
	Where results make sense

Topic: Finite morphism

	PlanetMath: finite morphism
		Both of these affine morphisms are of finite type, but are not finite.
		The overall morphism is of finite type, but again is not finite.
		This is version 6 of finite morphism, born on 2002-07-24, modified 2006-06-08.
	planetmath.org /encyclopedia/FiniteType.html (195 words)

	Function (mathematics) - Wikipedia, the free encyclopedia
		If the domain X is finite, a function f may be defined by simply tabulating all the arguments x and their corresponding function values f(x).
		The morphisms are the relationships between the objects.
		There are a few restrictions on the morphisms which guarantee that they are analogous to functions (for these, see the article on categories).
	en.wikipedia.org /wiki/Function_(mathematics) (3457 words)

	Glossary of scheme theory - Wikipedia, the free encyclopedia
		A morphism f is finite if, locally on X, it is represented by a finitely generated integral extension of commutative rings.
		A morphism f is an open immersion if locally on the target it is of the form of an inclusion of an affine open subset.
		A separated morphism is a morphism f such that the fiber product of Y with itself along f has its diagonal as a closed subscheme — in other words, the diagonal map is a closed immersion.
	en.wikipedia.org /wiki/Glossary_of_scheme_theory (1028 words)

	Proper map - Wikipedia, the free encyclopedia
		A morphism f : X → Y of algebraic varieties or is called proper if it is separated and universally closed.
		Projective morphisms are proper, but not all proper morphisms are projective.
		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.
	en.wikipedia.org /wiki/Proper_map (625 words)

	Direct Limit, Inverse Limit
		Morphisms are functions from one set into another, and inclusion is a special kind of morphism.
		Suppose there are different finite sets, and in one, x maps to y in t, and in another, x maps to z.
		Follow the morphism from r directly to v, and x is in the preimage of w.
	www.mathreference.com /cat,dirlim.html (1015 words)

	Homomorphisms
		The image C of the morphism phi of abelian varieties, which is a modular abelian subvariety contained in the codomain of phi, a morphism from C to the codomain of phi, and a surjective morphism from the domain of phi to C. G @@ phi : ModAbVarSubGrp, MapModAbVar -> ModAbVarSubGrp
		The cokernel of the morphism phi of abelian varieties and a morphism from the codomain of phi to the cokernel.
		Return the morphism n * phi where phi is a morphism of abelian varieties and n is positive and the smallest such that n * phi is a genuine homomorphism.
	www.math.lsu.edu /magma/text1323.htm (3250 words)

	[No title]
		An equivalent wsr of expressing the properties (l.8.a) and (l.8.c) is to say that A 0 B (together with the inclusions 22 a,- a 1, b i- 1 0 b) is the coproduct of A and B in the category red of k-reduced algebras, having (k-algebra) homomorphisms as its morphisuis.
		A k-algebra A is finitely generated iff there exists a finite subset (a1,..., a) of A such that each element of A can be expressed as a finite combination of a1,..., a using sums, products, and multiplication by elements of k.
		FINITENESS CONDITIONS We have shown in the previous chapter that any polynomial response map f is realizable by a canonical k-system.
	www.math.rutgers.edu /~sontag/FTP_DIR/polynomial_response_maps.txt (13591 words)

	[No title]
		S is a finite morphism in K and as* sume that ker() as well as coker() are locally finite* Ap-modules.
		Using Corollary (2.4) and observing that if R is finitely generated as a ring t* hen Q(R) is locally finite* (even finite as an abelian group), we obtain the following theor* *em concerning the reconstruction of certain objects of K from their structure in high degrees.
		It follows then from [11] in the case of finite groups G and H, a* nd [12, Corollary 2.4] in the general case, that ae maps, as claimed, a Sylow p-subgrou *p of G isomorphically onto a Sylow p-subroup of H. (3.5) Remark.
	hopf.math.purdue.edu /Mislin/lannes.txt (3093 words)

	Encyclopedia :: encyclopedia : Finite (Site not responding. Last check: 2007-11-01)
		This page is about the sporadic finite simple group Th.
		In mathematics, the finite Thompson group Th is one of the 26 sporadic finite simple groups.
		Finite -- to fail, but infinite to Venture -- (Source)
	www.hallencyclopedia.com /Finite (318 words)

	definition of mappings (Site not responding. Last check: 2007-11-01)
		The whole point of morphisms is to send points to points in a way that respects the structure from which the points arise...
		A morphism from V to W is a function f from A* to A*, such that
		Show that a morphism from R to R corresponds precisely to a continuous function f from the real numbers to the real numbers.
	home.hetnet.nl /~sufra/morphisms.htm (218 words)

	Models of curves and finite covers, by Qing Liu and Dino Lorenzini (Site not responding. Last check: 2007-11-01)
		Let f : X \to Y be a finite morphism of curves over K. In this article, we study some possible relationships between the models over O_K of X and of Y. Three such relationships are listed below.
		Let f : X \to Y be a finite morphism, with g(Y) \ge 2.
		Finally, given any finite morphism f : X \to Y, is it possible to choose suitable regular models {\cal X} and {\cal Y} of X and Y over O_K such that f extends to a finite morphism {\cal X} \to {\cal Y} ?
	front.math.ucdavis.edu /ANT/0065 (307 words)

	Abstracts (Site not responding. Last check: 2007-11-01)
		A differential groupoid is a groupoid whose space of morphisms and space of units are differentiable manifolds (possibly with corners) such that all structural maps are differentiable and the domain map is a submersion.
		A finite type algebra is an algebra $A$ whose center $Z$ is the algebra of regular functions on a complex algebraic variety and which is finitely generated as a $Z$-module.
		Every finite type algebra has a composition series with two-sided ideals such that the subquotients are ideals in Azumaya algebras.
	www.math.psu.edu /nistor/abstracts.html (2189 words)

	Publications (Site not responding. Last check: 2007-11-01)
		Abstract: The generalized Mordell-Lang conjecture (GML) is the statement that the irreducible components of the Zariski closure of a subset of a group of finite rank inside a semi-abelian variety are translates of closed algebraic subgroups.
		Abstract: We prove that the existence of an automorphism of finite order on a ${\bf Q}$-variety $X$ implies the existence of algebraic linear relations between the logarithm of certain periods of $X$ and the logarithm of special values of the $\Gamma$-function.
		This implies that a slight variation of results by Anderson, Colmez and Gross on the periods of CM abelian varieties is valid for a larger class of CM motives.
	www.math.jussieu.fr /~dcr/publi.html (1256 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-01)
		A flat morphism of finite type corresponds to the intuitive concept of a continuous family of varieties.
		Flat morphisms are used also in descent theory.
		A morphism of schemes is called faithfully flat if it is flat and surjective.
	eom.springer.de /F/f040600.htm (251 words)

	SEP: Category Theory
		Morphism : For every pair X, Y of objects, there is a set Hom(X, Y), called the morphisms from X to Y in C.
		Note that a category is characterized by its morphisms, and not by its objects.
		Secondly, the notion of adjointness is formally equivalent to the notion of a universal morphism (or construction) and to that of representable functor.
	plato.stanford.edu /entries/category-theory (11786 words)

	Hironaka: Alexander stratifications of character varieties
		In this paper, we give an exposition of Fox calculus in the language of group cohomology and in the language of finite abelian coverings of CW complexes.
		is the fundamental group of a compact Kähler manifold, then the strata are finite unions of translated affine subtori.
		As we will show, this is not the case for general finitely presented groups.
	math-doc.ujf-grenoble.fr /numdam-bin/item?id=AIF_1997__47_2_555_0 (292 words)

	[No title]
		Of particular interest are its finite subgroups; these are precisely t* he finite* groups which occur as groups of symmetries of the surface Sg equipped with a complex structure (a Riemann surface).
		According to Kerckhoff [Ke], the finite subgroups of g are precisely the subgroups of the form o(F) for F a finite gro* up of holomorphic automorphisms of (Sg; o) for some o, amounting to a positive solu tion of the Nielsen realization problem; for an account on the long history of this *problem 2.
		She proved that g is residually finite (which means that g admits an embedding into a prod- uct of finite groups).
	hopf.math.purdue.edu /Mislin/bernoulli.txt (11294 words)

	Boolean algebra (Site not responding. Last check: 2007-11-01)
		The set of all subsets of S that are either finite or cofinite is a Boolean algebra.
		The class of all Boolean algebras, together with this notion of morphism, forms a category.
		Therefore, the number of elements of every finite Boolean algebra is a power of two.
	boolean-algebra.iqnaut.net (1640 words)

	[No title]
		As a corollary, it is proved that there are only finitely many nonisomorphic minimal-rank realizations of a response map over the integers, while for delay -dif f erential systems these are classified by a lattice of subspaces of a finite-dimensional real vector space.
		Dual systems Y appear here for purely technical purposes, 2 3 but they are of fundamental importance in studying questions of regulation (duality of reachability and observability); see Ching and Wyman [19781 and Sontag [19781 for further discussions of duality.
		In this case, since a finitely generated torsion module is finite, one concludes from the results presented here that there are only finitely many nonisomorphic minimal realizations of any givenf.
	www.math.rutgers.edu /~sontag/FTP_DIR/lat.txt (1545 words)

	[No title]
		A morphism or bundle is a maple table which, like variety, grows dynamically as more data is needed.
		The morphism object i.Z is created to represent the inclusion of Z in the ambient variety.
		For example, if you have an inclusion morphism called B and blow it up, the blow up will be called BB, the exceptional divisor class will be EB and so on.
	www.math.sunysb.edu /~sorin/online-docs/schubert/schubertmanual.txt (5882 words)

	Richard Pink: Recent Preprints
		Let k be either a finite field of characteristic p or a local field of residue characteristic p.
		Abstract: Consider a finitely generated Zariski dense subgroup \Gamma of a connected simple algebraic group G over a global field F. An important aspect of strong approximation is the question of whether the closure of \Gamma in the group of points of G with coefficients in a ring of partial adeles is open.
		While this statement can be deduced from the classification of finite simple groups, our proof is self-contained and uses methods only from algebraic geometry and the theory of linear algebraic groups.
	www.math.ethz.ch /~pink/preprints.html (1657 words)

	IDA: Interactive Document on Algebra
		6.3, Proposition: the kernel of a morphism is a normal subgroup
		5.3, Theorem: the kernel of a morphism is an ideal
		5.2, Proposition: there is a morphism from a group to a quotient group
	www.win.tue.nl /~ida/demo/indextb.html (934 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		There are relative versions of this problem: It IS decidable if a finite graph is embeddable into the Cayley graph of a finite commutative group, but NOT decidable if a finite graph is embeddable into the Cayley graph of a finite nilpotent group.
		One of the first easy theorems we learn in an algebra course is that a subsemigroup of a finite group is itself a group.
		However the generalization to groupoids is undecidable: The class of subcategories of finite groupoids is undecidable.That is there is no algorithm to decide if a given finite category is a subcategory of a finite groupoid.
	www.math.technion.ac.il /~techm/20021114123020021114mar (284 words)

	The Dimensional Ladder
		Functors Definition Examples: a functor from the free category on an object, a morphism, an isomorphism, an endo, an auto Example: a functor from a group G to Set is a set acted on by G, or G-set.
		Equivalence of Categories Definition Examples from duality: Vect is equivalent to Vect^{op} some sort of finite posets are equivalent to distributive lattices^{op} Compact Hausdorff spaces is equivalent to commutative C*-algebras^{op} Locally compact abelian groups is equivalent to itself^{op} Skeletal categories Theorem: a category is equivalent to any of its skeleta.
		Example: a functor from a group to Top is a continuous action Example: more generally, a functor from a monoid to C is an action of the monoid on some object of C. example: category of representations of various quivers (free categories on graphs) i.
	math.ucr.edu /home/baez/hda/dimensional_ladder.html (2262 words)

	[No title]
		This usage differs from that of other authors, who reserve the word variety for an irreducible scheme of finite type.
		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.
		But this is also easy, since the natural transformation is defined by composing with the morphism, and we can test it on an identity morphism.
	odin.mdacc.tmc.edu /~krc/agathos/point.html (844 words)

	Springer Online Reference Works
		A cochain complex is defined in a dual manner (as a graded object with a morphism
		is an exact sequence of complexes, then there exists a connecting morphism
		) that is natural with respect to morphisms of exact sequences and is such that the long homology sequence (that is, the sequence
	eom.springer.de /c/c024110.htm (210 words)

	week53
		An n-category is a mathematical structure containing not only objects, which one might think of as "things", and morphisms, which one might think of as "processes leading from one thing to another", but also 2-morphisms, which are "processes leading from one process to another", and then 3-morphisms, etc., on up to n-morphisms.
		Moreover, we should think of Th as a category with all finite limits, that is, one in which all finite diagrams have limits.
		Well, if a model is a sort of functor, a morphism between them should be a sort of natural transformation between functors; that's how it usually goes.
	math.ucr.edu /home/baez/week53.html (2436 words)

	Creation and Basic Functions
		A morphism with finite kernel from the modular abelian variety A to a modular abelian variety attached to modular symbols.
		A list of morphisms from the modular abelian variety A into abelian varieties, which are used in making sense of intersections, sums, etc. The embeddings at the beginning of the list take precedence over those that occur later.
		Abelian varieties can have their base ring set to a finite field, but there is very little that is implemented for abelian varieties over finite fields.
	www.math.lsu.edu /magma/text1321.htm (5832 words)

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