Derived Category....................
	www.loc.gov /catdir/toc/fy036/2002190826.html (118 words)

	Dmitri Orlov at MSRI - Equivalences of derived categories of coherent sheaves (Site not responding. Last check: 2007-09-06)
		Dmitri Orlov at MSRI - Equivalences of derived categories of coherent sheaves
		Dmitri Orlov - Equivalences of derived categories of coherent sheaves
		A PDF version of the lecture notes is available here.
	www.msri.org /publications/ln/msri/2002/hodgetheory/orlov/1/index.html (32 words)

	Derived Categories (Site not responding. Last check: 2007-09-06)
		This program is an intensive two-week class on derived categories.
		It is open to graduate students and postdocs in both mathematics and physics.
		Support comes from Utah's NSF VIGRE grant and the Mathematical Sciences Research Institute, and the National Science Foundation.
	www.math.utah.edu /dc (41 words)

	CiteULike: Tag derived (Site not responding. Last check: 2007-09-06)
		posted to derived geometry moduli quivers stability by lieven as
		Improving self-interpretation of XML-based business documents by introducing derived elements
		posted to applications b2b business data derived documents xml-schema by jipibanf as
	www.citeulike.org /tag/derived (40 words)

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