Timeline for Is Morava K-theory of a classifying space of a compact Lie group a Noetherian ring?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 12 at 17:00 | vote | accept | Daniel Pomerleano | ||
| Sep 12 at 9:04 | answer | added | Oscar Randal-Williams | timeline score: 8 | |
| Sep 11 at 20:21 | comment | added | Dave Benson | @DanielPomerleano It's exactly what Ravenel does in his paper. A simpler case where this was the issue is Lenny Evens' proof of finite generation of group cohomology. There's a good discussion of the issue in his paper, "The cohomology ring of a finite group". The issue is that even if each $E_r$ in the spectral sequence is finitely generated (or Noetherian), this does not prove that this holds for $E_\infty$. | |
| Sep 11 at 20:05 | comment | added | Daniel Pomerleano | @DaveBenson Sure, there's a difference. Could you say why this stronger condition would actually be needed? That sounds pretty hopeless in this case. | |
| Sep 11 at 19:27 | comment | added | Dave Benson | There's a difference between converging and stopping at a finite page. | |
| Sep 11 at 19:24 | comment | added | Daniel Pomerleano | @DaveBenson The basis for Ravenel's convergence argument seems to be that the cohomology spectral sequence is dual to the homology spectral sequence, which always converges. This seems to be a general claim: the Serre spectral sequence always converges in Morava K-theory? | |
| Sep 11 at 19:17 | comment | added | Daniel Pomerleano | @DrewHeard In your reference, the AS spectral sequence is invoked as well and I'm not sure how the convergence question I have is dealt with. | |
| Sep 11 at 19:08 | comment | added | Drew Heard | If K_p(n) was a ring spectrum, this would follow from Corollary 4.5 of arxiv.org/pdf/2106.08669...I wonder if it is possible to make this type of argument work out | |
| Sep 11 at 16:38 | comment | added | Daniel Pomerleano | @skd The spectral sequence does not a priori converge. This was the argument I initially considered. | |
| Sep 11 at 16:28 | comment | added | Dave Benson | For finite groups, Ravenel proved that the spectral sequence stops, and therefore the Morava $K$-theory of $BG$ is finitely generated as a $K(n)^*(pt)$-module, a stronger statement than you are asking for. But I don't know of a place where this is generalised to compact Lie groups. | |
| Sep 11 at 16:03 | comment | added | Dave Benson | @skd Doesn't that argument require that the spectral sequence stops at some finite page? | |
| Sep 11 at 14:54 | comment | added | skd | If G is a compact Lie group, then H^*(BG; F_p) is finitely generated as an algebra for any prime p. Recall that there is a spectral sequence going from E_2 = H^*(BG; F_p)[v_n^{\pm 1}] to K(n)^*(BG), so that the E_2-page is also finitely generated as an algebra. This implies that K(n)^*(BG) is also noetherian. | |
| Sep 11 at 13:54 | history | asked | Daniel Pomerleano | CC BY-SA 4.0 |