Smooth scheme

Last updated September 15, 2024

In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smooth variety over a field. Smooth schemes play the role in algebraic geometry of manifolds in topology.

Definition

First, let X be an affine scheme of finite type over a field k. Equivalently, X has a closed immersion into affine space Aⁿ over k for some natural number n. Then X is the closed subscheme defined by some equations g₁ = 0, ..., g_r = 0, where each g_i is in the polynomial ring k[x₁,..., x_n]. The affine scheme X is smooth of dimension m over k if X has dimension 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 of the matrix of derivatives are zero is the empty set. Equivalently, the ideal 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 Fⁿ → 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, 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 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. In particular, a smooth scheme over a field is reduced.

Define a variety over a field k to be an integral 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, 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, possibly empty.

For any scheme X that is locally of finite type over a field k, there is a coherent sheaf Ω¹ of differentials on X. The scheme X is smooth over k if and only if Ω¹ is a vector bundle of rank equal to the dimension of X near each point.^[2] In that case, Ω¹ is called the cotangent bundle of X. The tangent bundle of a smooth scheme over k can be defined as the dual bundle, TX = (Ω¹)^*.

Smoothness is a geometric property, 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 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.^[3]

Examples

Affine space and projective space are smooth schemes over a field k.
An example of a smooth hypersurface in projective space Pⁿ over k is the Fermat hypersurface x₀^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 x² = 0 in the affine line A¹ over k.
An example of a singular (non-smooth) variety over k is the cuspidal cubic curve x² = y³ in the affine plane A², 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 are in general not smooth.

Notes

↑ 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).
↑ Theorem 30.3, Matsumura, Commutative Ring Theory (1989).
↑ Lemma 1 in section 28 and Corollary to Theorem 30.5, Matsumura, Commutative Ring Theory (1989).

Related Research Articles

In commutative algebra, the prime spectrum of a commutative ring R is the set of all prime ideals of R, and is usually denoted by $; in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings .$

In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring a topological space.

Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers $; and p -adic integers .$

In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for research on other topics in algebraic geometry and number theory.

In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.

Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.

In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings, as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.

In algebraic geometry, an affine algebraic set is the set of the common zeros over an algebraically closed field $k$ of some family of polynomials in the polynomial ring $An affine variety or affine algebraic variety, is an affine algebraic set such that the ideal generated by the defining polynomials is prime.$

In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space $over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of .$

In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities and allowing "varieties" defined over any commutative ring.

In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central role in commutative algebra: they form a very broad class, and yet they are well understood in many ways.

In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be any Noetherian local ring with unique maximal ideal m, and suppose a₁, ..., a_n is a minimal set of generators of m. Then Krull's principal ideal theorem implies that n ≥ dim A, and A is regular whenever n = dim A.

In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point P on an algebraic variety V. It does not use differential calculus, being based directly on abstract algebra, and in the most concrete cases just the theory of a system of linear equations.

In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces.

In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods from calculus and geometry over the complex numbers to contexts where such methods are not available.

In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively, 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 étale 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 étale topology.

In algebraic geometry, an algebraic variety or scheme X is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety X (understood to be irreducible) is normal if and only if the ring O(X) of regular functions on X is an integrally closed domain. A variety X over a field is normal if and only if every finite birational morphism from any variety Y to X is an isomorphism.

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the existence of sections of line bundles or of more general coherent sheaves; such sections can be viewed as generalized functions. Cohomology provides computable tools for producing sections, or explaining why they do not exist. It also provides invariants to distinguish one algebraic variety from another.

This is a glossary of algebraic geometry.

References

D. Gaitsgory's notes on flatness and smoothness at http://www.math.harvard.edu/~gaitsgde/Schemes_2009/BR/SmoothMaps.pdf
Hartshorne, Robin (1977), Algebraic Geometry , Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
Matsumura, Hideyuki (1989), Commutative Ring Theory, Cambridge Studies in Advanced Mathematics (2nd ed.), Cambridge University Press, ISBN 978-0-521-36764-6, MR 1011461