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Topic: Locally ringed space

	Locally ringed space - Wikipedia, the free encyclopedia
		In mathematics, a locally ringed space (or local ringed space) is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as "functions" defined on that open set.
		Locally ringed spaces appear throughout analysis and are also used to define the schemes of algebraic geometry.
		A locally ringed space is a topological space X, together with a sheaf F of commutative rings on X, such that all stalks of F are local rings.
	en.wikipedia.org /wiki/Locally_ringed_space (812 words)

	PlanetMath: locally ringed space (Site not responding. Last check: 2007-11-07)
		The utility of this definition lies in the fact that one can then form constructions in familiar instances of locally ringed spaces which readily generalize in ways that would not necessarily be obvious without this framework.
		Another useful application of locally ringed spaces is in the construction of schemes.
		This is version 10 of locally ringed space, born on 2002-05-01, modified 2005-03-05.
	planetmath.org /encyclopedia/LocallyRingedSpace.html (349 words)

	Talk:Locally ringed space - Wikipedia, the free encyclopedia
		In topos theory it is shown how the theory therefore qualifies for a classifying topos, which parametrises local rings.
		The structure of a locally ringed space is equivalent to the right kind of morphism to this topos - which can also be identified via algebraic geometry.
		What should happen is that the structure of X as a locally ringed space of R-algebras should be equivalent to a morphism to the classifying topos for local R-algebras, which is (known to algebraic geometers as) the Zariski topos for Spec(R).
	en.wikipedia.org /wiki/Talk:Locally_ringed_space (335 words)

	Locally ringed space
		In topology, a locally ringed space is a topological space X, together with a sheaf F on X, such that the stalks of X are commutative local rings.
		The idea of a locally-ringed space is that of a rather general geometrical object.
		It is not at all the case that F(U) is a local ring for open sets U of X: a ring, yes.
	ebroadcast.com.au /lookup/encyclopedia/lo/Locally_ringed_space.html (245 words)

	Ringed space (Site not responding. Last check: 2007-11-07)
		In mathematics, a locally ringed space (or localringed space) is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of whichare thought of as "functions" defined on that open set.
		A locally ringed space is a topological space X,together with a sheaf F of commutative rings on X, such that all stalks ofF are local rings.
		Schemes are locally ringed spaces obtained by "gluing together"spectra of commutative rings.
	www.therfcc.org /ringed-space-206647.html (679 words)

	Encyclopedia: Compact space
		In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space R
		In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.
		In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers.
	www.nationmaster.com /encyclopedia/Compact-space (3557 words)

	dual space
		Given any vector space V over some field F, we define the dual space V* to be the set of all linear functionals on F, i.e., scalar-valued linear transformations on V (in this context, a "scalar" is a member of the base-field F).
		f produces an injective homomorphism between the space of linear operators from V to W and the space of linear operators from W* to V*; this homomorphism is an isomorphism iff W is finite-dimensional.
		When dealing with a normed vector space V (e.g., a Banach space or a Hilbert space), one typically is only interested in the continuous linear functionals from the space into the base field.
	www.fact-library.com /dual_space.html (921 words)

	Spectrum of a ring: Definition and Links by Encyclopedian.com - All about Spectrum of a ring (Site not responding. Last check: 2007-11-07)
		In abstract algebra and algebraic geometry, the spectrum of a commutative ring R is defined to be the set of all prime ideals of R.
		Spec(R) can be turned into a topological space as follows: a subset V of Spec(R) is closed if and only if there exists a subset I of R such that V consists of all those prime ideals in R that contain I.
		at P of this sheaf is equal to the localization of R at P, which is a local ring.
	www.encyclopedian.com /sp/Spectrum-of-a-ring.html (666 words)

	Science Fair Projects - Algebraic variety
		The quotient of the polynomial ring by this ideal is the coordinate ring of the affine algebraic variety.
		A scheme is a locally ringed space such that every point has a neighbourhood, which, as a locally ringed space, is isomorphic to a spectrum of a ring.
		Basically, a variety is a scheme whose structure sheaf is a sheaf of K-algebras with the property that the rings R that occur above are all domains and are all finitely generated K-algebras, i.e., quotients of polynomial algebras modulo prime ideals.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Algebraic_variety (774 words)

	PlanetMath: scheme (Site not responding. Last check: 2007-11-07)
		In fact, the points of such an object take a secondary role: this is neccesary because, for example, over a finite field most curves have no points at all until you pass to a suitable field extension.
		An affine scheme is a locally ringed space
		An affine variety corresponds to the prime spectrum of its coordinate ring, and a projective variety has an open cover by affine pieces each of which is an affine variety, and hence an affine scheme.
	planetmath.org /encyclopedia/Scheme.html (654 words)

	Gluing axiom -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
		This, however, is not quite the same thing; one speaks instead of a (Click link for more info and facts about locally ringed space) locally ringed space, because it is not true, except in trite cases, that such a sheaf is a functor into a category of local rings.
		It is the stalks of the sheaf that are local rings, not the collections of sections (which are (Gymnastic apparatus consisting of a pair of heavy metal circles (usually covered with leather) suspended by ropes; used for gymnastic exercises) rings, but in general are not close to being local).
		This is a logical matter: axioms for a local ring require use of (Click link for more info and facts about existential quantification) existential quantification, in the form that for any r in the ring, one of r and 1 − r is (Click link for more info and facts about invertible) invertible.
	www.absoluteastronomy.com /encyclopedia/g/gl/gluing_axiom.htm (1276 words)

	ipedia.com: Scheme (mathematics) Article (Site not responding. Last check: 2007-11-07)
		Technically, a scheme is a topological space together with commutative rings for all its open sets, which arises from "gluing together" spectra (spaces of prime ideals) of commutative rings.
		Around 1942 Oscar Zariski had defined an abstract Zariski space from the function field of an algebraic variety, for the needs of birational geometry: this is like a direct limit of ordinary varieties (under 'blowing up'), and the construction, reminiscent of locale theory, used valuation rings as points.
		He defines the spectrum of a commutative ring as the space of prime ideals with Zariski topology, but augments it with a sheaf of rings: to every Zariski-open set he defines a commutative ring, thought of as the ring of "polynomial functions" defined on that set.
	www.ipedia.com /scheme__mathematics_.html (1167 words)

	Spectrum of a ring -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
		If P is a point in Spec(R), that is, a prime ideal, then the stalk at P equals the (A determination of the location of something) localization of R at P, and this is a (Click link for more info and facts about local ring) local ring.
		The functor Spec yields a contravariant equivalence between the category of commutative rings and the category of affine schemes; each of these categories is often thought of as the (Click link for more info and facts about opposite category) opposite category of the other.
		By studying spectra of polynomial rings instead of algebraic sets with Zariski topology, one can generalize the concepts of algebraic geometry to non-algebraically closed fields and beyond, eventually arriving at the language of (An elaborate and systematic plan of action) schemes.
	www.absoluteastronomy.com /encyclopedia/s/sp/spectrum_of_a_ring.htm (1157 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		A morphism of locally ringed spaces from $(X,\O_X)$ to $(Y,\O_Y)$ is a \htmladdnormallink{continuous map}{http://planetmath.org/encyclopedia/Continuous.html} $f: X \lra Y$ together with a \htmladdnormallink{morphism of sheaves}{http://planetmath.org/encyclopedia/Sheaf.html} $\phi: \O_Y \lra \O_X$ with respect to $f$ such that, for every point $p \in X$, the \htmladdnormallink{induced}{http://planetmath.org/encyclopedia/Subgraph.html} \htmladdnormallink{ring homomorphism}{http://planetmath.org/encyclopedia/RingHomomorphism.html} on stalks $\phi_p: (\O_Y)_{f(p)} \lra (\O_X)_p$ is a local \htmladdnormallink{homomorphism}{http://planetmath.org/encyclopedia/TypesOfHomomorphisms.html}.
		We then see that, in general, for {\bf any} locally ringed space $X$, the space of tangent vectors at $p$ should be defined as the $k$--vector space $(\m_p/\m_p^2)^$, where $k$ is the \htmladdnormallink{residue field}{http://planetmath.org/encyclopedia/ResidueField.html} $(\O_X)_p / \m_p$ and $^$ denotes dual with respect to $k$ as before.
		Another useful application of locally ringed spaces is in the construction of \htmladdnormallink{schemes}{http://planetmath.org/encyclopedia/Scheme.html}.
	www.ma.utexas.edu /~jcorneli/e/work%20folder/massive/LocallyRingedSpace.tex (386 words)

	Algebraic variety - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-11-07)
		An affine algebraic variety was an irreducible algebraic set in some affine space, over an algebraically closed field K. It therefore was given by a co-ordinate ring that was an integral domain, a quotient of a polynomial ring over K by a prime ideal.
		A projective algebraic variety was the closure in projective space of an affine variety.
		An abstract algebraic variety would be a particular kind of locally ringed space, namely such that every point has a neighbourhood, as ringed space, of type Spec(R) (spectrum of a ring) with R the co-ordinate ring of an affine algebraic variety of the kind discussed in the first paragraph.
	encyclopedia.learnthis.info /a/al/algebraic_variety.html (383 words)

	Encyclopedia: Cartan's theorems A and B
		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 OX_modules OXm → OXn.
		In mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a closed, complex submanifold of the vector space of n complex dimensions.
		In mathematics, a sheaf spanned by global sections is a sheaf F on a locally ringed space X, with structure sheaf OX that is of a rather simple type.
	www.nationmaster.com /encyclopedia/Cartan%27s-theorems-A-and-B (436 words)

	Encyclopedia: Grothendieck-Riemann-Roch theorem (Site not responding. Last check: 2007-11-07)
		In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety).
		In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain...
		Using this isomorphism, consider the Chern character 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 OX_modules OXm → OXn.
	www.nationmaster.com /encyclopedia/Grothendieck_Riemann_Roch-theorem (1200 words)

	Ahmed Abbes, TBA
		Abstract: Coleman's iterated integral theory can be formulated as saying that the space of paths between two points on a scheme X over a field of characteristic p, has a canonical element which is fixed under Frobenius.
		The local factors depend, just as in the l-adic case, on a choice of non-zero meromorphic 1-form on the curve.
		There is a global product formula, and the local factors satisfy exactly the same formulae as in the l-adic case, with q (number of elements in the finite field) replaced by 2\pi i.
	www.ms.u-tokyo.ac.jp /%7Et-saito/conf/ag/abstract.html (1268 words)

	[No title]
		Remark We have already seen that the spectrum of a commutative ring is a ringed space; that's the motivation for this definition.
		be a homomorphism of rings, inducing a morphism of ringed spaces
		The properties of localization show that this is a bijection; by looking at properties of containment, one also sees that it is a homeomorphism.
	odin.mdacc.tmc.edu /~krc/agathos/schem2.html (1105 words)

	math lessons - Spectrum of a ring
		In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R.
		It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.
		By studying spectra of polynomial rings instead of algebraic sets with Zariski topology, one can generalize the concepts of algebraic geometry to non-algebraically closed fields and beyond, eventually arriving at the language of schemes.
	www.mathdaily.com /lessons/Spectrum_of_a_ring (978 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		As pointed out last time, any affinoid space is a locally $G$-ringed space: the stalk at a point coincides with the local ring of the affinoid algebra at the corresponding maximal ideal.
		A \emph{(very weak, weak, somewhat weak, strong) affinoid space} is a locally $G$-ringed space of the form $\Max A$, for some affinoid algebra $A$, equipped with the corresponding $G$-topology.
		A \emph{coherent} (resp.\ \emph{coherent locally free}) sheaf on a rigid analytic space is one which on the elements of some admissible affinoid covering looks like the sheaf associated to a finitely generated (resp.\ finite free) module over the structure sheaf.
	www-math.mit.edu /~kedlaya/18.727/rigid-spaces.tex (2157 words)

	Re: Cell Complexes
		An R-scheme for some commutative ring R is a locally ringed space locally isomorphic as a locally ringed space to Spec(R).
		For a lot of differential geometry, you just need to be able to define tangent vectors, and for a locally ringed space, you can stretch the definition of tangent vectors and define them as elements of m/m^2, where m is the maximal ideal at the stalk of the sheaf at a given point.
		Unfortunately for you, and fortunately for people who actually want to use them, schemes are generally taken to be Hausdorff, given that Spec(R) certainly always is. It sounds to me like you should just study simplicial sets and leave it at that, since they're closest to the things you've been saying.
	www.lns.cornell.edu /spr/2002-08/msg0043352.html (426 words)

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