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Resolution Of Singularities
In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety ''V'' has a resolution, which is a non-singular variety ''W'' with a Proper morphism, proper birational map ''W''→''V''. For varieties over fields of characteristic 0, this was proved by Heisuke Hironaka in 1964; while for varieties of dimension at least 4 over fields of characteristic ''p'', it is an open problem. Definitions Originally the problem of resolution of singularities was to find a nonsingular model for the function field of a variety ''X'', in other words a complete variety, complete non-singular variety ''X′'' with the same function field. In practice it is more convenient to ask for a different condition as follows: a variety ''X'' has a resolution of singularities if we can find a non-singular variety ''X′'' and a Proper morphism, proper birational map from ''X′'' to ''X''. The condition that the map is proper is needed to exclude trivia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functorial Resolution Of Singularities
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous function, continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a Linguistics, linguistic context; see function word. Definition Let ''C'' and ''D'' be category (mathematics), categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each Mathematical object, object X in ''C'' to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals the height of the defining ideal. For this reason, the height of an ideal is often called its codimension. The dual concept is relative dimension. Definition Codimension is a ''relative'' concept: it is only defined for one object ''inside'' another. There is no “codimension of a vector space (in isolation)”, only the codimension of a vector ''sub''space. If ''W'' is a linear subspace of a finite-dimensional vector space ''V'', then the codimension of ''W'' in ''V'' is the difference between the dimensions: :\operatorname(W) = \dim(V) - \dim(W). It is the complement of the dimension of ''W,'' in that, with the dimension of ''W,'' it adds up to the dimension of the ambient space ''V:'' :\dim(W) + \operatorname(W) = \dim(V). Simi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert–Samuel Function
In commutative algebra the Hilbert–Samuel function, named after David Hilbert and Pierre Samuel,H. Hironaka, Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I. Ann. of Math. 2nd Ser., Vol. 79, No. 1. (Jan., 1964), pp. 109-203. of a nonzero finitely generated module M over a commutative Noetherian local ring A and a primary ideal I of A is the map \chi_^:\mathbb\rightarrow\mathbb such that, for all n\in\mathbb, :\chi_^(n)=\ell(M/I^M) where \ell denotes the length over A. It is related to the Hilbert function of the associated graded module \operatorname_I(M) by the identity : \chi_M^I (n)=\sum_^n H(\operatorname_I(M),i). For sufficiently large n, it coincides with a polynomial function of degree equal to \dim(\operatorname_I(M)), often called the Hilbert-Samuel polynomial (or Hilbert polynomial).Atiyah, M. F. and MacDonald, I. G. ''Introduction to Commutative Algebra''. Reading, MA: Addison–Wesley, 1969. Examples For the ring of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exceptional Divisor
In mathematics, specifically algebraic geometry, an exceptional divisor for a regular map :f: X \rightarrow Y of varieties is a kind of 'large' subvariety of X which is 'crushed' by f, in a certain definite sense. More strictly, ''f'' has an associated exceptional locus which describes how it identifies nearby points in codimension one, and the exceptional divisor is an appropriate algebraic construction whose support is the exceptional locus. The same ideas can be found in the theory of holomorphic mappings of complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...s. More precisely, suppose that :f: X \rightarrow Y is a regular map of varieties which is birational (that is, it is an isomorphism between open subsets of X and Y). A codimension-1 subvariety Z \subset ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal Sheaf
In algebraic geometry and other areas of mathematics, an ideal sheaf (or sheaf of ideals) is the global analogue of an ideal (ring theory), ideal in a ring (mathematics), ring. The ideal sheaves on a geometric object are closely connected to its subspaces. Definition Let ''X'' be a topological space and ''A'' a sheaf (mathematics), sheaf of rings on ''X''. (In other words, is a ringed space.) An ideal sheaf ''J'' in ''A'' is a subobject of ''A'' in the category (mathematics), category of sheaves of ''A''-modules, i.e., a sheaf (mathematics), subsheaf of ''A'' viewed as a sheaf of abelian groups such that : for all open subsets ''U'' of ''X''. In other words, ''J'' is a sheaf of modules, sheaf of ''A''-submodules of ''A''. General properties * If ''f'': ''A'' → ''B'' is a homomorphism between two sheaves of rings on the same space ''X'', the kernel of ''f'' is an ideal sheaf in ''A''. * Conversely, for any ideal sheaf ''J'' in a sheaf of rings ''A'', there is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-excellent
In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the problem of finding a natural class of "well-behaved" rings containing most of the rings that occur in number theory and algebraic geometry. At one time it seemed that the class of Noetherian rings might be an answer to this problem, but Masayoshi Nagata and others found several strange counterexamples showing that in general Noetherian rings need not be well-behaved: for example, a normal Noetherian local ring need not be analytically normal. The class of excellent rings was defined by Alexander Grothendieck (1965) as a candidate for such a class of well-behaved rings. Quasi-excellent rings are conjectured to be the base rings for which the problem of resolution of singularities can be solved; showed this in characteristic 0, but ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zariski–Riemann Space
In algebraic geometry, a Zariski–Riemann space or Zariski space of a subring ''k'' of a field ''K'' is a locally ringed space whose points are valuation rings containing ''k'' and contained in ''K''. They generalize the Riemann surface of a complex curve. Zariski–Riemann spaces were introduced by who (rather confusingly) called them Riemann manifolds or Riemann surfaces. They were named Zariski–Riemann spaces after Oscar Zariski and Bernhard Riemann by who used them to show that algebraic varieties can be embedded in complete ones. Local uniformization (proved in characteristic 0 by Zariski) can be interpreted as saying that the Zariski–Riemann space of a variety is nonsingular in some sense, so is a sort of rather weak resolution of singularities. This does not solve the problem of resolution of singularities because in dimensions greater than 1 the Zariski–Riemann space is not locally affine and in particular is not a scheme. Definition The Zariski–Riemann ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excellent Scheme
In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the problem of finding a natural class of "well-behaved" rings containing most of the rings that occur in number theory and algebraic geometry. At one time it seemed that the class of Noetherian rings might be an answer to this problem, but Masayoshi Nagata and others found several strange counterexamples showing that in general Noetherian rings need not be well-behaved: for example, a normal Noetherian local ring need not be analytically normal. The class of excellent rings was defined by Alexander Grothendieck (1965) as a candidate for such a class of well-behaved rings. Quasi-excellent rings are conjectured to be the base rings for which the problem of resolution of singularities can be solved; showed this in characteristic 0, but ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytically Normal
In algebra, an analytically normal ring is a local ring whose completion is a normal ring, in other words a domain that is integrally closed in its quotient field. proved that if a local ring of an algebraic variety is normal, then it is analytically normal, which is in some sense a variation of Zariski's main theorem. gave an example of a normal Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ... local ring that is analytically reducible and therefore not analytically normal. References * * * * * Commutative algebra {{commutative-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Uniformization
In algebraic geometry, local uniformization is a weak form of resolution of singularities, stating that a variety can be desingularized near any valuation, or in other words that the Zariski–Riemann space In algebraic geometry, a Zariski–Riemann space or Zariski space of a subring ''k'' of a field ''K'' is a locally ringed space whose points are valuation rings containing ''k'' and contained in ''K''. They generalize the Riemann surface of a c ... of the array is in some sense non-singular. Local uniformization was introduced by , who separated the problem of resolving the singularities of a variety into the problem of local uniformization and the problem of combining the local uniformizations into a global desingularization. Local uniformization of a variety at a valuation of its function field means finding a projective model of the variety such that the center of the valuation is non-singular. It is weaker than resolution of singularities: if there is a resolution of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excellent Ring
In commutative algebra, a quasi-excellent ring is a Noetherian ring, Noetherian commutative ring that behaves well with respect to the operation of completion of a ring, completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the problem of finding a natural class of "well-behaved" ring (mathematics), rings containing most of the rings that occur in number theory and algebraic geometry. At one time it seemed that the class of Noetherian rings might be an answer to this problem, but Masayoshi Nagata and others found several strange counterexamples showing that in general Noetherian rings need not be well-behaved: for example, a normal Noetherian local ring need not be analytically normal. The class of excellent rings was defined by Alexander Grothendieck (1965) as a candidate for such a class of well-behaved rings. Quasi-excellent rings are conjectured to be the base rings for which the problem of resolution of singularities ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
