Last modified date: <%+ tp.file.last_modified_date() %>

• Tags
  • #todo/untagged
• Refs:
  • #todo/add-references
• Links:
  • scheme

smooth scheme

For general schemes over a field

A scheme $X\in {\mathsf{Sch}}_{/ {k}} $ is smooth over \(k\) iff

• The base change \(X_{ { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu } }\) is regular
• The base changes \(X_{L}\) are regular for every (arbitrary) extension \(L/k\)
• The base changes \(X_{L}\) are regular for every finite extension \(L/k\).
• There exists a perfect extension \(L/k\) such that the base change \(X_{L}\) is regular.
• \(X\to \operatorname{Spec}k\) satisfies the infinitesimal criterion for smoothness.

Note that smoothness over a field implies regularity, but the converse is false if \(k\) is not perfect.

Geometric defiition, for finite type schemes

Let $X\in {\mathsf{Sch}}^{\mathrm{ft}}_{/ {k}} $, then there is a closed immersion \(X\hookrightarrow{\mathbf{A}}^N_{/k}\) and \(X = V(f_1, \cdots, f_m)\) for some \(f_i\in k[x_1, \cdots, x_{N}]\) . Say \(X\) is smooth of dimension \(n\) iff - There exist neighborhoods of each point \(U_x\ni x\) with \(\dim U_x \geq n\), and - \(\operatorname{rank}Df \geq N-n\) everywhere, where \(Df = {\left[ { {\frac{\partial f_i}{\partial x_j}\,}} \right]}\) is the matrix of partial derivatives. - Equivalently, the dimension of the Zariski tangent space at every point is equal to \(n\). - At singular points, the dimension of the tangent space increases.

For derived schemes

Recall that if \(X\) and \(S\) are smooth, then \(\pi : X \to S\) is smooth if and only the differential is everywhere surjective.

Smoothable schemes

Notes

• If a variety $X\in {\mathsf{Var}}_{/ {k}} $ is smooth then any coherent sheaf has a finite resolution by locally free sheaves of finite type and the subcategory of perfect complexes coincides with the entire bounded derived category \(\mathbf{D} {{\mathsf{Coh}}(X)} ^b\).