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In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, a smooth scheme over a field is a
scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
which is well approximated by
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
near any point. Smoothness is one way of making precise the notion of a scheme with no
singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular homology * SINGULAR, an open source Computer Algebra System (CAS) * Singular or sounder, a group of boar ...
points. A special case is the notion of a smooth
variety Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film), ...
over a field. Smooth schemes play the role in algebraic geometry of
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s in topology.


Definition

First, let ''X'' be an affine scheme of finite type over a field ''k''. Equivalently, ''X'' has a
closed immersion In algebraic geometry, a closed immersion of schemes is a morphism of schemes f: Z \to X that identifies ''Z'' as a closed subset of ''X'' such that locally, regular functions on ''Z'' can be extended to ''X''. The latter condition can be formaliz ...
into affine space ''An'' over ''k'' for some natural number ''n''. Then ''X'' is the closed subscheme defined by some equations ''g''1 = 0, ..., ''g''''r'' = 0, where each ''gi'' is in the polynomial ring ''k'' 'x''1,..., ''x''''n'' The affine scheme ''X'' is smooth of dimension ''m'' over ''k'' if ''X'' has
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
at least ''m'' in a neighborhood of each point, and the matrix of derivatives (∂''g''''i''/∂''x''''j'') has rank at least ''n''−''m'' everywhere on ''X''. (It follows that ''X'' has dimension equal to ''m'' in a neighborhood of each point.) Smoothness is independent of the choice of immersion of ''X'' into affine space. The condition on the matrix of derivatives is understood to mean that the closed subset of ''X'' where all (''n''−''m'') × (''n'' − ''m'') minors of the matrix of derivatives are zero is the empty set. Equivalently, the
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
in the polynomial ring generated by all ''g''''i'' and all those minors is the whole polynomial ring. In geometric terms, the matrix of derivatives (∂''g''''i''/∂''x''''j'') at a point ''p'' in ''X'' gives a linear map ''F''''n'' → ''F''''r'', where ''F'' is the residue field of ''p''. The kernel of this map is called the Zariski tangent space of ''X'' at ''p''. Smoothness of ''X'' means that the dimension of the Zariski tangent space is equal to the dimension of ''X'' near each point; at a
singular point Singularity or singular point may refer to: Science, technology, and mathematics Mathematics * Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiab ...
, the Zariski tangent space would be bigger. More generally, a scheme ''X'' over a field ''k'' is smooth over ''k'' if each point of ''X'' has an open neighborhood which is a smooth affine scheme of some dimension over ''k''. In particular, a smooth scheme over ''k'' is locally of finite type. There is a more general notion of a
smooth morphism In algebraic geometry, a morphism f:X \to S between schemes is said to be smooth if *(i) it is locally of finite presentation *(ii) it is flat, and *(iii) for every geometric point \overline \to S the fiber X_ = X \times_S is regular. (iii) mea ...
of schemes, which is roughly a morphism with smooth fibers. In particular, a scheme ''X'' is smooth over a field ''k'' if and only if the morphism ''X'' → Spec ''k'' is smooth.


Properties

A smooth scheme over a field is regular and hence
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
. In particular, a smooth scheme over a field is reduced. Define a variety over a field ''k'' to be an
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
separated scheme of finite type over ''k''. Then any smooth separated scheme of finite type over ''k'' is a finite disjoint union of smooth varieties over ''k''. For a smooth variety ''X'' over the complex numbers, the space ''X''(C) of complex points of ''X'' is a
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a ...
, using the classical (Euclidean) topology. Likewise, for a smooth variety ''X'' over the real numbers, the space ''X''(R) of real points is a real
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, possibly empty. For any scheme ''X'' that is locally of finite type over a field ''k'', there is a
coherent sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with ref ...
Ω1 of differentials on ''X''. The scheme ''X'' is smooth over ''k'' if and only if Ω1 is a
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
of rank equal to the dimension of ''X'' near each point. In that case, Ω1 is called the
cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. Th ...
of ''X''. The
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
of a smooth scheme over ''k'' can be defined as the dual bundle, ''TX'' = (Ω1)*. Smoothness is a
geometric property This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. ...
, meaning that for any field extension ''E'' of ''k'', a scheme ''X'' is smooth over ''k'' if and only if the scheme ''XE'' := ''X'' ×Spec ''k'' Spec ''E'' is smooth over ''E''. For a
perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds: * Every irreducible polynomial over ''k'' has distinct roots. * Every irreducible polynomial over ''k'' is separable. * Every finite extension of ''k' ...
''k'', a scheme ''X'' is smooth over ''k'' if and only if ''X'' is locally of finite type over ''k'' and ''X'' is regular.


Generic smoothness

A scheme ''X'' is said to be generically smooth of dimension ''n'' over ''k'' if ''X'' contains an open dense subset that is smooth of dimension ''n'' over ''k''. Every variety over a perfect field (in particular an algebraically closed field) is generically smooth.Lemma 1 in section 28 and Corollary to Theorem 30.5, Matsumura, Commutative Ring Theory (1989).


Examples

*Affine space and
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
are smooth schemes over a field ''k''. *An example of a smooth
hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...
in projective space Pn over ''k'' is the Fermat hypersurface ''x''0''d'' + ... + ''x''''n''''d'' = 0, for any positive integer ''d'' that is invertible in ''k''. *An example of a singular (non-smooth) scheme over a field ''k'' is the closed subscheme ''x''2 = 0 in the affine line ''A''1 over ''k''. *An example of a singular (non-smooth) variety over ''k'' is the cuspidal cubic curve ''x''2 = ''y''3 in the affine plane ''A''2, which is smooth outside the origin (''x'',''y'') = (0,0). *A 0-dimensional variety ''X'' over a field ''k'' is of the form ''X'' = Spec ''E'', where ''E'' is a finite extension field of ''k''. The variety ''X'' is smooth over ''k'' if and only if ''E'' is a separable extension of ''k''. Thus, if ''E'' is not separable over ''k'', then ''X'' is a regular scheme but is not smooth over ''k''. For example, let ''k'' be the field of rational functions F''p''(''t'') for a prime number ''p'', and let ''E'' = F''p''(''t''1/''p''); then Spec ''E'' is a variety of dimension 0 over ''k'' which is a regular scheme, but not smooth over ''k''. *
Schubert varieties In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points. Like a Grassmannian, it is a kind of moduli space, whose points correspond to certain kinds of subspaces ''V'', specified using linea ...
are in general not smooth.


Notes


References

* D. Gaitsgory's notes on flatness and smoothness at http://www.math.harvard.edu/~gaitsgde/Schemes_2009/BR/SmoothMaps.pdf * * {{Citation , last1=Matsumura , first1=Hideyuki , title=Commutative Ring Theory , publisher=
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pr ...
, edition=2nd , series=Cambridge Studies in Advanced Mathematics , isbn=978-0-521-36764-6 , year=1989 , mr=1011461


See also

* Étale morphism *
Dimension of an algebraic variety In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways. Some of these definitions are of geometric nature, while some other are purely algebraic and rely on commut ...
*
Glossary of scheme theory This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometr ...
*
Smooth completion In algebraic geometry, the smooth completion (or smooth compactification) of a smooth affine algebraic curve ''X'' is a complete smooth algebraic curve which contains ''X'' as an open subset. Smooth completions exist and are unique over a perfect ...
*