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I added a clarifying clause to infinity-field so it now reads
The Morava K-theory A-∞ rings $K(n)$ are essentially the only $A_\infty$-fields. See at Morava K-theory – As infinity-Fields, where $K(0) \simeq H \mathbb{Q}$ and we define $K(\infty)$ as $H \mathbb{F}_p$.
This is from Lurie’s lectures. What precisely does he mean? He says in lecture 24 that for $k$ any field that its E-M spectrum $H k$ is an infinity-field, so the “essentially” is doing some work. Is the idea that all infinity-fields are $K(n)$-modules (cor 10, lecture 25), so the $K(n)$ essentially cover things?
On another point, would there be a higher form of the rational/p-adic fracturing of $\mathbb{Z}$, involving the $K(n)$?
Maybe it’s like how every ordinary field has prime subfield either $\mathbb{Q}$ or $\mathbb{F}_p$. But it would be rather misleading to say these are “essentially the only fields”, wouldn’t it?
Thanks. I suppose you are quite right. I have changed “essentially the only” to “the basic” now.
But the “essentially the only” comes from that lecture which you point to, which in its final remark 13 says:
We will later see that every field satisfies the hypotheses of Proposition 9 for some $0 \lt n \lt \infty$. In other words, the Morava K-theories $K(n)$ are essentially the only examples of fields in the stable homotopy category (provided that we allow the cases $n = 0$ and $n = \infty$).
But I agree with you, that what prop. 9 actually says is much more like what you say in #2.
(Unless I am missing something. I was confused about this point before. Maybe we are lucky and an expert sets us straigth, if necessary.)
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