## Monday, July 27, 2015

### UCI Summer School part 2: Duality Theorem (Monroe Eskew)

The Duality Theorem gives a general technique for forcing to make an ideal whose quotient algebra has various properties. It appears in Matthew Foreman's "Calculating quotient algebras of generic embeddings." My version of these notes omits a lot of the dots which indicate that certain objects are just names in a forcing extension. This is for aesthetic reasons, and hopefully does not lead to confusion.

Duality Theorem: Suppose $I$ is a precipitous ideal on $Z$ and $\mathbb{P}$ is any partial order. If: there is a further generic extension of the extension by $P(Z)/I$ so that if $j:V\rightarrow M\subseteq M\subseteq V[G]$ is the ultrapower embedding from $G\subseteq P(Z)/I$, there is $H\subseteq \mathbb{P}$ generic over $V$ and $\hat{H}\subseteq j(\mathbb{P})$ generic over $M$ and some extension of $j$ to $\hat{j}:V[H]\rightarrow M[\hat{H}]$.

Then: there is a $\mathbb{P}$-name for an ideal $J$ on $Z$ and a $P(Z)/I$-name for an ideal $K$ on $j(\mathbb{P})$ and a canonical isomorphism $$\iota:\mathcal{B}(\dot{\mathbb{P}\ast P(Z)/J})\equiv \mathcal{B}(P(Z)/I\ast j(\mathbb{P})/\dot{K}).$$

So a very general statement of lifting a generic ultrapower map to a forcing extension gives a useful isomorphism for computing $P(Z)/J$ in the generic extension by $\mathbb{P}$. We remark that in some cases, this will be an equivalence.

Note: In what follows, we tacitly identify all of the posets involved with their Boolean completions. Occasionally for emphasis, this identification will be explicit.

Proof: Assume (1). There is some $A\in I^+$ and some $P(Z)/I$-name for a forcing $\dot{\mathbb{Q}}$ so that
$$A\Vdash_{P(Z)/I}(\Vdash_\dot{\mathbb{Q}} \dot{H_0}\subseteq j(\mathbb{P}) \textrm{ is generic over }M \textrm{ and }H:=j^{-1}[H_0]\subseteq \mathbb{P}\textrm{ is generic over }V).$$

Note that the set $\{p\in\mathbb{P}:\quad \Vdash_{P(A)/I\ast \mathbb{Q}} j(p)\not\in H_0\}$ cannot be dense, since it is the complement of the generic $j^{-1}[H_0]$. So there is $p_0\in \mathbb{P}$ so that for all $p\le p_0$, $\| j(p)\in H_0 \|_{P(A)/I\ast \mathbb{Q}}\neq 0$. We will constrain ourselves to work below this $p_0$ in $\mathbb{P}$ and below $A$ in $P(Z)/I$. For simplicity, assume that $A=Z$ and $p_0=1_\mathbb{P}$.

In $V^{P(Z)/I}$, define
$$K=\{p\in j(\mathbb{P}):\quad \Vdash_{\mathbb{\dot{Q}^G}} p\not\in H_0\}.$$
Let $G\ast h$ be generic for $P(Z)/I\ast j(\mathbb{P})/K$. From the $j(\mathbb{P})/K$-generic $h$, we can define a $j(\mathbb{P})$-generic $\hat{H}=\{p:[p]_K\in h\}$.

Claim: The following properties of $H_0$ are also true of $\hat{H}$:

1. $\Vdash_{P(Z)/I\ast \mathbb{Q}} \hat{H}$ is $j(\mathbb{P})$-generic over $M$.
2. $\Vdash_{P(Z)/I\ast \mathbb{Q}} j^{-1}[\hat{H}]$ is $\mathbb{P}$-generic over $V$.
3. For all $p\in \mathbb{P}$, $\not\Vdash_{P(Z)/I\ast \mathbb{Q}} j(p)\not\in \hat{H}$.
Proof of Claim: For (1), suppose $D\in M$ is open dense in $j(\mathbb{P})$. Then $\{[d]_K:d\in D\textrm{ and }d\not\in K\}$ is dense in $j(\mathbb{P})/K$. Otherwise, there would exist $p\in j(\mathbb{P})/K$ so that $p\wedge d\in K$ for all $d\in D$. But this is impossible because we could then force with $\mathbb{Q}$ over $V[G]$ to get a generic $H_0$ containing $p$ (as $p\not\in K$), and then $H_0\cap D=\emptyset$, contradicting genericity of $H_0$ over $M$.

The remaining parts of the claim can be checked similarly, and are left as an exercise. $\Box$.

Now let $e:\mathbb{P}\rightarrow \mathcal{B}(P(Z)/I\ast \dot{j(\mathbb{P})/K})$ be defined by $e(p)=\|j(p)\in \hat{H}\|$.

By (3) of the claim above, $\mathrm{ker}(e)=0$. Also $e$ preserves Boolean operations simply by the elementarity of $j$. By (2) of the claim, $e$ is a regular embedding (maps maximal antichains pointwise to maximal antichains).

Exercise: $e:\mathbb{P}\rightarrow \mathbb{Q}$ is a regular embedding iff for every $q\in\mathbb{Q}$ there is $p\in \mathbb{P}$ so that for every $p'\le p$, $e(p')$ is compatible with $q$.

Thus, if $H\subseteq \mathbb{P}$ is generic over $V$, we can force with the quotient $\mathcal{B}(P(Z)/I\ast \dot{j(\mathbb{P})/K})/e"H$ over $V[H]$ to obtain a generic $G\ast h$ for $P(Z)/I\ast j(\mathbb{P})/K$. By the definition of $e$, we have $j_G"H\subseteq \hat{H}$, where $\hat{H}$ is defined from $h$ as before. So we can extend the embedding $j_G$ to $\hat{j}:V[H]\rightarrow M[\hat{H}]$.

In $V[H]$ we can finally define $J=\{A\subseteq Z:1\Vdash [\mathrm{id}]\not\in \hat{j}(A)\}$, where the forcing is with the quotient $\mathcal{B}(P(Z)/I\ast \dot{j(\mathbb{P})/K})/e"H$. In $V$, let
$$\iota(p,\dot{A})=e(p)\wedge \|[id]\in\hat{j}(\dot{A})\|.$$

Exercise: $\iota$ is order and incompatibility preserving.

It remains to show that the range of $\iota$ is dense. So take an arbitrary $(B,\dot{q})\in P(Z)/I\ast j(P)/K$. By strengthening this condition, we may assume without loss of generality that there is $f:Z\rightarrow \mathbb{P}$ in $V$ so that $B\Vdash [[f]_M]_K=\dot{q}$.

By regularity of $e$ (using the characterization in the exercise), there is a $p$ so that for all $p'\le p$, $e(p')\wedge (B,\dot{q})\neq 0$. Let $\dot{A}$ be a $\mathbb{P}$-name for a subset of $Z$ such that $p \Vdash \dot{A}=\{z\in B: f(z)\in H\}$ and $\neg p \Vdash \dot{A}\in J^+$.

We check that $(p,\dot{A})$ is actually a condition, which involves checking that $p\Vdash \dot{A}\in J^+$. So take a generic $G\ast \dot{h}$ containing $e(p)\wedge (B,\dot{q})$ (which is nonzero by choice of $p$). Then clearly $B\in G$ and since $[[f]_M]_K=q\in h$, we have $[f]_M=j(f)([\mathrm{id}])\in\hat{H}$. Therefore $[\mathrm{id}]\in \hat{j}(A)$, so this generic $G\ast \dot{h}$ shows that it is not forced by $\mathcal{B}(P(Z)/I\ast \dot{j(\mathbb{P})/K}/e"H$ that $[\mathrm{id}]\not\in \hat{j}(A)\}$.

By definition $\iota:=\iota(p,\dot{A})$ forces $j(p)\in \hat{H}$ and $[\mathrm{id}]\in\hat{j}(\dot{A})$. Since $\hat{j}$ extends $j$ and it's forced that $\dot{A}\subseteq B$, $\iota$ must force $B\in G$. And since $j(p)\Vdash_{j(\mathbb{P})} j(\dot{A})=j(\{z:j(f)(z)\in \hat{H}\})$, $\iota$ must force $\dot{q}=[j(f)(\mathrm{id})]_K\in h$. $\Box$

Remark: Suppose $K$ as in the Duality Theorem is forced to be principal, i.e., there is $m$ so that $$\Vdash K=\{p\in j(\mathbb{P}:p\le \neg m\}.$$
Then the Duality Theorem is easily seen to be an equivalence.

We can compute some nice properties of the ideal $J$ arising from the previous theorem.

Proposition: Using the notation of the previous theorem, $J$ is forced to be precipitous, with the same completeness as $I$. If $I$ is normal, then $J$ is also normal. Also, if $\bar{G}\subseteq P(Z)/J$ is generic over $V[H]$ and $G\ast h=\iota[H\ast \bar{G}]$ and $\hat{j}:V[H]\rightarrow M[\hat{H}]$ are as before, then $V[H]^Z/{\bar{G}}=M[\hat{H}]$ and $\hat{j}$ is the ultrapower embedding.

Finally, we relate $J$ to the ideal in $V[H]$ generated by $I$.

Proposition: Suppose $K$ as in the Duality Theorem is forced to be principal, with $m$ so that $\Vdash K=\{p\in j(\mathbb{P}):p\le \neg m\}.$ Suppose further that there exist $f$ and $A$ so that $A\Vdash \dot{m}=[f]_G$ and $\dot{B}$ is a $\mathbb{P}$-name for $\{z\in A: f(z)\in \dot{H}\}$. Then $\bar{I}\upharpoonright B=J\upharpoonright B$, and $A\setminus B\in J$, where $\bar{I}$ is the ideal in $V[H]$ generated by $I$.

## Friday, July 17, 2015

### UCI Summer School part 1: Basics (Monroe Eskew)

The UCI summer school in set theory just finished, and I'll be posting my notes on this blog. They will not be added in chronological order, and the final product will be very different from the actual presentation, assume that all errors were added by me.

There are some exercises which we solved during problem sessions during the day; I will preserve these in the text and can post solutions as well if there is interest.

We start off with Monroe Eskew's lectures on the duality theorem.

Basic facts: This section is meant to be supplemented by my earlier postings about ideal properties. The basic situation is that $I$ is an ideal on $Z\subseteq P(X)$, $G$ is a generic for the forcing $P(Z)/I$, and $j:V\rightarrow M=V^Z/G$ is the generic ultrapower by $G$.

Definition: An ideal $I$ has the disjointing property if every antichain $A$ in $P(Z)/I$ has a pairwise disjoint system of representatives.

Exercise: If $I$ is a $\kappa$-complete, $\kappa^+$-saturated ideal, then $I$ has the disjointing property.  If $I$ is a normal ideal on $Z\subseteq P(X)$, and $I$ is $|X|^+$-saturated, then $I$ has the disjointing property.

Lemma: Suppose $A\in I^+$ and $I$ has the disjointing property, and $[A]\Vdash \dot{\tau} \in V^Z/\dot{G}$. Then there is $f:Z\rightarrow V$ such that $A\Vdash \tau = [f]_G$.

Proof: Exercise.

Lemma: Suppose $I$ is a countably closed ideal with the disjointing property. Let $\kappa=\mathrm{crit}(j)$. Then $I$ is precipitous, and the generic ultrapower $M$ is closed under $\kappa$-sequences from $V[G]$.
Proof: Precipitousness follows from the combinatorial characterization of precipitousness. Suppose that $\Vdash \langle \tau_\alpha:\alpha<\kappa\rangle \subseteq V^Z/G$.  By the previous lemma, for all $\alpha<\kappa$, there is $f_\alpha:Z\rightarrow V$ in $V$ such that $\Vdash \tau_\alpha=[f_\alpha]_G$ and $k:Z\rightarrow \mathrm{ON}$ in $V$ such that $\Vdash \kappa=[k]_G$. Finally, take $f:Z\rightarrow V$ given by  $f(z)=\langle f_\alpha(z):\alpha<k(z)\rangle$. In $M$, $[f]_G=\langle \tau_\alpha:\alpha<\kappa\rangle$. $\Box$

Definition: $I$ is $(\lambda,\kappa)$-presaturated if for any sequence of antichains $\langle A_\alpha:\alpha<\gamma<\lambda\rangle$, the set $\{X:\forall\alpha<\gamma\, |\{a\in A_\alpha:a\cap x\in I^+\}|<\kappa\}$ is dense.

Exercise: $(\lambda^+, \kappa^+)$-presaturation and $\kappa$-completeness imply that the generic ultrapower $M$ is closed under $\lambda$-sequences from $V[G]$. Also show this for normal and $(\lambda^+,|X|^+)$-presaturated ideals.

We will use the notation $\mathcal{B}(\mathbb{P})$ to denote Boolean completion of a poset. As we have seen in previous posts, this is very handy when dealing with generic ultrapowers.

Exercise: $\mathbb{P}$ and $\mathbb{Q}$ are separative posets. The following are equivalent:

1. $\mathcal{B}(\mathbb{P})=\mathcal{B}(\mathbb{Q})$.
2. There is a $\mathbb{P}$-name $\dot{h}$ and a $\mathbb{Q}$-name $\dot{g}$ so that $$\Vdash_\mathbb{P} \dot{h} \textrm{ is }\mathbb{Q}-\textrm{generic over }V,$$ $$\Vdash_\mathbb{Q} \dot{g} \textrm{ is }\mathbb{P}-\textrm{generic over }V,$$ and$$\Vdash_\mathbb{P} \dot{g}^{\dot{h}^\dot{G}}=\dot{G} \textrm{ and } \Vdash_\mathbb{Q} \dot{h}^{\dot{g}^\dot{H}}=\dot{H}.$$