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Let $X$ be a projective curve over an algebraic closed field $k$ which characteristic zero. Define $K^0(X)$ as the Grothendieck group of the derived category $Perf(X)$ and $K_0(X)$ as the Grothendieck group of the derived category $D^b_{\text{coh}}(X)$.

If $X$ is a smooth curve, then it is well-known that $$ K^0(X)\cong K_0(X)\cong \mathbb{Z}\oplus Pic(X). $$

Now if $X$ is singular, do we have any explicit computations on $K^0(X)$ and $K_0(X)$?

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    $\begingroup$ Look at Weibel's K-Book, Ch. II, Ex. 8.2. For $K^0(X)$, the same formula holds, it's ${\Bbb Z}$ plus Pic. For $K_0$, you can compute via the normalization. (Neither characteristic zero nor projective are necessary.) $\endgroup$
    – Dave Anderson
    Commented Apr 23, 2015 at 23:12
  • $\begingroup$ @DaveAnderson Thank you very much! By the way where can I find the relation between the $K_0$ of $X$ and the normalization of $X$? Is it also in Weibel's book? $\endgroup$
    – Zhaoting Wei
    Commented Apr 24, 2015 at 1:16

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