## FANDOM

78 Pages

In stability theory and its generalizations, forking is a way of making precise the notion of "generic extensions" of types and of independence. A formula which forks over a set of parameters $C$ is thought of as being "non-generic." If $p$ is a complete type over some set $C$, the non-forking extensions of $p$ are those extensions which contain no formula forking over $C$; these are thought of as the "generic" extensions of $p$.

Forking is also related to a notion of independence: one says that tuples $a$ and $b$ are independent over a set $C$ if $\operatorname{tp}(a/bC)$ does not fork over $C$. This turns out to be a useful notion in the setting of stable theories, and their generalizations. In stable and simple theories, one has in particular the machinery of the forking calculus.

## Definitions Edit

Fix some complete theory $T$, and let $\mathbb{U}$ be a monster model. If $C$ is a small set of parameters and $\phi(x;b)$ is a formula with parameters from $\mathbb{U}$, one says that $\phi(x;b)$ divides over $C$ if there is a $C$-indiscernible sequence $\langle b_i \rangle_{i < \omega}$ of realizations of $\operatorname{tp}(b/C)$ such that $\bigwedge_{i < \omega} \phi(x;b)$ is inconsistent. Alternatively, without mentioning indiscernible sequences, one can define dividing as follows: a formula $\phi(x;b)$ is said to $k$-divide over $C$, for $k$ a positive integer, if there is a sequence $\langle b_i \rangle_{i < \omega}$ of realizations of $\operatorname{tp}(b/C)$ such that $\bigwedge_{i < \omega} \phi(x;b_i)$ is $k$-inconsistent, that is, such that any $k$-element subset of $\{\phi(x;b_i)\}$ is inconsistent.

A formula $\phi(x;b)$ is said to fork over $C$ if it implies a finite disjunction of formulas (with parameters from $\mathbb{U}$) each of which divides over $C$. That is, there is some $c_i \in \mathbb{U}$ and $\psi_i(x;y)$ such that $T \vdash \forall x : \phi(x;b) \rightarrow \bigvee_{i = 1}^n \psi_i(x;c_i)$ Any formula which divides over $C$ forks over $C$, though there are examples of the converse failing.

A partial type $\Sigma(x)$ is said to fork over $C$ (resp. divide over $C$) if it implies some formula which forks (resp. divides) over $C$. If $p$ is a complete type over $C$, a non-forking extension of $p$ is an extension which does not fork over $C$.

If $a$ is a tuple and $B$ and $C$ are small sets, one says that $a$ is (forking) independent from $B$ over $C$, written $a \downarrow_C B$, if $\operatorname{tp}(a/BC)$ doesn't fork over $C$, i.e., $\operatorname{tp}(a/BC)$ is a non-forking extension of $\operatorname{tp}(a/B)$. If $A$, $B$, and $C$ are small sets, then $A \downarrow_C B$ means that $a \downarrow_C B$ for every finite tuple $a$ from $A$. In general, this notion is not symmetric: $A \downarrow_C B$ need not be equivalent to $B \downarrow_C A$. Simple theories are exactly the theories for which this symmetry always holds; they include stable theories.

## The case of ACF Edit

In the theory of algebraically closed fields, forking turns out to be related to algebraic independence. If $L/K$ is an extension of fields (seen as substructures of the monster) and $a$ is a finite tuple, then it turns out that $a \downarrow_K L$ if and only if $tr.deg(a/K) = tr.deg(a/L)$.

## General properties of forking Edit

Forking is defined to ensure that it has the following extension property: if $C' \supset C$, any partial type over $C'$ which does not fork over $C$ can be extended to a complete type over $C'$ which does not fork over $C$. Dividing does not have this property. In fact, a partial type $\Sigma(x)$ can be extended to a global complete type which does not divide over $C$ if and only if the original type $\Sigma(x)$ does not fork over $C$. A global type divides over $C$ if and only if it forks over $C$.

Dividing can be defined more directly, in terms of indiscernible sequences. Specifically, the following are equivalent:

• $\operatorname{tp}(A/bC)$ doesn't divide over $C$
• For every $C$-indiscernible sequence $I$ beginning with $b$, there is some $I' \equiv_{bC} I$ such that $I'$ is $AC$-indiscernible.

Forking has the following "left transitivity" property: if $\operatorname{tp}(a/BC)$ doesn't fork over $C$ and $\operatorname{tp}(a'/BCa)$ doesn't fork over $Ca$, then $\operatorname{tp}(aa'/BC)$ doesn't fork over $C$. In terms of independence, this means that $a \downarrow_C B$ and $a' \downarrow_{Ca} B$ together imply $aa' \downarrow_C B$. This is the mirror image of the more familiar kind of "right transitivity" which holds in simple theories: if $\operatorname{tp}(a/BC)$ doesn't fork over $C$ and $\operatorname{tp}(a/BB'C)$ doesn't fork over $BC$, then $\operatorname{tp}(a/BB'C)$ doesn't fork over $C$. (In symbols: $a \downarrow_C B$ and $a \downarrow_{CB} B'$ imply $a \downarrow_C BB'$.)

TODO: monotonicity, base monotonicity, finite character, normality, "rigidity."

## Forking vs dividing Edit

In general, any formula or type which divides over $C$ also forks over $C$. The converse need not hold. The standard example of this is the theory of a dense circular order. If one considers the structure with underlying set $[0,1) \subset \mathbb{Q}$ and with a ternary predicate $C(x,y,z)$ interpreted as saying that $x < y < z$ or $y < z < x$ or $z < x < y$, then the formula $x = x$ does not divide over $\emptyset$, but does fork over $\emptyset$. In complete detail, the formula $x = x$ implies the disjunction of the two formulas $C(x,0,1/4)$ and $C(x,1/2,3/4)$. All tuples $(b,c)$ with $b \ne c$ have the same type over $\emptyset$, and the sequence of formulas $C(x,1/2,1/3), C(x,1/3,1/4), C(x,1/4,1/5), \ldots$ is $2$-inconsistent. So forking is not the same as dividing in this theory.

On the other hand, it is known that forking and dividing agree in simple theories (including stable theories), o-minimal theories, and C-minimal theories (such as ACVF). Furthermore, in an NIP setting, it is proven by Chernikov and Kaplan that if $C$ is a model, then a formula forks over $C$ if and only if it divides over $C$. That is, forking and dividing agree over models. This holds more generally in NTP${}_2$ theories (again by Chernikov and Kaplan).

One useful consequence of forking=dividing (when it holds) is that no type forks over its base. That is, $\operatorname{tp}(a/C)$ never forks over $C$. This comes from the easily verifiable fact that no formula over $C$ can divide over $C$.

## Forking in simple theories Edit

In simple theories, one has the full machinery of the forking calculus. Forking satisfies symmetry: $A \downarrow_C B$ is equivalent to $B \downarrow_C A$. This allows one to make definitions like Lascar rank, weight, independence, domination, and so on. Forking also satisfies local character: there is a cardinal $\kappa$ such that for every finite tuple $a$ and set $C$, there is some subset $C_0 \subset C$ such that $|C_0| < \kappa$ and $a \downarrow_{C_0} C$. In other words, every type doesn't fork over some subtype of size bounded by $\kappa$.

## Forking in NIP theories Edit

In NIP theories, non-forking has an equivalent description in terms of Lascar or compact (KP) strong types. As in any theory, $\operatorname{tp}(a/BC)$ doesn't fork over $C$ if and only if $\operatorname{tp}(a/BC)$ has an extension to a global type $p$ which doesn't divide over $C$. In the NIP setting, however, it turns out that the following are equivalent for a global type $p$:

• $p$ doesn't divide over $C$
• $p$ is Lascar $C$-invariant. That is, for every formula $\phi(x;y)$ and every $b$ and $b'$, if $b$ and $b'$ have the same Lascar strong type over $C$, then $\phi(x;b) \in p(x) \iff \phi(x;b') \in p(x)$. Equivalently, $p(x)$ is fixed by the group of Lascar strong automorphisms over $C$.
• $p$ is $bdd(C)$-invariant. Equivalently, if $\phi(x;y)$ is a formula and $b$ and $b'$ have the same compact strong type over $C$, then $\phi(x;b) \in p(x) \iff \phi(x;b') \in p(x)$.

For stable theories, Lascar strong type, compact strong type, and strong type all agree, so one recovers the characterization of non-forking over $C$ as having an $\operatorname{acl}^{eq}(C)$-invariant global extension.

## Forking in stable theories Edit

Main article: Forking in stable theories.