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Topic: Affine scheme

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	Scheme (mathematics) - Wikipedia, the free encyclopedia
		Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern algebraic geometry.
		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.
		One may think of a scheme as covered by "coordinate charts" of affine schemes: the above formal definition means exactly that schemes are obtained by gluing together affine schemes for the Zariski topology.
	www.wikipedia.org /wiki/Scheme_theory (1128 words)

	PlanetMath: scheme
		A scheme in some sense captures the equations defining an algebraic object, so that the points of that object can be examined over many different fields.
		However, frequently one wishes to work in a slightly different category, such as the category of ``complex schemes'', that is, schemes obtained from complex algebras.
		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)

	Glossary of scheme theory - Wikipedia, the free encyclopedia
		For example, we can speak of locally noetherian schemes, namely those which are covered by the spectra of Noetherian rings; and we say that a scheme is noetherian when it is covered by 'finitely' many spectra of noetherian rings.
		For example, one might say of a scheme that it is connected which simply means that the underlying topological space is connected.
		A scheme X is said to be irreducible when (as a topological space) it is not the union of two closed subsets except if one is equal to X.
	en.wikipedia.org /wiki/Glossary_of_scheme_theory (844 words)

	Proper map - Wikipedia, the free encyclopedia
		A morphism of schemes is called proper if it is separated, of finite type and universally closed ([EGA] II, 5.4.1 [1]).
		Affine varieties of non-zero dimension are never proper.
		A deep property of proper morphisms is the existence of a Stein factorization, namely the existence of an intermediate scheme such that a morphism can be expressed as one with connected fibres, followed by a finite morphism.
	www.wikipedia.org /wiki/Proper_morphism (468 words)

	Constructing Schemes (Site not responding. Last check: 2007-10-21)
		As shown in the examples in the introduction to this chapter, schemes are defined inside some ambient space, either affine or projective space, by a collection of polynomials from the coordinate ring associated with that space.
		Schemes may also be defined inside other schemes using polynomials from the coordinate ring of the bigger scheme or polynomials from the ambient space.
		The subscheme of X defined, for an affine scheme X by the trivial polynomial 1, or by maximal ideal (x_1,..., x_n) for a projective scheme X. The returned scheme is marked as saturated.
	magma.maths.usyd.edu.au /magma/htmlhelp/text1152.htm (856 words)

	PlanetMath: variety
		In the other direction, it identifies an affine variety with the prime spectrum of its coordinate ring: the variety in
		A projective variety is identified as the gluing together of the affine varieties obtained by taking the complements of hyperplanes.
		Using the theory of schemes, we can glue these affine varieties together to get a scheme; the result will be projective.
	planetmath.org /encyclopedia/Variety.html (270 words)

	Encyclopedia: Affine scheme (Site not responding. Last check: 2007-10-21)
		General schemes are obtained by "gluing together" several affine schemes.
		It is useful to use the language of category theory and observe that Spec is a functor.
		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 opposite category of the other.
	www.nationmaster.com /encyclopedia/Affine-scheme (1014 words)

	[No title]
		Our category of schemes is equivalent to the algebraic geometer's category of affine schemes, which in turn is equivalent (by Yoneda's lemma) to the opposite* * of the category of rings.
		This is a scheme because it is represented by Z[x1; : :;:xn].
		Schemes of t* *his form are called basic open subschemes of X. A locally closed subscheme is a basic open subscheme of a closed subscheme.
	hopf.math.purdue.edu /Strickland/st-fsfg.txt (9406 words)

	Re: Underlying topological space of the Grassmannian scheme G(2,4) over (Site not responding. Last check: 2007-10-21)
		> The affine scheme C^1 differs from the affine variety C^1 by having a > single, additional dense point corresponding to the ideal (0).
		As far as I understand, the main advantage in using schemes rather than varieties is the additional generality it provides.
		On the other hand, reading Hartshorne might still be good to learn algebraic geometry in general, schemes or not schemes.
	www.lns.cornell.edu /spr/2003-09/msg0054040.html (337 words)

	Schemes
		It is this resulting scheme which is the base change of X to L. If one has a number of schemes in the same ambient space and wants to base change them all at the same time, a little care is required.
		If F is a sequence of schemes lying in a common ambient space whose base ring admits automatic coercion to K or is the domain of a ring map m then this returns the base change of the elements of F as a new sequence.
		The ith affine patch of the scheme X. The number of affine patches is dependent on the type of projective ambient space in which X lies, but for instance, the standard projective space of dimension n has n + 1 affine patches.
	www.math.niu.edu /help/math/magmahelp/text973.html (7384 words)

	Introduction and First Examples
		Schemes may be defined quite generally over any ring k, although of course many functions require k to lie in some restricted class.
		Although closed points of schemes may be defined as schemes in terms of polynomials, there is a far more convenient way to define them: simply coerce the coordinates of the point into the scheme.
		This code computes the intersection of the scheme which is image of f with the conic S (although note that S is defined only as a scheme here and not as a conic).
	www.umich.edu /~gpcc/scs/magma/text1018.htm (2824 words)

	Function Fields and their Elements (Site not responding. Last check: 2007-10-21)
		Additionally, function field elements may be used in the definition of scheme maps (see Section Maps between Schemes) from the projective or affine schemes on which they are defined to other schemes and may be evaluated at points as described earlier.
		Coerce the element g into the function field F of a scheme where g is some function on the scheme of F, for example, g may be an element of the field of fractions of the coordinate ring of a scheme having F as its function field.
		Given an element f of a function field of a scheme, return f as an element of the field of fractions of the coordinate ring of the scheme f is a function on.
	magma.maths.usyd.edu.au /magma/htmlhelp/text1155.htm (606 words)

	Affine and Projective Spaces (Site not responding. Last check: 2007-10-21)
		The affine cone of P. This is the affine space whose functions are the polynomials of P with grading forgotten.
		Although schemes haven't been defined yet, there are a number of schemes that immediately come to mind when thinking of maps; the image of a map is perhaps the most obvious one.
		If f has an affine domain which is a standard patch on its projective closure, this returns the natural map from the projective closure of its domain to that of its codomain.
	www.dtr.isy.liu.se /Magma/text747.html (4050 words)

	Maps between Schemes
		The scheme in the domain of the map of schemes f given by the pullback of the equations defining the subscheme X of the codomain of f.
		The scheme X must be a subscheme of the domain of f and d must be a positive integer.
		The restriction of the map f, a map of schemes from an affine scheme to a projective scheme, to a rational map from its domain to the jth standard affine patch of its codomain.
	www.umich.edu /~gpcc/scs/magma/text1020.htm (5798 words)

	Affine Algebras which are Fields (Site not responding. Last check: 2007-10-21)
		If the ideal J of relations defining an affine algebra A = K[x_1,..., x_n]/J, where K is a field, is maximal, then A is a field and may be used with any algorithms in Magma which work over fields.
		Note that an affine algebra defined over a field which itself is a field also has finite dimension when considered as a vector space over its coefficient field, so all of the operations in the previous section are also available.
		Starting with the same affine algebra A = Q(a, b, x)F[y]/ as in the last example, we factor some univariate polynomials over A. A is of course isomorphic to an absolute field, but the presentation given may be much more convenient to the user.
	www.math.lsu.edu /magma/text1139.htm (441 words)

	[No title]
		Therefore, algebraic geometry is a theory which is based on the two funda* mental notions of affine* scheme and Grothendieck topology.
		For the purpose of schemes, the Zariski topology is enough, but* * 'etale or even faithfully flat and quasi-compact (for short ffqc) topologies also proves * very useful in order to define more general objects as algebraic spaces or algebraic stacks *.
		The category Aff of affine schemes is then the opposite of the categor* y of com- mutative and unital monoids in the symmetric monoidal category (Z-mod,), whic *h is a categorical notion.
	hopf.math.purdue.edu /Toen-Vezzosi/agmod-web.txt (18061 words)

	Learn more about Category theory in the online encyclopedia. (Site not responding. Last check: 2007-10-21)
		One of the central themes of algebraic geometry is the equivalence of the category C of affine schemes and the category D of commutative rings.
		The functor G associates to every commutative ring its spectrum, the scheme defined by the prime ideals of the ring.
		Another important duality occurs in functional analysis: the category of commutative C*-algebras with identity is contravariantly equivalent to the category of compact Hausdorff spaces.
	www.onlineencyclopedia.org /c/ca/category_theory.html (2963 words)

	Ambients
		For the purposes of this chapter, any scheme is contained in some ambient space, either an affine space or one of a small number of standard projective spaces: these are projective space itself, possibly weighted, and rational scrolls.
		There are affine patches on F_n --- a standard `first' patch is the affine plane where v not=0 and y not=0 --- and one can study them to get an idea of what this surface looks like.
		Points of schemes are handled in an extremely flexible way: their coordinates need not be elements of the base ring, for instance.
	www.math.lsu.edu /magma/text1151.htm (2167 words)

	Algebraic Geometry
		Projective spaces are typically regarded as a union of affine spaces, with affine coordinates.
		An affine variety is a k-space which is isomorphic to a closed subspace of an affine space over k.
		of schemes; the image is naturally equivalent to the category of affine schemes.
	www.risberg.ws /Hypertextbooks/Mathematics/Geometry/algebraic.htm (1388 words)

	SVD and log-log frequency sampling with Gabor kernels for invariant pictorial recognition (Site not responding. Last check: 2007-10-21)
		Affine invariance is obtained by a representation which is based on a new sampling configuration in the frequency domain.
		We discuss the decomposition of affine transform into slant, tilt, swing, scale and 2D translation by applying singular value decomposition (SVD).
		The affine invariant spectral signatures (AISS) are derived from a set of Cartesian logarithmic-logarithmic (log-log) sampling configuration in the frequency domain.
	csdl2.computer.org /persagen/DLAbsToc.jsp?resourcePath=/dl/proceedings/&toc=comp/proceedings/icip/1997/8183/03/8183toc.xml&DOI=10.1109/ICIP.1997.632037 (279 words)

	Affine Invariant Watermarks for 3D Polygonal and NURBS Based Models
		The scheme realizes affine invariant watermarks by displacing vertices (control points) and satisfies constrains regarding maximum tolerated vertex movements or, in the NURBS case, differences of original and watermarked surfaces.
		The scheme described can be stacked on more robust scheme allowing transmission of labeling information to the user or increasing blind detection capabilities of the underlying scheme.
		The second contribution of this paper is a general technique for reducing processing time of watermark (label) extraction satisfying impatient users and enhancing robustness with respect to affine transformations and, in particular, vertex randomization attacks.
	publica.fhg.de /documents/N-3773.html (291 words)

	CSM -- Examples
		The input to all functions is the defining ideal of S in projective space: it is expected that this is a homogeneous ideal in a polynomial ring.
		As the schemes are isomorphic, and the Fulton class is intrinsic, these computation agree.
		This is the ideal of the base scheme of a certain rational map encountered in studying the orbit closure of the 5-tuple of points on P
	www.math.fsu.edu /~aluffi/CSM/CSMexamples.html (1217 words)

	8.2 Functors of points (Site not responding. Last check: 2007-10-21)
		For those who have studied affine schemes earlier in a slightly different way we offer the following result which is proved in the second appendix.
		A scheme is said to be reducible if it can be written as the union of two distinct (but not necessarily disjoint!) proper closed subschemes.
		As a consequence of the Lasker-Noether Primary Decomposition theorem any scheme can be written as the union of a finite collection of irreducible closed subschemes; moreover, the underlying reduced schemes of these closed subschemes are uniquely determined.
	www.imsc.ernet.in /~kapil/crypto/notes/node40.html (1505 words)

	Alexey Kuznetsov - Homepage
		We introduce a new class of lattice models based on a Poisson approximation scheme for affine processes, whereby the approximant process itself is affine.
		We introduce a Poisson approximation scheme for jump processes and use it to construct numerical discretizations for the corresponding partial integro-differential equations.
		We propose a new classification scheme for stochastic diffusion processes which are integrable by reduction to hypergeometric equations and discover rich new families of integrable processes.
	www.math.toronto.edu /kuznecov/articles (295 words)

	Scheme (Site not responding. Last check: 2007-10-21)
		term prescheme to describe a scheme that is not separated), but we will not impose this requirement.
		affine variety corresponds to the prime spectrum of its
		projective variety has an open cover by affine pieces each of which is an affine variety, and hence an affine scheme.
	simba.cs.uct.ac.za /~hussein/PlanetMath-snapshot_2004-01-12/entries/Scheme/Scheme.html (112 words)

	[No title]
		We want to work over arbitrary fields, and the property of being irreducible is not stable under the operation of extending the base field.) The most primitive notion of a point is also the rarest usage among algebraic geometers.
		This is another scheme that has only one closed point.
		In particular, we have shown that the functor of points determines the scheme.
	odin.mdacc.tmc.edu /~krc/agathos/point.html (844 words)

	NSDL Metadata Record -- example of functor of points of a scheme
		Let X be an affine scheme of finite type over a field k.
		T For an example of why schemes contain much more information than the list of points over their base field, take...
		This suggests that schemes may be the appropriate adaptation of varieties to deal with non-algebraically closed fields.
	nsdl.org /mr/1035232 (253 words)

	The K-Theory of Vector bundles with Endomorphisms over a Scheme (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		Abstract: Using Thomason and Trobaugh's Localization theorem [Th-Tr] for the K-theory of a scheme, we study the K-theory of the category of vector bundles with endomorphisms over a scheme.
		The results generalize those of D. Grayson [Gr1] when the scheme is affine.
		We also give an example showing that the Mayer-Vietoris sequence does not hold for the K-theory of vector bundles with endomorphisms, which indicates that the K-theory of vector bundles with endomorphisms is a global theory (not determined by...
	citeseer.ist.psu.edu /yao95ktheory.html (272 words)

	[No title]
		The proof of injectivity is then similar to the argument we gave in the affine case.
		R → Spec(R) is a fully faithful contravariant functor from the category RG of commutative rings into the category SCH of schemes; the image is naturally equivalent to the category of affine schemes.
		The previous proposition says that every morphism between affine schemes arises in a unique way from a ring homomorphism, yielding the result.
	odin.mdacc.tmc.edu /~krc/agathos/schem2.html (1105 words)

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