Equivariant K-theory

In model theory, a branch of mathematical logic, the Hrushovski construction generalizes the Fraïssé limit by working with a notion of strong substructure $\le$ rather than $\subseteq$. It can be thought of as a kind of "model-theoretic forcing", where a (usually) stable structure is created, called the generic. The specifics of $\le$ determine various properties of the generic, with its geometric properties being of particular interest. It was initially used by Ehud Hrushovski to generate a stable structure with an "exotic" geometry, thereby refuting Zil'ber's Conjecture.

Three conjectures

The initial applications of the Hrushovski construction refuted two conjectures and answered a third question in the negative. Specifically, we have:

• Lachlan's Conjecture Any stable ${\mathrm{\aleph }}_0$-categorical theory is totally transcendental.

• Zil'ber's Conjecture Any uncountably categorical theory is either locally modular or interprets an algebraically closed field.

• Cherlin's Question Is there a maximal (with respect to expansions) strongly minimal set?

The construction

Let L be a finite relational language. Fix C a class of finite L-structures which are closed under isomorphisms and substructures. We want to strengthen the notion of substructure; let $\le$ be a relation on pairs from C satisfying:

• $A\le B$ implies $A\subseteq B$.
• $A\subseteq B\subseteq C$ and $A\le C$ implies $A\le B$
• $\varnothing \le A$ for all $A\in \mathbb{ℂ}$.
• $A\le B$ implies $A\cap C\le B\cap C$ for all $C\in \mathbb{ℂ}$.
• If $f:A\to A^{\prime }$ is an isomorphism and $A\le B$, then $f$ extends to an isomorphism $B\to B^{\prime }$ for some superset of $B$ with $A^{\prime }\le B^{\prime }$.

An embedding $f:A\hookrightarrow D$ is strong if $f(A)\le D$.

We also want the pair (C, $\le$) to satisfy the amalgamation property: if $A\le B_1,A\le B_2$ then there is a $D\in \mathbb{ℂ}$ so that each $B_i$ embeds strongly into $D$ with the same image for $A$.

For infinite $D$, and $A\in \mathbb{ℂ}$, we say $A\le D$ iff $A\le X$ for $A\subseteq X\subseteq D$, $X\in \mathbb{ℂ}$. For any $A\subseteq D$, the closure of $A$ (in $D$), ${\mathrm{cl}}_D(A)$ is the smallest superset of $A$ satisfying $\mathrm{cl}(A)\le D$.

Definition A countable structure $G$ is a (C, $\le$)-generic if:

• For $A{\subseteq }_{\omega }G$, $A\in \mathbb{ℂ}$.
• For $A\le G$, if $A\le B$ then $B$ there is a strong embedding of $B$ into $G$ over $A$
• $G$ has finite closures: for every $A{\subseteq }_{\omega }G$, ${\mathrm{cl}}_G(A)$ is finite.

Theorem If (C, $\le$) has the amalgamation property, then there is a unique (C, $\le$)-generic.

The existence proof proceeds in imitation of the existence proof for Fraïssé limits. The uniqueness proof comes from an easy back and forth argument.

References

• E. Hrushovski. A stable ${\mathrm{\aleph }}_0$-categorical pseudoplane. Preprint, 1988
• E. Hrushovski. A new strongly minimal set. Annals of Pure and Applied Logic, 52:147–166, 1993.
• Slides on Hrushovski construction from Frank Wagner