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Suppose $j : V \to M$ is $\lambda$-supercompactness embedding derived from an $\kappa$-complete normal ultrafilter $U$ on $P_\kappa(\lambda)$, $\lambda$ regular. Suppose $\eta$ is an ordinal such that $\sup(j[\lambda]) \leq \eta < j(\lambda)$. Let $W$ be the uniform ultrafilter on $\lambda$ defined by $X \in W$ iff $\eta \in j(X)$. Let $i : V \to N$ be the ultrapower embedding by $W$. There is a factor map $k : N \to M$ given by $$k([f]_W) = j(f)(\eta).$$

Question: What is the relationship between $N$ and $M$? What is the critical point of $k$? How closed is $N$?

We can ask similar questions about $\lambda$-strong embeddings.

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2 Answers 2

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You will always have $N = M$ and $k = \text{id}$. As Joel mentions, this uses Solovay's lemma that $M = H^M(\text{ran}(j)\cup \{\sup j[\lambda]\})$. We can use this to show that $k$ is surjective, by proving that in fact $k(\sup i[\lambda]) = \sup j[\lambda]$. If not, $k(\sup i[\lambda]) > \sup j[\lambda]$ is a generator of $j$, but $j$ has no generators above $\sup j[\lambda]$ (since $\sup j[\lambda]$ generates everything by Solovay's lemma). Since $\sup j[\lambda] \in\text{ran}(k)$ and $\text{ran}(j)\subseteq \text{ran}(k)$ and $M = H^M(\text{ran}(j)\cup\{\sup j[\lambda]\})$, $k$ is surjective, and so $k = \text{id}$ and $N = M$.

Here are the facts about generators I'm using. Recall that an ordinal $\nu$ is a generator of an elementary embedding $i : W\to N$ if $\nu \in N$ and $\nu\notin H^N(\text{ran}(i)\cup [\nu]^{<\omega})$. I need:

  1. If $i : W\to N$ is an elementary embedding that is discontinuous at a regular cardinal $\lambda$ of $W$, then $\sup j[\lambda]$ is a generator of $j$.
  2. If $k : N \to M$ is a further embedding, then for any generator $\nu$ of $i$, $k(\nu)$ is a generator of $k\circ i$.
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  • $\begingroup$ Why is $\sup j[\lambda] \in \ran(k)$ if we select $\eta \not= \sup j[\lambda]$? $\endgroup$ Commented Dec 15, 2023 at 15:33
  • $\begingroup$ Because $k(\sup i[\lambda]) = \sup j[\lambda]$! $\endgroup$ Commented Dec 15, 2023 at 15:34
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If you take $\eta=\sup j[\lambda]$, then $N=M$ by a theorem of Solovay (the sup function is one-to-one on a measure one set), used in his proof that the SCH holds above any supercompact cardinal. In this case, the measure $W$ is isomorphic as an ultrafilter to the original supercompactness measure.

It follows that there are also many other values of $\eta$ for which $N=M$, since whenever $\eta$ codes the ordinal $\sup j[\lambda]$ in some canonical way, then $\sup j[\lambda]$ will be placed into the seed hull of $\eta$, and consequently everything will be there, causing $N=M$ again.

For other values of $\eta$, however, we do not necessarily have N=M.

The critical point of $k$ is the first ordinal missing from the seed hull $\{j(f)(\eta)\mid f:\lambda\to V\}$, so of course when it exists it will be at most $(2^\lambda)^{+V}$, simply because there only $2^\lambda$ many functions to represent such ordinals below $j(\lambda)$.

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