Let $\mathcal{T}$ be a triangulated category that is generated by one object, say $A$ in the sense that the smallest triangulated subcategory containing $A$ and closed under coproducts and isomorphisms is $\mathcal{T}$ itself. Denote by $K_0(\mathcal{T})$ the Grothendieck group of $\mathcal{T}$. Is $K_0(\mathcal{T})$ generated by the equivalence class of $A$?
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3$\begingroup$ Yes. Also, triangulated subcategories are closed under finite coproducts. Since you stress coproducts, I think you mean infinite ones too. In that case the Grothendieck group is trivial. $\endgroup$– Fernando MuroCommented Apr 30, 2023 at 9:07
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$\begingroup$ @FernandoMuro I am a little concerned how to generate the shifts of $A$ in the Grothendieck group!! $\endgroup$– user45397Commented Apr 30, 2023 at 9:08
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1$\begingroup$ Shifting is multiplication by -1. $\endgroup$– Fernando MuroCommented Apr 30, 2023 at 9:25
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1$\begingroup$ Yes, that’s what I mean. $\endgroup$– Fernando MuroCommented Apr 30, 2023 at 9:34
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2$\begingroup$ $A\to 0\to A[1] \to A[1]$ $\endgroup$– Fernando MuroCommented Apr 30, 2023 at 9:44
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1 Answer
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Fernando Muro has already answered this in the comments, but perhaps a reference would help. This is all spelled out in Neeman's book on triangulated categories. In particular, see Definition 4.5.8 and Proposition 4.5.11, and their proofs.
answered May 3, 2023 at 3:17