Timeline for What are the higher homotopy groups of Spec Z ?
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| Nov 4, 2009 at 4:11 | history | edited | David Treumann | CC BY-SA 2.5 |
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| Nov 4, 2009 at 0:35 | comment | added | Tyler Lawson | 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. | |
| Nov 3, 2009 at 22:55 | comment | added | David Treumann | Are you really saying that there are square-zero extensions of the sphere spectrum that are unramified? This isn't possible with plain rings. | |
| Nov 3, 2009 at 21:52 | comment | added | Tyler Lawson | 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. | |
| Nov 3, 2009 at 17:49 | history | answered | David Treumann | CC BY-SA 2.5 |