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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) are essentially the only A-fields. See at Morava K-theory – As infinity-Fields, where K(0)H and we define K() as H𝔽p.

    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 Hk 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 , involving the K(n)?

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 21st 2014

    Maybe it’s like how every ordinary field has prime subfield either or 𝔽p. 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<. 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=).

    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.)