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| In [[mathematics]], an '''algebraic space''' is a generalization of the [[scheme (mathematics)|schemes]] of [[algebraic geometry]] introduced by {{harvs|txt|authorlink=Michael Artin|last=Artin|year1=1969|year2=1971}} for use in [[deformation theory]]. Intuitively, an algebraic space is something that looks locally like an [[affine scheme]] in the [[etale topology]], in the same way that a scheme is something that looks locally like an affine scheme in the [[Zariski topology]]. The resulting [[category (mathematics)|category]] of algebraic spaces extends the category of schemes and allows one to carry out several natural constructions that are used in the construction of [[moduli space]]s but are not always possible in the smaller category of schemes, such as taking the quotient of a [[free action]] by a [[finite group]].
| | They call me Emilia. Managing people is what I do and the wage has been really satisfying. Years in the past we moved to North Dakota. What I adore performing is to gather badges but I've been taking on new issues lately.<br><br>Stop by my weblog: [http://nfldev.com/index.php?do=/profile-12302/info/ http://nfldev.com/] |
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| ==Definition==
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| There are two common ways to define algebraic spaces: they can be defined as either quotients of schemes by etale equivalence relations, or as sheaves on a big etale site that are locally isomorphic to schemes. These two definitions are essentially equivalent.
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| ===Algebraic spaces as quotients of schemes===
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| An '''algebraic space''' ''X'' comprises a scheme ''U'' and a closed subscheme ''R'' ⊂ ''U'' × ''U'' satisfying the following two conditions:
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| :1. ''R'' is an [[equivalence relation]] as a subset of ''U'' × ''U''
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| :2. The projections ''p<sub>i</sub>'': ''R'' → ''U'' onto each factor are [[étale morphism|étale map]]s.
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| Some authors, such as Knutson, add an extra condition that an algebraic space has to be quasi-[[Quasi-separated|separated]], meaning that the diagonal map is quasi-compact.
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| One can always assume that ''R'' and ''U'' are [[affine scheme]]s. Doing so means that the theory of algebraic spaces is not dependent on the full theory of schemes, and can indeed be used as a (more general) replacement of that theory.
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| If ''R'' is the trivial equivalence relation over each connected component of ''U'' (i.e. for all ''x'', ''y'' belonging to the same connected component of ''U'', we have ''xRy'' if and only if ''x''=''y''), then the algebraic space will be a scheme in the usual sense. Since a general algebraic space does not satisfy this requirement, it allows a single connected component of ''U'' to [[covering space|cover]] ''X'' with many "sheets". The point set underlying the algebraic space ''X'' is then given by |''U''| / |''R''| as a set of [[equivalence class]]es.
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| Let ''Y'' be an algebraic space defined by an equivalence relation ''S'' ⊂ ''V'' × ''V''. The set Hom(''Y'', ''X'') of '''morphisms of algebraic spaces''' is then defined by the condition that it makes the [[descent (category theory)|descent sequence]]
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| :<math>\mathrm{Hom}(Y, X) \rightarrow \mathrm{Hom}(V, X) {{{} \atop \longrightarrow}\atop{\longrightarrow \atop {}}} \mathrm{Hom}(S, X)</math>
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| exact (this definition is motivated by a descent theorem of [[Grothendieck]] for surjective étale maps of affine schemes). With these definitions, the algebraic spaces form a [[category (mathematics)|category]].
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| Let ''U'' be an affine scheme over a field ''k'' defined by a system of polynomials '''''g'''''('''''x'''''), '''''x''''' = (''x''<sub>1</sub>, …, ''x<sub>n</sub>''), let
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| :<math>k\{x_1, \ldots, x_n\}\ </math>
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| denote the [[ring (mathematics)|ring]] of [[algebraic function]]s in '''''x''''' over ''k'', and let ''X'' = {''R'' ⊂ ''U'' × ''U''} be an algebraic space.
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| The appropriate '''stalks''' ''Õ<sub>X</sub>''<sub>, ''x''</sub> on ''X'' are then defined to be the [[local ring]]s of algebraic functions defined by ''Õ<sub>U</sub>''<sub>, ''u''</sub>, where ''u'' ∈ ''U'' is a point lying over ''x'' and ''Õ<sub>U</sub>''<sub>, ''u''</sub> is the local ring corresponding to ''u'' of the ring
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| :''k''{''x''<sub>1</sub>, …, ''x<sub>n</sub>''} / ('''''g''''')
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| of algebraic functions on ''U''.
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| A point on an algebraic space is said to be '''smooth''' if ''Õ<sub>X</sub>''<sub>, ''x''</sub> ≅ ''k''{''z''<sub>1</sub>, …, ''z<sub>d</sub>''} for some [[indeterminate (variable)|indeterminate]]s ''z''<sub>1</sub>, …, ''z<sub>d</sub>''. The dimension of ''X'' at ''x'' is then just defined to be ''d''.
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| A morphism ''f'': ''Y'' → ''X'' of algebraic spaces is said to be '''étale''' at ''y'' ∈ ''Y'' (where ''x'' = ''f''(''y'')) if the induced map on stalks
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| :''Õ<sub>X</sub>''<sub>, ''x''</sub> → ''Õ<sub>Y</sub>''<sub>, ''y''</sub>
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| is an isomorphism.
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| The '''structure sheaf''' ''O<sub>X</sub>'' on the algebraic space ''X'' is defined by associating the ring of functions ''O''(''V'') on ''V'' (defined by étale maps from ''V'' to the affine line '''A'''<sup>1</sup> in the sense just defined) to any algebraic space ''V'' which is étale over ''X''.
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| ===Algebraic spaces as sheaves===
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| An algebraic space ''X'' over a scheme ''S'' can also be defined as a sheaf over the big etale site of ''S'' such that the diagonal map from ''X'' to ''X''×<sub>''S''</sub>''X'' is representable by schemes and such that there is a surjective etale morphism from some scheme to ''X''. Here a morphism of sheaves from ''X'' to ''Y'' is called representable by schemes if the pullback of any morphism of a scheme to ''Y'' is also a scheme. Some authors, such as Knutson, add an extra condition that an algebraic space has to be [[quasi-separated]], meaning that the diagonal map is quasi-compact.
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| ==Algebraic spaces and schemes==
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| Algebraic spaces are similar to schemes, and much of the theory of schemes extends to algebraic spaces. For example, most properties of morphisms of schemes also apply to algebraic spaces, one can define cohomology of quasicoherent sheaves, this has the usual finiteness properties for proper morphisms, and so on.
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| Schemes are given by gluing together affine schemes using the Zariski topology, while algebraic spaces are given by gluing together affine schemes using the finer etale topology. Alternatively one can think of schemes as being locally isomorphic to affine schemes in the Zariski topology, while algebraic spaces are locally isomorphic to affine schemes in the etale topology. If one uses the even finer classical topology over the complex numbers, one gets analytic spaces rather than algebraic spaces.
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| * Proper algebraic spaces over a field of dimension one (curves) are schemes.
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| * Non-singular proper algebraic spaces over a field of dimension two (smooth surfaces) are schemes.
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| * [[Quasi-separated]] group objects in the category of algebraic spaces over a field are schemes, though there are non quasi-separated group objects that are not schemes.
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| * Commutative-group objects in the category of algebraic spaces over an arbitrary scheme which are proper, locally finite presentation, flat, and cohomologically flat in dimension 0 are schemes.
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| * Not every singular algebraic surface is a scheme.
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| * [[Hironaka's example]] can be used to give a non-singular 3-dimensional proper algebraic space that is not a scheme, given by the quotient of a scheme by a group of order 2 acting freely. This illustrates one difference between schemes and algebraic spaces: the quotient of an algebraic space by a discrete group acting freely is an algebraic space, but the quotient of a scheme by a discrete group acting freely need not be a scheme (even if the group is finite).
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| * Every algebraic space contains a dense open affine subscheme, and the complement of such a subscheme always has [[codimension]] ≥ 1. Thus algebraic spaces are in a sense "close" to affine schemes.
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| *The quotient of the complex numbers by a lattice is an algebraic space, but is not an elliptic curve, even though the corresponding analytic space is an elliptic curve (or more precisely is the image of an elliptic curve under the functor from complex algebraic spaces to analytic spaces). In fact this algebraic space quotient is not a scheme, is not complete, and is not even quasi-separated. This shows that although the quotient of an algebraic space by an infinite discrete group is an algebraic space, it can have strange properties and might not be the algebraic space one was "expecting". Similar examples are given by the quotient of the complex affine line by the integers, or the quotient of the complex affine line minus the origin by the powers of some number: again the corresponding analytic space is a variety, but the algebraic space is not.
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| ==Algebraic spaces and analytic spaces==
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| Algebraic spaces over the complex numbers are closely related to analytic spaces and [[Moishezon manifold]]s.
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| Roughly speaking, the difference between complex algebraic spaces and analytic spaces is that complex algebraic spaces are formed by gluing affine pieces together using the etale topology, while analytic spaces are formed by gluing with the classical topology. In particular there is a functor from complex algebraic spaces of finite type to analytic spaces. Hopf manifolds give examples of analytic surfaces that do not come from a proper algebraic space (though one can construct non-proper and non-separated algebraic spaces whose analytic space is the Hopf surface). It is also possible for different algebraic spaces to correspond to the same analytic space: for example, an elliptic curve and the quotient of '''C''' by the corresponding lattice are not isomorphic as algebraic spaces but the corresponding analytic spaces are isomorphic.
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| Artin showed that proper algebraic spaces over the complex numbers are more or less the same as Moishezon spaces.
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| ==See also==
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| * [[Algebraic stack]]
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| ==References==
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| *{{Citation | last1=Artin | first1=Michael | author1-link=Michael Artin | editor1-last=Abhyankar | editor1-first=Shreeram Shankar | title=Algebraic geometry: papers presented at the Bombay Colloquium, 1968 | url=http://books.google.com/books/?id=3evuAAAAMAAJ | publisher=[[Oxford University Press]] | series=of Tata Institute of Fundamental Research studies in mathematics, | mr=0262237 | year=1969 | volume=4 | chapter=The implicit function theorem in algebraic geometry | pages=13–34}}
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| *{{Citation | last1=Artin | first1=Michael | author1-link=Michael Artin | title=Algebraic spaces | url=http://books.google.com/books/about/Algebraic_spaces.html?id=Gl7vAAAAMAAJ | publisher=Yale University Press | series=Yale Mathematical Monographs | isbn=978-0-300-01396-2 | mr=0407012 | year=1971 | volume=3}}
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| *{{Citation | last1=Knutson | first1=Donald | title=Algebraic spaces | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-05496-2 | doi=10.1007/BFb0059750 | mr=0302647 | year=1971 | volume=203}}
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| ==External links==
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| * {{springer|id=a/a011630|title=Algebraic space|first=V.I.|last= Danilov}}
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| *[http://stacks.math.columbia.edu/chapter/44 Algebraic space] in the stacks project
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| [[Category:Algebraic geometry]]
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They call me Emilia. Managing people is what I do and the wage has been really satisfying. Years in the past we moved to North Dakota. What I adore performing is to gather badges but I've been taking on new issues lately.
Stop by my weblog: http://nfldev.com/