What is the $K$-theory of the category of coherent sheaves on the infinite (countable) dimensional projective space over a field? As far as I know, $K$-theory is oriented; hence this theory should be "simple" (power series in one variable over the $K$-theory of the base field). Yet I am not sure that this answer gives those $K$-groups that I am interested in. Also, does the answer depend on the choice of a definition of the infinite dimensional projective space (how many possible definitions are there?)? I would be deeply grateful for an explanation!
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asked Dec 19, 2017 at 21:40
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2$\begingroup$ What precisely is your definition of the infinite dimensional projective space over a field? $\endgroup$– Jason StarrCommented Dec 19, 2017 at 22:24
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$\begingroup$ Actually, I am reading a short text of somebody else, and there is no definition included. This should be an ind-variety, and one should be able to apply the Barth-Van de Ven-Tyurin-Sato theorem to it. $\endgroup$– Mikhail BondarkoCommented Dec 20, 2017 at 7:17
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1$\begingroup$ The Barth-Van de Ven-Tyurin-Sato theorem implies that $K^0$, the $K$-ring of locally free sheaves on ind-projective space, is the subring $\mathbb{Z}[x,x^{-1}] \subset \mathbb{Q}[c]_{\langle c \rangle}$, where $x$ is the class of $[\mathcal{O}(-1)]$ and $c$ equals $1-x$. The ind-projective space "approximates" the Artin stack $B\mathbb{G}_m$, and the $K$-group of this Artin stack is $\mathbb{Q}[[c]]$, cf. the thesis of Toen. So your guess is plausible. You might check Gaitsgory's work on ind-coherent sheaves. $\endgroup$– Jason StarrCommented Dec 20, 2017 at 11:13
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