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Apr 8, 2017 at 5:49 comment added jdc @OscarRandal-Williams: This is an extremely nice example. I really did want to assume $G$ connected; do you happen to have an analogous example at hand for that case too?
Apr 7, 2017 at 17:27 comment added Oscar Randal-Williams The representation ring $R(C_2)$ is $\mathbb{Z}[T]/(T^2-1)$ with $T$ the sign representation. Taking additive basis $1$ and $H= T-1$, so the augmentation ideal $I$ consists of the multiples of $H$, we have $H^2 = -2H$. Thus the associated graded for the $I$-adic filtration is $\mathbb{Z} \oplus (\mathbb{Z}/2)^\infty$. This has torsion.
Apr 7, 2017 at 2:58 vote accept jdc
Apr 6, 2017 at 23:34 comment added jdc @DenisNardin: The filtration is by the kernels of $K^0 BG \to K^0 (BG)_n$ for $n$-skeleta $(BG)_n$; these kernels contain $\widehat I(G)^n$ and induce the same topology on $\widehat R(G)$. My inclination is that this means the associated graded shouldn't have torsion, but I don't have anything stronger than that.
Apr 6, 2017 at 23:11 comment added Denis Nardin @jdk the classical example is the Bockstein spectral sequence, that is build for this. Another one is the Adams spectral sequence, where 2∊π_0S is represented by an element h_0 in Adams filtration 1.
Apr 6, 2017 at 22:49 comment added jdc Filtration: Could you give me an example of the type of situation you might expect torsion in the associated graded? I mean, $\widehat R(G)$ is a power series ring over $\mathbb Z$ in finitely many indeterminates; I do not have intuition about this, but would you expect it to be possible, say, that $2x$ lies in a lower filtrand than $x$ for some $x$?
Apr 6, 2017 at 22:45 comment added jdc Convergence: Pick a CW-structure on $BG$ and look at $n$-skeleta; the AHSS unambiguously converges for these, and the $\varprojlim^1$ term in the Milnor exact sequence vanishes, so it is enough to look at these finite approximations. Their torsion primes will be amongst those of $BG$. I thought about putting this in the question but decided it would just take up space.
Apr 6, 2017 at 22:32 answer added Tom Goodwillie timeline score: 27
Apr 6, 2017 at 22:23 comment added Denis Nardin I haven't thought about this very much, but a priori it is possible that the $E_∞$-page contains torsion while the $K^*BG$ does not (remember: the $E_∞$-page is just the associated graded of a filtration on $K^*BG$). Moreover it is not even clear that the AHSS converges in this case...
Apr 6, 2017 at 21:40 history asked jdc CC BY-SA 3.0