## What are the higher homotopy groups of Spec Z ?

The homotopy groups of the étale topos of a scheme were defined by Artin and Mazur. Are these known for Spec Z? Certainly π1 is trivial because Spec Z has no unramified étale covers, but what is known about the higher homotopy groups?

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If etale pi_1 classifies obstructions to trivializing finite flat unramified Z-algebras, it would be nice if the whole etale homotopy type classified obstructions to trivializing simplicial commutative Z-algebras that were finite, flat, and unramified in a homotopy sense. I think all of these notions make sense: "finite" means that the homotopy groups vanish in high degrees and are finitely generated, "flat" means that these homotopy groups have no torsion, and "unramified" means that the cotangent complex is zero. Is that right?

Presumably algebraic topologists have thought about the sphere spectrum version of this question. Are there any connective E-infinity ring spectra that are finite, flat, and unramified over the sphere?

After Tyler's comments, I see that this is a bad analogy. The dictionary between etale locally constant sheaves of sets and finite flat unramified algebras (which in one direction takes an algebra and associates the sheaf of sections of its spectrum over Spec Z) just doesn't extend to a dictionary between homotopy-style locally constant sheaves and homotopy-style finite flat unramified algebras.

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With regards to your second question: This appears in John Rognes' paper on Galois theory of structured ring spectra, at least in part. There are none if you ask that the generators all live in pi_0. If you allow new generators in positive degrees, there are square-zero extensions and their ilk, and more following those that are harder to classify. I am not sure about your first question. – Tyler Lawson Nov 3 2009 at 21:52
Are you really saying that there are square-zero extensions of the sphere spectrum that are unramified? This isn't possible with plain rings. – David Treumann Nov 3 2009 at 22:55
I guess I was thinking by analogy with your "homotopy vanishing in high degrees" examples above. No, those can't ever have trivial cotangent complex, and thickenings of a commutative Z-algebra to something with positive homotopy groups usually can't either; e.g. if R -> S is an isomorphism on pi_0 then the first nonvanishing relative homotopy group coincides with the first nonzero homotopy group of the cotangent complex. – Tyler Lawson Nov 4 2009 at 0:35