I'm reading Carlsson's Derived Representation Theory and the Algebraic $K$-theory of Fields, and in section 4.2 he gives an example which I'm not fully understanding.
The example is as follows: $k$ is an algebraically closed field of characteristic $0$, $A= k[[x]]$, $F=Frac(A)= k[[x]][x^{-1}] = k((x))$.
He says that we then get a fiber sequence on $K$-theory: $Kk\to KA\to KF$. I don't understand this part.
Indeed, let $C = \mathrm{fib}(Perf(A)\to Perf(F))$. If I understand correctly, $C$ is the full subcategory of $Perf(A)$ on those complexes on whose homology $x$ acts nilpotently.
This implies by devissage that the inclusion $Perf(k)\to C$ induces an equivalence on $K$-theory, so it looks like we're on the right track: we're looking for a fiber sequence $K(C)\to KA\to KF$, and we have an exact sequence $C\to Perf(A)\to Perf(F)$.
Right, but we're dealing with (as far as I can tell) connective $K$-theory, which is an additive invariant, not a localizing one. In particular, if we want a fiber sequence from an exact sequence of small stable $\infty$-categories, we want the latter to be split-exact: here this doesn't seem to be the case.
And in fact, given what Carlsson does next, there is no splitting (the result he gets on $\ell$-completion contradicts the existence of such a splitting)
So he's probably using some other result to get this fiber sequence : what's that result ? What am I missing ?
(it may also be the case that he's looking at nonconnective $K$-theory, but then wouldn't he mention that ? And the description he gets later seems to indicate that he is in fact looking at connective $K$-theory)
I think there are two ways to solve this:
1- Use nonconnective $K$-theory and the fact that $K\simeq \tau_{\geq 0}\mathbb K$ : $K$-theory is the connective cover of nonconnective $K$-theory. I wasn't sure about this, but it seems to be true. Since $\tau_{\geq 0}$ is right adjoint, it preserves fiber sequences, and thus our sequence $K(C)\to KA\to KF$ is a fiber sequence of connective spectra.
You simply then check by hand that it's surjective on $\pi_0$, which is easy here, which implies that it's also a fiber sequence of spectra (and then by the theorem of the heart and devissage, $K(C)\simeq Kk$ and we get the desired result, if I'm not mistaken)
2- In Blumberg-Gepner-Tabuada's paper about the universal property of $K$-theory, they state that $K$-theory sends strict exact sequences of small stable $\infty$-categories to fiber sequences of spectra. This is theorem 9.10 of their paper:
$$\mathrm{Map}(\mathcal U^\kappa_{wloc}(Sp^\omega), \mathcal U^\kappa_{wloc}(\mathcal A))\simeq K(\mathcal A)$$ Since $\mathrm{Map}$ preserves fiber sequences, and $\mathcal U^\kappa_{wloc}$ satisfies this property (proposition 8.5 in the same paper), it follows that $K$ does too.
But $C\to Perf(A)\to Perf(F)$ is strict exact, not only exact; so we get the result again.