Smooth scheme
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
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 gi and all those minors is the whole polynomial ring.
In geometric terms, the matrix of derivatives (∂gi/∂xj) at a point p in X gives a linear map Fn → Fr, 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
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
Define a variety over a field k to be an
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
Smoothness is a
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 x0d + ... + xnd = 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 Fp(t) for a prime number p, and let E = Fp(t1/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).
References
- D. Gaitsgory's notes on flatness and smoothness at http://www.math.harvard.edu/~gaitsgde/Schemes_2009/BR/SmoothMaps.pdf
- MR 0463157
- Matsumura, Hideyuki (1989), Commutative Ring Theory, Cambridge Studies in Advanced Mathematics (2nd ed.), MR 1011461
See also
- Étale morphism
- Dimension of an algebraic variety
- Glossary of scheme theory
- Smooth completion