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Suppose $\mathcal{C}\subset \mathcal{C}'$ is a pair of pre-triangulated smooth DG categories over a characteristic-zero basefield (say, $\mathbb{C}$), such that $\mathcal{C}$ is faithfully embedded in $\mathcal{C}'$ and $\mathcal{C}'$ is the Karoubi completion of $\mathcal{C}.$ Then it is classically known that the map on Grothendieck groups $K^0(\mathcal{C})\to K^0(\mathcal{C}')$ is an embedding, and a rational equivalence: in particular, if $K^0(\mathcal{C}')$ is torsion-free, then there is no splitting of the map, $K^0(\mathcal{C})\leftarrow K^0(\mathcal{C}').$

My question is whether it is possible for there to be a non-trivial splitting over the Quillen K-theory coefficient ring spectrum $K^*(\mathbb{C})$ of the map of module spectra $K^*(\mathcal{C})\to K^*(\mathcal{C}').$ This question came up in studying the algebraic K theory of a p-adic group.

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  • $\begingroup$ Could you give an example when the map on $K^0$ is not an isomorphism? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Dec 26, 2015 at 7:45
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    $\begingroup$ Sure. Take complexes of vector spaces of even total dimension. $\endgroup$
    – Dmitry Vaintrob
    Commented Dec 26, 2015 at 17:30
  • $\begingroup$ I doubt that any property of pre-triangulated smooth DG categories (that is not valid in some more general context) may be classically known.:) So, could you give a reference? $\endgroup$
    – Mikhail Bondarko
    Commented Dec 26, 2015 at 18:00
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    $\begingroup$ Your premise is false: the map on $K^0$ is not a rational isomorphism. Just take the subcategory of objects with trivial class in $K^0$. See Thomason's Classification of triangulated subcategories. $\endgroup$
    – Ben Wieland
    Commented Dec 30, 2015 at 1:49

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