algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, a smooth scheme over a field is a scheme 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 relat ...

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 or sounder, a group of boar, see List of animal names * Singular (band), a Thai jazz pop duo *'' Singula ...

points. A special case is the notion of a smooth variety 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 formali ...

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 ''g_i'' 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 coo ...

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 In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another 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 In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point ''P'' on an algebraic variety ''V'' (and more generally). It does not use differential calculus, being based directly on abstract algebra, an ...

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, 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 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. ...

. 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 In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

the morphism ''X'' → Spec ''k'' is smooth.

Properties

A smooth scheme over a field is

regular Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...

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 Reduction, reduced, or reduce may refer to: Science and technology Chemistry * Reduction (chemistry), part of a reduction-oxidation (redox) reaction in which atoms have their oxidation state changed. ** Organic redox reaction, a redox reacti ...

. Define a variety over a field ''k'' to be an

integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...

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 number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s, 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 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 ...

, 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

, 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 refer ...

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 eve ...

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. This m ...

of ''X''. The

tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...

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 ''X_E'' := ''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 no multiple roots in any field extension ''F/k''. * Every irreducible polynomial over ''k'' has non-zero f ...

''k'', a scheme ''X'' is smooth over ''k'' if and only if ''X'' is locally of finite type over ''k'' and ''X'' is

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 In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...

) 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 In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form : y^2 - a^2 x^3 = 0 (with ) in some Cartesian coordinate system. Solving for leads to the ''explicit form'' : y = \ ...

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, \mathbf_k(V) of k-dimensional subspaces of a vector space V, usually with singular points. Like the Grassmannian, it is a kind of moduli space, whose elements sati ...

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 was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

, edition=2nd , series=Cambridge Studies in Advanced Mathematics , isbn=978-0-521-36764-6 , year=1989 , mr=1011461