Stationary type (stability theory)

In a stable theory $$T$$, a type $$p$$ over a set $$C$$ is said to be stationary if $$p$$ has a unique non-forking extension to every $$D \supseteq C$$, or equivalently, to $$\mathbb{U}$$. In other words, $$p$$ is stationary if for every $$D \supset C$$, if $$a \downarrow_D C$$ and $$a' \downarrow_D C$$ and $$a, a' \models p$$, then $$a \equiv_D a'$$.

An equivalent condition is that $$\operatorname{tp}(a/C)$$ is stationary if and only if there is some global $$C$$-definable type $$p$$ such that $$a \models p | C$$.

Types over models are always stationary. More generally, if C is a set such that C = acleq(C) (or dcleq(C) = acleq(C)), then types over $$C$$ are stationary. That is, strong types are stationary.

In the setting of ACF, types correspond to irreducible varieties. Stationary types are exactly the types corresponding to geometrically irreducible varieties. Over algebraically closed sets, irreducible varieties are already geometrically irreducible.

Two stationary types are said to be parallel if they have the same nonforking global extension. Parallelism is an equivalence relation, the equivalence relation on stationary types generated by the relation "$$p$$ is a nonforking extension of $$q$$."

In the setting of ACF, two types are parallel if they have the same associated variety.

Canonical bases
If $$p$$ is a stationary type, the canonical basis of $$p$$ is the definable closure of the set of codes for the formulas occurring in the definition of the unique global non-forking extension of $$p$$. That is, if $$\mathbb{U}$$ denotes the monster, and $$p|\mathbb{U}$$ denotes the unique non-forking extension of $$p$$ to $$\mathbb{U}$$, then for each formula $$\phi(x;y)$$, there is a $$\mathbb{U}$$-definable set $$D_\phi$$ such that $$\phi(x;a) \in p|\mathbb{U}$$ if and only if $$a \in D_\phi$$. The canonical base $$Cb(p)$$ is $$\operatorname{dcl}^{eq}(\{[D_\phi] : \phi\})$$, where $$[D_\phi] \in \mathbb{U}^{eq}$$ denotes the code for the definable set $$D_\phi$$. The canonical base is always a small set, and depends only on $$p|\mathbb{U}$$, i.e., on the parallelism class of $$p$$. Moreover, $$Cb(p)$$ is uniquely determined by the following fact: an automorphism $$\sigma \in \operatorname{Aut}(\mathbb{U})$$ fixes $$Cb(p)$$ pointwise if and only if $$\sigma$$ fixes the parallelism class of $$p$$. The canonical base is the smallest (definably closed) set over which $$p|\mathbb{U}$$ is defined. It can also be characterized as the unique smallest definably closed set $$C$$ such that some type over $$C$$ is stationary and parallel to $$p$$.

If $$\operatorname{tp}(a/C)$$ is stationary, $$Cb(\operatorname{tp}(a/C))$$ is always contained in $$\operatorname{dcl}^{eq}(C)$$, essentially because any automorphism which fixes $$C$$ pointwise must send $$\operatorname{tp}(a/C)$$ to itself, and must therefore also fix the unique nonforking extension to $$\mathbb{U}$$. One always has $$a \downarrow_{Cb(a/C)} C$$.

Since strong types are always stationary, $$Cb(\operatorname{stp}(a/C))$$ makes sense without any assumptions. This set is always contained in $$\operatorname{acl}^{eq}(C)$$, and it is also in $$\operatorname{dcl}^{eq}(Ca)$$.

If $$T$$ is superstable, then the canonical base of any stationary type is in the definable closure of a finite set. Moreover, this property characterizes superstability. (Or does it?)