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Topic: Sheaf cohomology

	Sheaf cohomology - Wikipedia, the free encyclopedia
		In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F.
		Its development was rapid in the years after 1950, when it was realised that sheaf cohomology was connected with more classical methods applied to the Riemann-Roch theorem, the analysis of a linear system of divisors in algebraic geometry, several complex variables, and Hodge theory.
		From the sheaf point of view, the Čech theory is the restriction to the theory of sheaves of locally constant functions with values in A.
	en.wikipedia.org /wiki/Sheaf_cohomology (776 words)

	PlanetMath: sheaf cohomology
		Sheaf cohomology can be explicitly calculated using Čech cohomology.
		In fact in [2], this is how the cohomology of projective space is explicitly calculated.
		This is version 10 of sheaf cohomology, born on 2003-08-14, modified 2005-05-15.
	planetmath.org /encyclopedia/SheafCohomology.html (221 words)

	Springer Online Reference Works
		The universality property enables one to compare cohomologies arising in concrete situations with sheaf cohomology (and consequently also with each other), to discern for them the natural bounds within which their application is effective, and also to apply sheaf-theoretic methods to the solution of concrete problems.
		Cohomology with coefficients in a sheaf was first defined by the Aleksandrov–Čech method.
		He also showed that to construct a cohomology theory it is entirely sufficient to use his canonical flabby resolution, which, from the point of view of homological algebra, turns out to be simply one of the acyclic resolutions of a sheaf.
	eom.springer.de /s/s084840.htm (2238 words)

	Cohomology - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab1.netlab.uky.edu) (Site not responding. Last check: 2007-11-03)
		Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology.
		Cohomology arises from the algebraic dualization of the construction of homology.
		A cohomology theory is a family of contravariant functors from the category of pairs of topological spaces and continuous functions (or some subcategory thereof such as the category of CW complexes) to the category of Abelian groups and group homomorphisms that satisfies the Eilenberg-Steenrod axioms.
	en.wikipedia.org.cob-web.org:8888 /wiki/Cohomology (714 words)

	Springer Online Reference Works
		A cohomology theory with values in a sheaf and with supports contained in a given subset.
		In terms of local cohomology one can define hyperfunctions, which have important applications in the theory of partial differential equations [5].
		An analogue of local cohomology also exists in étale cohomology theory [3].
	eom.springer.de /L/l060090.htm (348 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).
		II.6.3 in [G], esp. the footnote); and as application the cohomology of the sheaves associated to divisors of bidegree (a,b).
	www.math.leidenuniv.nl /~bogaart/seminarfall05.html (641 words)

	[No title]
		They also discovered that the process of turning a ``presheaf'' into a sheaf was implicitly involved in the notion of forcing used by Cohen in his proof of the independence of the Continuum Hypothesis -- and by Scott and Solovay in the corresponding ``Boolean valued'' models of set theory.
		In Section the sheaf cohomology groups of an arbitrary topos are introduced.
		Homotopy and cohomology of topoi are discussed extensively in [] (for the fundamental group), [] (vol.
	www1.elsevier.com /homepage/saj/523281/h14.htm (667 words)

	coherent sheaves
		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.stanford.edu /~mluciano/M2-help/0585.html (155 words)

	Applications of Sheaf Cohomology in Twistor Theory
		For example, in the representation of zero rest mass fields on spacetime, by means of twistor functions, the field equations are essentially replaced merely by a condition of holomorphicity (Penrose 1975, 1969) and in the nonlinear graviton (Penrose 1976) curved vacuum spacetimes are generated by deforming the complex structure of flat twistor space.
		One of the natural paths along which complex analysis and contour integral techniques can be developed leads to sheaf theory and sheaf cohomology which, as a result, takes up a fundamental role in the mathematical apparatus of twistor theory.
		Perhaps these results depending essentially on the structure of complex numbers give some clues towards an under- standing of the fundamental role of complex structures in physics but a full understanding, which perhaps incorporates the union of quantum mechanics and relativity, is yet to emerge.
	users.ox.ac.uk /~tweb/00003/index.shtml (584 words)

	PlanetMath: acyclic sheaf
		A sheaf is acyclic if it is acyclic on
		See Also: sheaf, sheaf cohomology, De Rham-Weil theorem, sheaf cohomology
		This is version 5 of acyclic sheaf, born on 2004-10-09, modified 2004-11-22.
	planetmath.org /encyclopedia/AcyclicSheaf.html (66 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		Question: Is this statement true for sheaf cohomology with (trivial) coefficients in Z? Assume that X,Y are paracompact and hence H* is isomorphic to Alexander-Spanier and Cech cohomologies.
		It has a wedge axiom - the homology of the strong wedge is the product of the homology of the individual pieces, and there is a UCT with Cech cohomology as well as a lim - lim^1 sequence for the inverse limit of finite complexes.
		Milnor wrote about this theory about 35 years ago and his paper was published fairly recently (no - it was not sitting on an editor's desk for that long.) There is some work by some Russian topologists over the years re extending this to more general topological spaces - I don't know the details.
	www.lehigh.edu /~dmd1/mg123.txt (316 words)

	Sheaf (mathematics) - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab1.netlab.uky.edu) (Site not responding. Last check: 2007-11-03)
		In mathematics, a sheaf is the basic tool for expressing relationships between small regions of a space and large regions.
		This sheaf is especially important when f is the projection of a fiber bundle onto its base space.
		The only natural topology on such a variety, however, is the Zariski topology, but sheaf cohomology in the Zariski topology is badly behaved because there are very few open sets.
	en.wikipedia.org.cob-web.org:8888 /wiki/Sheaf_(mathematics) (5356 words)

	Roberts on Nonabelian Cohomology \| The String Coffee Table
		Local cohomology with values in some local net of groups is pretty much like sheaf cohomology with values in a sheaf of groups.
		In his paper, Roberts briefly notes that the second local cohomology associated to the net of solutions of the free vacuum Maxwell equations and of the free vector particle provides nontrivial information about the sheaf of Cauchy data for these fields.
		In this paper, Duskin works out the theory of descent (=nonabelian sheaf cohomology) in detail for the 2-truncated case, and promises a sequel, never written, which does the full story.
	golem.ph.utexas.edu /string/archives/000842.html (1302 words)

	[No title]
		The purpose of the Wednesday morning session is twofold: to show how to compute with coherent sheaves and their cohomology, and to apply this to some important applications: the Hodge diamond of a projective variety, and computing with divisors on curves and surfaces.
		Part I. We use a 'working persons' definition of sheaf on projective space and an algebraic version of the Cech complex to define cohomology of a (coherent) sheaf on projective space.
		The proofs are doable given the definitions and are mostly left as an exercise for the afternoon.
	www.math.umn.edu /~ayong/Macaulay_week1_abstracts.html (1028 words)

	Math 748S, Spring 2005 (Site not responding. Last check: 2007-11-03)
		In particular, we will discuss sheaf cohomology and derived functors.
		We will relate the "classical" cohomology theories to each other by proving the abstract de Rham theorem, which is really just a motivation for the main topic: derived categories.
		If we have time, we will explain the approach to intersection cohomology via intersection complexes, and then give some applications (to Schubert varieties, or to the Springer correspondence, or whatever seems interesting -- assuming we have time, of course).
	www.math.umd.edu /~tjh/748S_spr05.html (510 words)

	Periodic cyclic homology as sheaf cohomology, by Guillermo Corti~nas (Site not responding. Last check: 2007-11-03)
		Periodic cyclic homology as sheaf cohomology, by Guillermo Corti~nas
		A theorem of Grothendieck establishes that the cohomology of the structure sheaf on the infinitesimal topology of a scheme of characteristic zero is de Rham cohomology.
		We prove that, for the noncommutative infinitesimal topology of an associative algebra over a field of characteristic zero, the cohomology of the structure sheaf modulo commutators is periodic cyclic cohomology.
	www.math.uiuc.edu /K-theory/0307 (120 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		The proof of Kunneth formula for singular cohomology relies on the Eilenberg-Zilber theorem (C_(X x Y) and C_(X)\otimes C_*(Y) are homotopic chain complexes).
		The problem with sheaf cohomology is that the corresponding homology (Borel-Moore homology) is very messy and there is no obvious relation between C(X x Y) and C(X)\otimes C*(Y) (except when X,Y are locally compact and compact supports are considered).
		Therefore, I suspect that this may be a difficult question showing that sometimes singular cohomology has better properties than sheaf-Cech- Alexander-Spanier cohomology.
	www.lehigh.edu /dmd1/public/www-data/as122.txt (191 words)

	Local Cohomology - Cambridge University Press (Site not responding. Last check: 2007-11-03)
		This book provides a careful and detailed algebraic introduction to Grothendieck's local cohomology theory, and provides many illustrations of applications of the theory in commutative algebra and in the geometry of quasi-affine and quasi-projective varieties.
		Topics covered include Castelnuovo-Mumford regularity, the Fulton-Hansen connectedness theorem for projective varieties, and connections between local cohomology and both reductions of ideals and sheaf cohomology.
		It is designed for graduate students who have some experience of basic commutative algebra and homological algebra, and also for experts in commutative algebra and algebraic geometry.
	www.cup.cam.ac.uk /catalogue/catalogue.asp?isbn=0521372860 (197 words)

	CoherentSheaf -- the class of all coherent sheaves (Site not responding. Last check: 2007-11-03)
		Each object of class CoherentSheaf is called a coherent sheaf.
		Each coherent sheaf is also a member of class MutableHashTable.
		CoherentSheaf ZZ -- canonical twist of a coherent sheaf
	www.msri.org /about/computing/docs/macaulay/2-0.9/0241.html (119 words)

	funny definition of dimension of a topological space (Site not responding. Last check: 2007-11-03)
		for example, riemann himself would not understand the definition of the genus of a curve in hartshorne, as the dimension of the sheaf cohomology group h^1(O), on page 294, unless he saw the comment on line three of page 414, that it equals the dimension of the space H^0(K).
		This relates the topology H^1(C) and the analysis H^0(K) (both due to riemann), to the sheaf cohomology group H^1(O), which illustrates the relation between hartshorne's definition and riemann's definition.
		246 the "one point at a time" version of the sheaf sequence as in hartshorne, which proves the result inductively but avoids relating the two proofs.
	www.physicsforums.com /showthread.php?t=81747 (1501 words)

	Not Even Wrong » Blog Archive » The Kostant Dirac Operator
		For weights that are not dominant, one gets not holomorphic sections, but elements in higher cohomology groups.
		These can be expressed either in terms of the sheaf cohomology of G/T with coefficients in the sheaf of holomorphic sections of the line bundle, or in terms of Lie algebra cohomology.
		Instead of using complex manifold methods and the Dolbeaut operator to construct cohomology classes, one can use spinors and the Dirac operator, with the representation appearing as the kernel of the Dirac operator (or, more accurately, its index).
	www.math.columbia.edu /~woit/wordpress/?p=20 (1367 words)

	Problems on groupoids (Site not responding. Last check: 2007-11-03)
		2) Theory agrees with Moore's cohomology if the groupoid is a locally compact group.
		3) An equivalence of groupoids induces an isomorphism of cohomology.
		4) The second cohomology with the circle as coefficients may be identified with the Brauer group of the groupoid.
	unr.edu /homepage/ramazan/groupoid/open_prb/gop.html (629 words)

	cohomology -- general cohomology functor (Site not responding. Last check: 2007-11-03)
		ScriptedFunctor -- the class of all scripted functors
		HH^ZZ ChainComplex -- cohomology of a chain complex
		HH^ZZ ChainComplexMap -- cohomology of a chain complex map
	www.math.temple.edu /computing/Macaulay2/0225.html (52 words)

	Vanishing Sheaf Cohomology
		Hartshorne proves in his book that if X is a noetherian topological
		space of dimension n, then H^i(X,F)=0 for any abelian sheaf F on X and
		comparing assertions of cohomology theories on complex analytic spaces
	sci4um.com /about6227.html (244 words)

	Singular Manual: sheafcoh_lib
		cohomology of sheaf associated to coker(M) D.4.14.4 sheafCoh
		cohomology of sheaf associated to coker(M) D.4.14.5 dimH
		compute h^i(F(d)), F sheaf associated to coker(M) Auxiliary procedures:
	www.math.lsu.edu /singular/sing_803.htm (43 words)

	Publisher description for Library of Congress control number 97029059 (Site not responding. Last check: 2007-11-03)
		Publisher description for Local cohomology : an algebraic introduction with geometric applications / M.P. Brodmann, R.Y. Sharp.
		Bibliographic record and links to related information available from the Library of Congress catalog
		Library of Congress subject headings for this publication: Algebra, Homological, Sheaf theory, Commutative algebra
	www.loc.gov /catdir/description/cam028/97029059.html (149 words)

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