Open subscheme/Affineness/Varying with base/Introduction/Section

We consider a base scheme ${\displaystyle {}B}$ and a morphism

${\displaystyle Z\longrightarrow B}$

together with an open subscheme ${\displaystyle {}W\subseteq Z}$. For every base point ${\displaystyle {}b\in B}$ we get the open subset

${\displaystyle {}W_{b}\subseteq Z_{b}\,}$

inside the fiber ${\displaystyle {}Z_{b}}$. It is a natural question to ask how properties of ${\displaystyle {}W_{b}}$ vary with ${\displaystyle {}b}$. In particular, we may ask how the cohomological dimension of ${\displaystyle {}W_{b}}$ varies and how the affineness (the cohomological dimension of a scheme ${\displaystyle {}X}$ is the maximal number ${\displaystyle {}i}$ such that ${\displaystyle {}H^{i}(X,{\mathcal {F}})\neq 0}$ for some quasicoherent sheaf ${\displaystyle {}{\mathcal {F}}}$. A noetherian scheme is affine if and only if its cohomological dimension is ${\displaystyle {}0}$. Tight closure can be characterized by the cohomological dimension of torsors.). may vary.

In the algebraic setting, we have a commutative ${\displaystyle {}K}$-algebra ${\displaystyle {}D}$, a commutative ${\displaystyle {}D}$-algebra ${\displaystyle {}S}$ and an ideal ${\displaystyle {}{\mathfrak {a}}\subseteq S}$ (so ${\displaystyle {}B=\operatorname {Spec} {\left(D\right)}}$, ${\displaystyle {}Z=\operatorname {Spec} {\left(S\right)}}$ and ${\displaystyle {}W=D({\mathfrak {a}})}$) which defines for every prime ideal ${\displaystyle {}{\mathfrak {p}}\in \operatorname {Spec} {\left(D\right)}}$ the extended ideal ${\displaystyle {}{\mathfrak {a}}_{\mathfrak {p}}}$ in ${\displaystyle {}S\otimes _{D}\kappa ({\mathfrak {p}})}$. Then in this situation, ${\displaystyle {}D({\mathfrak {a}}_{\mathfrak {p}})\subseteq \operatorname {Spec} {\left(S\otimes _{D}\kappa ({\mathfrak {p}})\right)}}$ is the fiber over ${\displaystyle {}{\mathfrak {p}}}$.

This question is already interesting when ${\displaystyle {}B=\operatorname {Spec} {\left(D\right)}}$ is an affine one-dimensional integral scheme, in particular in the following two situations.

1. ${\displaystyle {}B=\operatorname {Spec} {\left(\mathbb {Z} \right)}}$. Then we speak of an arithmetic deformation and want to know how affineness varies with the characteristic and what the relation is to characteristic zero.
2. ${\displaystyle {}B={\mathbb {A} }_{K}^{1}=\operatorname {Spec} {\left(K[t]\right)}}$, where ${\displaystyle {}K}$ is a field. Then we speak of a geometric deformation and want to know how affineness varies with the parameter ${\displaystyle {}t}$, in particular how the behavior over the special points where the residue class field is algebraic over ${\displaystyle {}K}$ is related to the behavior over the generic point.

It is fairly easy to show that if the open subset in the generic fiber is affine, then also the open subsets are affine for almost all special points.