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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 21st 2014

    I added a clarifying clause to infinity-field so it now reads

    The Morava K-theory A-∞ rings K(n)K(n) are essentially the only A A_\infty-fields. See at Morava K-theory – As infinity-Fields, where K(0)HK(0) \simeq H \mathbb{Q} and we define K()K(\infty) as H𝔽 pH \mathbb{F}_p.

    This is from Lurie’s lectures. What precisely does he mean? He says in lecture 24 that for kk any field that its E-M spectrum HkH k is an infinity-field, so the “essentially” is doing some work. Is the idea that all infinity-fields are K(n)K(n)-modules (cor 10, lecture 25), so the K(n)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)K(n)?

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 21st 2014

    Maybe it’s like how every ordinary field has prime subfield either \mathbb{Q} or 𝔽 p\mathbb{F}_p. But it would be rather misleading to say these are “essentially the only fields”, wouldn’t it?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 21st 2014
    • (edited Aug 21st 2014)

    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<n<0 \lt n \lt \infty. In other words, the Morava K-theories K(n)K(n) are essentially the only examples of fields in the stable homotopy category (provided that we allow the cases n=0n = 0 and n=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.)