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Topic: Ext functor of sheaves

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Ext functors

ext links

Ext functor of sheaves

derived functor

additive functor

representable functor

Tor functors

Ext functors

Enriched functor

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	Derived functor - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-26)
		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)

	PlanetMath: derived functor
		Sheaf cohomology arises as the right derived functors of the global section functor on sheaves.
		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)

	Coherent duality - Wikipedia, the free encyclopedia
		Coherent duality in mathematics refers to a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the 'local' theory.
		Higher direct images are a sheafified form of sheaf cohomology in this case with proper (compact) support; they are bundled up into a single functor by means of the derived category formulation of homological algebra (introduced with this case in mind).
		In order to get a statement in more classical language, but still wider than Serre duality, Hartshorne (Algebraic Geometry) uses the Ext functor of sheaves; this is a kind of stepping stone to the derived category.
	en.wikipedia.org /wiki/Coherent_duality (560 words)

	[No title]
		In particular, my definition of sheaves and quasi-coherent sheaves ov* *er presheaves of groupoids is quite different from the definition I have heard from Hopkins, though the two definitions are presumably equivalent.
		Sheaves over functors The object of this section is to define the notion of a sheaf of modules M ov* *er a sheaf of sets X on Aff.
		Sheaves of groupoids have been much studied in the literature; a stack is a special kind of sheaf of groupoids, and stacks are essential in modern algebraic geometry [FC90 ].
	www.math.purdue.edu /research/atopology/Hovey/hopfalgebroids.txt (11184 words)

	Derived functor
		Suppose we are given a covariant left exact functor F : A → B between two abelian categories A and B.
		The functor which assigns to each such sheave 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
		Second, suppose η : F → G is a natural transformation from the left exact functor F to the left exact functor G.
	www.teachtime.com /en/wikipedia/d/de/derived_functor.html (949 words)

	The Rising Sea » Mathematics Notes (Site not responding. Last check: 2007-10-26)
		Sheaves of graded modules over sheaves of graded rings, quasi-structures, modules over schemes, sheaves of algebras and sheaves of graded algebras (Quite rough in places, I’m in the process of typing written 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.
		Ext: (EXT) Ext in general abelian categories, using injectives and projectives and balancing the two, Ext for linear categories, dimension shifting, Ext and coproducts, Ext for commutative rings, another characterisation of derived functors.
	therisingsea.org /?page_id=3 (1462 words)

	[No title]
		Functors preserve the arrows* , while cofunctors reverse the arrows, i.e., a cofunctor is a functor* on C OP, the cate* *gory opposite to C.
		A functor is an equivalence iff it is full* , faithful, and has a representative_image_, i.e., for any Y 2 Ob D there exists an X 2 Ob *C such that F X is isomorphic to Y.
		In a* ny category, a morphism is an isomorphism iff it is both a monomorphism and an ext *remal epimorphism iff it is both an extremal monomorphism and an epimorphism.
	hopf.math.purdue.edu /WarnerG/warner-book.txt (13171 words)

	[No title]
		Ab are the restrictions of representable functors on T, and all natural transformations are the restrictions of morphisms in T. It has been something of a mystery, to what extent this generalises to other trian- gulated categories.
		Since the functor y commutes with filtered colimits, the image of the sequence is also a filtered colimit of split sequences.
		We suppose therefore that the short exact sequence of functors is given, and that F is the restriction of a representable.
	jdc.math.uwo.ca /papers/purity.txt (7106 words)

	[No title]
		The functor $Loc$ from the previous subsection restricts to a functor $Loc^\lambda: mod^{fg}(U^\lambda) \to mod^c(\DD^\lambda)$.
		It follows from the definitions that if $S$ is a Serre functor for $D$ (relative to some commutative algebra $\O$ such that $D$ is equipped with a strong $\O$-linear structure) then $S^{-1}$ sends the right orthogonal to a full subcategory $C$ into the left orthogonal to the same subcategory.
		An $H$-category is a category $\mathcal C$ with functors %extra data $([g], \tau_{g,h})$ where $[g]: {\mathcal C} \ra {\mathcal C}, \ g\in H$, such that $[e_H]$ is isomorphic to the identity functor, and $[gh]$ to $[g]\circ [h]$ for $g,h\in H$.
	www.math.northwestern.edu /~bezrukav/30 (10229 words)

	[No title]
		D be an additive functor, and denote by D0 the full subcategory of* * D formed by the objects in the image of F.
		E be an exact functor satisfy* ing Ann F Ann G. Then the composite HE OG with the Yoneda functor* factors through F because F is a cohomological quotient functor.
		Moreover, The* orem 4.4 implies that F is a cohomological quotient functor*.
	www.math.purdue.edu /research/atopology/KrauseH/quotient.txt (17394 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.
		These complexes are of course precisely the objects in the derived category of coherent sheaves, as described in part I.
	golem.ph.utexas.edu /string/archives/000538.html (1550 words)

	[No title]
		The functor ffl* is ju* st the forgetful functor* from -comodules to A-modules.
		Howeve* r, there is a symmetric monoidal trivial comodule functor* from the category of abe* *lian groups to -comodules that takes the abelian group M to A Z M with the trivial comodule structure given by jL Z 1.
		This means that the inclusion functor from -comodules to -modules has a right adjoint R. Indeed, if M is a -module, let us denote* * by ~: M -!Hom A (, M) the adjoint to the structure map of M. Then RM = {x 2 M~*(x) = ff(y) for somey}.
	hopf.math.purdue.edu /Hovey/comodule.txt (11965 words)

	[No title]
		We show that $\pi_\ast A$ is a functor of the Andr\'e-Quillen homology of $A$, where $A$ is regarded as an $F_2$ algebra.
		We show that a certain functor S --> A is the universal example of a homology theory with values in an AB 5 category, and compare this with some results of Freyd.
		We show that the localization of $BG$ with respect to a multiplicative complex oriented homology theory $h_*$ is again a space of type $K(\pi,1)$; in fact, it is the same as the localization of $BG$ with respect to the ordinary homology theory determined by the ring $h_0$.
	claude.math.wesleyan.edu /~mhovey/archive/all97 (9156 words)

	Colloquium, Oct 21, 2005, Mathematics and Statistics, Penn State Altoona,
		Abstract: Homological Algebra is essentially the study of derived functors such as Ext and Tor.
		Today the derived functors are studied using the idea of derived category.
		One benefit is that this puts the derived functor Tor into a logical framework and also gives an alternate definition of Ext.
	math.aa.psu.edu /research/seminars/colloquium/2005-06/col051021.htm (180 words)

	(R)CFT on more general 2-Categories \| The String Coffee Table
		Anyway, as I understand your idea, it would mean that one should generalise the modular functor such that the domain category might possibly be thought of as C-decorated somethings, where C is still modular, but the somethings are no longer surfaces.
		In the RCFT case it is possible to define a RCFT quite abstractly, in purely categorical language, such that, essentially, the only reference to the geometrical notion of sewing is in the axioms of a modular functor.
		as I understand your idea, it would mean that one should generalise the modular functor such that the domain category might possibly be thought of as C-decorated somethings, where C is still modular, but the somethings are no longer surfaces.
	golem.ph.utexas.edu /string/archives/000792.html (6368 words)

	[No title]
		The framework we use is the concept of multicategory, a generalization of symmetric monoidal category that precisely captures the multiplicative structure we have present at all stages of the construction.
		Thus, Morava E-theory is "n derived functors away from being a homology theory".
		They are an introduction to the algebraic aspects of the theory of unstable modules over the Steenrod algebra and to the relations of the related category to functor categories.
	www.lehigh.edu /~dmd1/h3104.txt (1977 words)

	[No title]
		However the global sections functor on sheaves is no* t usually right exact, and the sequence of sections continues with the sheaf cohomology groups *H1(G;.).
		The answer is to view the circle of* * equivariance of EG as T=T[n], and then to use the inflation functor studied in Chapters 10 a* *nd 24 of [6] to obtain a T-spectrum.
		0 of sheaves to define the quotient sheaf Q(aD) for 0 a 1.
	hopf.math.purdue.edu /Greenlees-Hopkins-Rosu/ellT.txt (8922 words)

	ScriptedFunctor -- the class of all scripted functors (Site not responding. Last check: 2007-10-26)
		The type ScriptedFunctor is a member of the class Type.
		Each scripted functor is also a member of class MutableHashTable.
		A scripted functor accepts a subscript or a superscript: the primary example is HH.
	www.stanford.edu /~mluciano/M2-help/0031.html (114 words)

	Edgar E. Enochs - Page 2
		Flat and cotorsion quasi-coherent sheaves (with Sergio Estrada, Juan Ramon Garcia Rozas and Luis Oyonarte), to appear in J.Algebra Represent.
		Torsion free covers of a generalization of quasi-coherent sheaves (with Sergio Estrada, Juan Rammon Garcia Rozas and Luis Oyonarte), proceedings of the first Moroccan-Andalusian conference on algebras and their applications, Tetouan, Morocco (2001), 150-160.
		Relative homological algebra in the category of quasi-coherent sheaves (with Sergio Estrada), to appear in Adv.
	www.ms.uky.edu /~enochs/info.html (2476 words)

	[No title]
		We show in this paper that a comodule is the same thing as a quasi-coherent sheaf over this presheaf of groupoids.
		We prove the general theorem that internal equivalences of presheaves of groupoids with respect to a Grothendieck topology on Aff give rise to equivalences of categories of sheaves in that topology.
		We then show using faithfully flat descent that an internal equivalence in the flat topology gives rise to an equivalence of categories of quasi-coherent sheaves.
	claude.math.wesleyan.edu /~mhovey/archive/letter119 (1020 words)

	2001-2002 Graduate Courses (Site not responding. Last check: 2007-10-26)
		Completeness and compactness; elimination of quantifiers; omission of types; elementary chains; homogeneous models; two cardinal theorems by Vaught, Chang, and Keisler; categories and functors; inverse systems of compact Hausdorff spaces; applications of model theory to algebra.
		PQ: Math 25500 or Math 27900, or consent of instructor.
		Selected topics from EGA: functor of points, Zariski tangent spaces of functors, the standard deformation examples, flat descent, principal bundles, Serre vanishing theorems and applications, e.g.
	www.math.uchicago.edu /2001-2002.html (2746 words)

	Math JS Milne Preprints
		Introduces the sheaves \nu (now called the logarithmic de Rham-Witt sheaves), proves the flat duality theorem for surfaces (conjectured by M. Artin), and, for sheaves killed by p, generalizes it to all smooth projective varieties.
		Consequently, one obtains a morphism of gerbes of fibre functors with certain properties.
		Unfortunately, since the results are generally negative or inconclusive, they are of little interest except perhaps for the question they raise on the existence of a cyclic extension of Q having certain properties (see Question 6.5).
	www.jmilne.org /math/Preprints (2485 words)

	[No title]
		There is a global variable DIM, the dimension, which controls the power of t above which sheaves will be truncated.
		For locally free sheaves the answer should have a straightforward interpretation.
		For general coherent sheaves, however, the operations must be understood to be "virtual".
	www.math.sunysb.edu /~sorin/online-docs/schubert/schubertmanual.txt (5882 words)

	18.726 Lecture Plan
		The Proj functor, projective space; the functor from abstract algebraic varieties to schemes; morphisms of schemes; the functor of points; properties of schemes: reduced
		Separated, quasi-separated morphisms; morphisms between affine schemes are separated; the diagonal morphism is always a locally closed immersion; image of a quasi-compact map is closed iff it is stable under specialization; a non-quasi-separated morphism
		Relationship between sheaves on a Proj and graded modules; twisting sheaves; Serre's theorem on coherence of pushforwards (also in the proper case); very ample, relatively globally generated, ample, relatively ample sheaves
	math.mit.edu /~kedlaya/18.726/calendar.html (467 words)

	[No title]
		We give a sufficient condition for an Ext-finite triangulated category to satisfy an analogue of the Brown representability theorem, which claims that every contravariant cohomological functor to the category of vector spaces is representable.
		The condition consists in existence of a strong generator.
		Such a generator exists for the derived categories of coherent sheaves on algebraic schemes but not for analitic surfaces with no curves.
	www.math.ksu.edu /~zlin/alg-sem/alg01s.html (135 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		The restriction functor from a certain category of modules over the big quantum group (known to be equivalent to a certain category of perverse sheaves on affine Grassmannian) to the category of modules over the small quantum group corresponds to the convolution functor.
		An action of the Langlands dual Lie algebra $g^L$ on the cohomology of the convolution of a G[[z]]-equivariant sheaf on the affine Grassmanian is given by the local Ext algebra.
		In case of G=SL(n) the action of $g^L$ can be given in a very geometric way by correspondences.
	www.math.uchicago.edu /seminars/Kuznetsov.txt (91 words)

	Koszul duality for operads, Victor Ginzburg, Mikhail Kapranov
		[5] A. Beilinson, Coherent sheaves on $P^{n}$ and problems in linear algebra, Funct.
		[30] M. Kapranov, On the derived categories and the $K$-functor of coherent sheaves on intersections of quadrics, Math.
		[34] M. Kashiwara and P. Schapira, Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.dmj/1077286744 (1043 words)

	Citebase - Ample filters of invertible sheaves
		Authors: Keeler, Dennis S. Let X be a scheme, proper over a commutative noetherian ring A.
		We introduce the concept of an ample filter of invertible sheaves on X and generalize the most important equivalent criteria for ampleness of an invertible sheaf.
		We also prove the Theorem of the Base for X and generalize Serre's Vanishing Theorem.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0108068 (181 words)

	some bibliography: B
		Beilinson, A.A. and Manin, Yu.I. and Schechtman, V.V. Sheaves of the Virasoro and Neveu-Schwarz algebras
		The equivariant $K$-homology is equipped with a canonical base formed by the classes of simple equivariant perverse coherent sheaves.
		The first step is to observe that the derived functor of global sections provides an equivalence between the derived category of $D$-modules (with no divided powers) on the flag variety and the appropriate derived category of modules over the corresponding Lie algebra.
	www.justpasha.org /math/bib/b.html (8846 words)

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