Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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.)