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Topic: Coherent sheaf

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	Coherent sheaf - Wikipedia, the free encyclopedia
		In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a sheaf isomorphic with the cokernel of a morphism of O
		The role played by coherent sheaves is as a class of sheaves, say on an algebraic variety or complex manifold, that is more general than the locally free sheaf — such as invertible sheaf, or sheaf of sections of a (holomorphic) vector bundle — but still with manageable properties.
		In the basic work of Serre, it was shown first that compact complex manifolds have the property that their sheaf cohomology for any coherent sheaf consists of vector spaces of finite dimension.
	en.wikipedia.org /wiki/Coherent_sheaf (700 words)

	Coherence - Wikipedia, the free encyclopedia
		Coherence is from Latin cohaerere = stick together, to be connected with, logically consistent.
		Coherence is an attribute of physical quantities that can be described in terms of waves when a well-defined wavefront can be defined, as in classical optics.
		Cache coherence and (more generally) memory coherence are concepts in computer architecture.
	en.wikipedia.org /wiki/Coherent (165 words)

	PlanetMath: line bundle (Site not responding. Last check: 2007-10-08)
		In algebraic geometry, the term line bundle refers to a locally free coherent sheaf of rank 1, also called an invertible sheaf.
		sheaf of holomorphic sections is locally free and of rank 1.
		Cross-references: stalk, map, obvious, dimension, continuous functions, topology, sections, holomorphic, sheaf, variety, algebraic, non-singular, equivalent, vector bundle, complex, real, theory, manifold, invertible sheaf, rank, coherent sheaf, locally free, term, algebraic geometry
	planetmath.org /encyclopedia/LineBundle.html (136 words)

	Caroline Melles (Site not responding. Last check: 2007-10-08)
		A sheaf on an analytic variety is a structure which encodes local information over the entire variety.
		Coherent sheaves are locally finitely generated and are particularly useful.
		A coherent ideal sheaf on M determines a subvariety of M by encoding the local defining functions for that subvariety.
	web.usna.navy.mil /~wdj/talk99_29.htm (162 words)

	Seminar on Cohomology of Quasi-Coherent Sheaves
		The vanishing of Cech cohomology of a quasi-coherent sheaf on an affine scheme (lemma 2.17 and thm.
		Corollary: the vanishing of cohomology of a quasi-coherent sheaf on an affine scheme (using the result about the vanishing of Cech cohomology from the previous lecture, and a result in Godement's book, which we will use as a fact; see [EGA] III.1.3.1).
		The construction of the sheaf of differentials on a scheme and some of its properties, most importantly the one having to do with smooth varieties (see chapter 6 in [Liu]).
	www.math.leidenuniv.nl /~bogaart/seminarfall05.html (641 words)

	PlanetMath: coherent sheaf (Site not responding. Last check: 2007-10-08)
		, and a sheaf of this form on
		Cross-references: subset, open, scheme, finitely generated module, sheaf, sheafification, open set, sections, presheaf, prime spectrum, ring
		This is version 5 of coherent sheaf, born on 2003-08-15, modified 2003-11-15.
	planetmath.org /encyclopedia/CoherentSheaf.html (73 words)

	[No title]
		Then a sheaf of modules over X, often called just a sheaf over X, is a sheaf of O-modu* les on AffT=X. More concretely, a sheaf* M is a functorial assignment of an R-module Mx to each point (R, x), satisfying the sheaf condition.
		A sheaf M over X would be as assignment of a graded R-module Mx to each point (R, x) of X(R), satisfying the functoriality and sheaf conditions.
		A quasi-coherent sheaf over (X0, X1) is a sheaf M over (X0, X1) in the trivial topology such that M is quasi-coheren* t as a sheaf* over X0.
	hopf.math.purdue.edu /Hovey/hopfalgebroids.txt (11184 words)

	coherent sheaves (Site not responding. Last check: 2007-10-08)
		The main reason to implement algebraic varieties is support the computation of sheaf cohomology of coherent sheaves, which doesn't have an immediate description in terms of graded modules.
		In this example, we use cotangentSheaf to produce the cotangent sheaf on a K3 surface and compute its sheaf cohomology.
		Use the function sheaf to convert a graded module to a coherent sheaf, and module to get the graded module back again.
	www.math.uiuc.edu /Macaulay2/Manual/0411.html (155 words)

	[No title]
		Because of this stalk issue, and because the stalk of a presheaf is the stalk of the associated sheaf, psi^{-1} is an exact functor from the category of sheaves on Y to the category of sheaves on X. +++ 3.8 Sheaves of pseudo-discrete spaces.
		Thus, the more abstract notion of coherence as in FAC is a better point of view than the defn as in Hartshorne (which he makes applicable to both schemes and formal schemes, but in a sort of ad hoc manner).
		If O_X is coherent, and F is a coherent O_X-module, then any finitely generated (=finite type) submodule of a stalk of F extends, in a neighbourhood of the point, to a coherent submodule of F. This isn't proven, just taken from Tohoku.
	math.stanford.edu /~vakil/ega0 (5238 words)

	[No title]
		This is a coherent analytic sheaf over a complex variety ET const* ructed using an elliptic curve E. The stalk of this sheaf* is defined in terms of equiv* *ariant cohomology.
		Construction of the Sheaf The purpose of this section is to define a sheaf valued T -equivariant cohomo* logy theory, which we denote by KT(-).
		It follows that F is a sheaf concentrated at the elements of Zn, where it h* as stalk equal to C. Then the global sections of F are KS1(X) = C : : :C, n copies, one for each element of Zn.
	hopf.math.purdue.edu /Rosu-Knutson/ioanidkt.txt (4601 words)

	Mathematics (Site not responding. Last check: 2007-10-08)
		There is a general definition, due to Grothendieck, of the sheaf of differential operators on a general variety X. This is a coherent sheaf with many nice properties when X is a smooth variety, but in general it is only quasi-coherent.
		One line of investigation is to study this sheaf, or its ring of sections in the local case, and the corresponding representations or D-modules.
		Although nice results are known about this sheaf in a variety of singular situations, it is known to behave quite badly "mostly".
	www.maths.warwick.ac.uk /~eriksen/Math/math.html (404 words)

	God's Calendar (No. 156)
		They kept the sacrifice on 14 Nisan in the evening at the end of the day of the 14th and commenced the meal on the evening of 15 Nisan, all determined according to the conjunction.
		The sacrifice of the Wave Sheaf, when coupled with the feasts and New Moons, has a great significance, which was dealt with in the paper The Harvests of God, the New Moon Sacrifices, and the 144,000 (No. 120).
		The Wave Sheaf Offering is included in Leviticus 23 even though it is not a Holy Day, because it is integral to the Feast of Unleavened Bread and is the primary element of the Harvest of God.
	www.ccg.org /English/s/p156.html (17903 words)

	[No title]
		The sheaf G we have just described is isomorphic to F, thus a* llowing an alternative definition of ES1(X).
		Therefore * F ~=G. As it is the case with any coherent* sheaf of OE-modules over an elliptic curv* e, ES1(X) splits (noncanonically) into a direct sum of a locally free sheaf, i.e.
		Notice that ES1(DW, SW) is an invertible sheaf, because it is the same as the * structure sheaf E S1() = OE twisted by the cocycle ~[W]fffi.
	www.math.purdue.edu /research/atopology/Rosu/ellc.txt (11110 words)

	[No title]
		Denote* * by O the sheaf of algebraic functions on CT, and by Oh the sheaf of holomorphic func* *tions.
		A sheaf-valued cohomology theory The purpose of this section is to define a sheaf valued T -equivariant cohomo* logy theory, which we denote by KT(-).
		It follows that F* * is a sheaf concentrated at the elements of Zn, where it has the stalk equal to C. Th* en the global sections of F are KS1(X) = C.
	www.math.purdue.edu /research/atopology/Rosu/kt.txt (5154 words)

	Category Theory
		Moreover, when developed in a categorical framework, traditional boundaries between disciplines are shattered and reconfigured; to mention but one important example, topos theory provides a direct bridge between algebraic geometry and logic, to the point where certain results in algebraic geometry are directly translated into logic and vice versa.
		The structure of this specific area, in a sense, might not need to rest on anything, that is, on some solid soil, for it might very well be just one part of a larger network that is without any Archimedean point, as if floating in space.
		Coherent and geometric logic, so-called, whose practical and conceptual significance has yet to be explored (Makkai and Reyes 1977, Mac Lane and Moerdiejk 1992, Johnstone 2002);
	plato.stanford.edu /entries/category-theory (11780 words)

	Citebase - Mumford-Thaddeus Principle on the Moduli Space of Vector Bundles on an Algebraic Surface
		A torsion-free coherent sheaf E is said to be L-twisted A-Gieseker-semistable for L ∈ Pic(X) ⊗ Q if and only if for all F ⊂ E χ(E ⊗ L ⊗ An) χ(F ⊗ L ⊗ An) ≤ rk(F) rk(E) for n >> 0, where we compute the Euler characteristics formally using the Riemann-Roch formula.
		A coherent torsion-free sheaf E of rank rk(E) is H-slope-semistable (resp.
		We denote by S((r, c1, c2) ⊗ L, H) the set of all coherent torsion free sheaves E of rk(E) = r, c1 (E) = c1, c2 (E) = c2, that are L-twisted H-Gieseker-semistable.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:alg-geom/9410016 (6279 words)

	Pourcin: Deformations of coherent foliations on a compact normal space (Site not responding. Last check: 2007-10-08)
		An universal analytic structure is construted on the set of (singular) holomorphic foliations on a normal compact space.
		Such a foliation is by definition a coherent subsheaf of the holomorphic tangent sheaf stable by the Lie-bracket
		TRAUTMANN, Gap-sheaves and extension of coherent analytic subsheaves, Lect.
	www.numdam.org /numdam-bin/item?id=AIF_1987__37_2_33_0 (114 words)

	[No title]
		\vvv \hhh In this paper, exploiting the theory of Fourier transforms, we prove that a coherent sheaf which is generically of rank $1$ on an abelian variety and is "cohomologically" a principal polarization, must in fact be a line bundle and hence a principal polarization.
		Given a log-resolution of the pair $(X,D)$, we can define the multiplier ideal sheaf associated to the divisor $D$, $$\II (D):=f_* (\OO _Y (K_{Y/X}-\lfloor f^* D \rfloor))\.$$ The definition is independent of the choice of log-resolution.
		The goal here is to show that the sheaf $\OO _{{X}}({\Theta})\ot \II (\Theta)$ is a cohomological principal polarization and then to apply proposition 2.2.
	www.math.utah.edu /~hacon/question (2496 words)

	UC Berkeley Mathematics
		Describing the direct image complex of a coherent sheaf under a projective morphism
		The ordinary direct image sheaf is simply the degree 0 part of a suﬃciently nice representative module.
		The higher direct images measure the failure of exactness of this construction (coming from the fact that the nice representatives of an exact sequence of sheaves do not form an exact sequence of modules.) However, base change (tensoring with an A-algebra) is badly behaved for these functors.
	math.berkeley.edu /index.php?module=calendar&calendar[view]=event&id=673 (212 words)

	CoherentSheaf ZZ -- canonical twist of a coherent sheaf (Site not responding. Last check: 2007-10-08)
		CoherentSheaf ZZ -- canonical twist of a coherent sheaf
		Operator: symbol " " -- blank operator for adjacent expressions
		-- twist a coherent sheaf F on a projective variety by the n-th power of the hyperplane line bundle.
	www.math.uiuc.edu /Macaulay2/Manual/0983.html (70 words)

	What's Up in the GST \| Musings
		If you want to be fancy, you can call that data a (particular example of a) coherent sheaf on
		In this context, Eric Sharpe and collaborators have written a series of papers recently which prove by explicit calculation that the open-string spectrum stretched between “sheaf
		The “exotic” D-branes are the ones not quasi-isomorphic to a (direct sum of) coherent sheaves.]
	golem.ph.utexas.edu /~distler/blog/archives/000217.html (576 words)

	Abstract (Site not responding. Last check: 2007-10-08)
		Abstract: (Joint work with David Eisenbud) The higher direct image complex of a coherent sheaf (or finite complex of coherent sheaves) under a projective morphism is a fundamental construction that can be defined via a Cech complex or an injective resolution, both inherently infinite constructions.
		Using exterior algebras and relative versions of theorems of Beilinson and Bernstein-Gel'fand-Gel'fand, we give an alternate description in finite terms.
		Our approach is so explicit that it yields an algorithm suited for computer algebra systems.
	www.nd.edu /~magic05/abstracts/schreyer.html (141 words)

	Atlas: Computing the higher direct image complex of a coherent sheaf by Frank-Olaf Schreyer (Site not responding. Last check: 2007-10-08)
		Atlas: Computing the higher direct image complex of a coherent sheaf by Frank-Olaf Schreyer
		The higher direct image complex of a coherent sheaf (or finite complex of coherent sheaves) under a projective morphism is a fundamental construction that can be defined via a Cech complex or an injective resolution, both inherently infinite constructions.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caql-07.
	atlas-conferences.com /cgi-bin/abstract/caql-07 (188 words)

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