# Smooth scheme > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Smooth_scheme > Markdown URL: https://mediated.wiki/source/Smooth_scheme.md > Source: https://en.wikipedia.org/wiki/Smooth_scheme > Source revision: 1350326771 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) Concept in algebraic geometry In [algebraic geometry](/source/Algebraic_geometry), a **smooth scheme** over a [field](/source/Field_(mathematics)) is a [scheme](/source/Scheme_(mathematics)) which is well approximated by [affine space](/source/Affine_space) near any point. Smoothness is one way of making precise the notion of a scheme with no [singular](/source/Singular_point_of_an_algebraic_variety) points. A special case is the notion of a smooth [variety](/source/Algebraic_variety) over a field. Smooth schemes play the role in algebraic geometry of [manifolds](/source/Manifold) in [topology](/source/Topology). ## Definition First, let *X* be an affine scheme of [finite type](/source/Glossary_of_scheme_theory#finite) over a field *k*. Equivalently, *X* has a [closed immersion](/source/Closed_immersion) 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](/source/Dimension_of_an_algebraic_variety) 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*.[1] (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](/source/Minor_(linear_algebra)) of the matrix of derivatives are zero is the empty set. Equivalently, the [ideal](/source/Ideal_(ring_theory)) in the [polynomial ring](/source/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](/source/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](/source/Singular_point_of_an_algebraic_variety), 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](/source/Glossary_of_scheme_theory#locally_of_finite_type). There is a more general notion of a [smooth morphism](/source/Smooth_morphism) 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](/source/If_and_only_if) the morphism *X* → Spec *k* is smooth. ## Properties A smooth scheme over a field is [regular](/source/Glossary_of_scheme_theory#regular) and hence [normal](/source/Normal_scheme). In particular, a smooth scheme over a field is [reduced](/source/Glossary_of_scheme_theory#reduced). Define a **variety** over a field *k* to be an [integral](/source/Glossary_of_scheme_theory#integral) [separated](/source/Glossary_of_scheme_theory#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](/source/Complex_number), the space *X*(**C**) of complex points of *X* is a [complex manifold](/source/Complex_manifold), 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](/source/Differentiable_manifold), possibly empty. For any scheme *X* that is locally of finite type over a field *k*, there is a [coherent sheaf](/source/Coherent_sheaf) Ω1 of [differentials](/source/Kahler_differentials) on *X*. The scheme *X* is smooth over *k* if and only if Ω1 is a [vector bundle](/source/Vector_bundle) of rank equal to the dimension of *X* near each point.[2] In that case, Ω1 is called the [cotangent bundle](/source/Cotangent_bundle) of *X*. The [tangent bundle](/source/Tangent_bundle) of a smooth scheme over *k* can be defined as the dual bundle, *TX* = (Ω1)*. Smoothness is a [geometric property](/source/Geometric_property), meaning that for any [field extension](/source/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](/source/Perfect_field) *k*, a scheme *X* is smooth over *k* if and only if *X* is locally of finite type over *k* and *X* is [regular](/source/Regular_scheme). ## 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](/source/Algebraically_closed_field)) is generically smooth.[3] ## Examples - Affine space and [projective space](/source/Projective_space) are smooth schemes over a field *k*. - An example of a smooth [hypersurface](/source/Hypersurface) in projective space **P***n* over *k* is the [Fermat hypersurface](/source/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](/source/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](/source/Separable_extension) 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](/source/Schubert_variety) are in general not smooth. ## Notes 1. **[^](#cite_ref-1)** The definition of smoothness used in this article is equivalent to Grothendieck's definition of smoothness by Theorems 30.2 and Theorem 30.3 in: Matsumura, Commutative Ring Theory (1989). 1. **[^](#cite_ref-2)** Theorem 30.3, Matsumura, Commutative Ring Theory (1989). 1. **[^](#cite_ref-3)** Lemma 1 in section 28 and Corollary to Theorem 30.5, Matsumura, Commutative Ring Theory (1989). ## References - [D. Gaitsgory](/source/Dennis_Gaitsgory)'s notes on flatness and smoothness at [http://www.math.harvard.edu/~gaitsgde/Schemes_2009/BR/SmoothMaps.pdf](http://www.math.harvard.edu/~gaitsgde/Schemes_2009/BR/SmoothMaps.pdf) - [Hartshorne, Robin](/source/Robin_Hartshorne) (1977), *[Algebraic Geometry](/source/Algebraic_Geometry_(book))*, [Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics), vol. 52, New York: Springer-Verlag, [ISBN](/source/ISBN_(identifier)) [978-0-387-90244-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-90244-9), [MR](/source/MR_(identifier)) [0463157](https://mathscinet.ams.org/mathscinet-getitem?mr=0463157) - [Matsumura, Hideyuki](/source/Hideyuki_Matsumura) (1989), *Commutative Ring Theory*, Cambridge Studies in Advanced Mathematics (2nd ed.), [Cambridge University Press](/source/Cambridge_University_Press), [ISBN](/source/ISBN_(identifier)) [978-0-521-36764-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-36764-6), [MR](/source/MR_(identifier)) [1011461](https://mathscinet.ams.org/mathscinet-getitem?mr=1011461) ## See also - [Étale morphism](/source/%C3%89tale_morphism) - [Dimension of an algebraic variety](/source/Dimension_of_an_algebraic_variety) - [Glossary of scheme theory](/source/Glossary_of_scheme_theory) - [Smooth completion](/source/Smooth_completion) --- Adapted from the Wikipedia article [Smooth scheme](https://en.wikipedia.org/wiki/Smooth_scheme) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Smooth_scheme?action=history)). 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