This question is related to the MO questions What is the difference between Grothendieck groups K_0(X) vs K^0(X) on schemes? and Does a fully faithful functor between triangulated categories induce embedding of their Grothendieck groups?

Let $X$ be a noetherian scheme over a field $k$. We have the derived category of perfect complexes on $X$, $Perf(X)$, and the derived category of bounded complex of coherent sheaves on $X$, $D^b(coh(X))$. It is clear that $Perf(X)$ is a strictly full triangulated subcategory of $D^b(coh(X))$ and for non-regular $X$, these two categories are not equivalent.

Now we define $K^0(X)$ be the Grothendieck group of $Perf(X)$ and $K_0(X)$ be the Grothendieck group of $D^b(coh(X))$. Then the embedding $Perf(X)\to D^b(coh(X))$ induces a homomorphism $$ K^0(X)\to K_0(X). $$

$\textbf{My question}$ is: is the above homomorphism an momomorphism? (under some assumptions on $X$ or the base field $k$, if necessary) If not, is there any counter-example?


No, this is not always a monomorphism. For the underlying reduced scheme $X_{\text{red}}$ of $X$, the pushforward homomorphism $$ K_0(X_{\text{red}})\to K_0(X) $$ is an isomorphism (via devissage). If you read Manin's "Lectures on the K-functor", you will see that the natural map $$ \text{Pic}(X) \to K^0(X) $$ is an injection. Yet the composition to $K_0(X)$ factors through the pullback homomorphism $$ \text{Pic}(X)\to \text{Pic}(X_{\text{red}}), $$ which may easily have a kernel, cf. Hartshorne, Chapter III, Exercise 4.6.

  • $\begingroup$ Thank you very much! By the way do we have the injection result if we assume that the scheme $X$ is reduced? $\endgroup$ – Zhaoting Wei Apr 20 '15 at 13:51
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    $\begingroup$ @ZhaotingWei: "... do we have the injection result if we assume that the scheme $X$ is reduced?" Unfortunately not. There are examples as above where $X$ is projective, and both $X$ and $X_{\text{red}}$ are local complete intersection schemes over a field $k$. Let $i:X\to \mathbb{P}^n_k$ be a closed immersion. Let $\nu: Y \to \mathbb{P}^n_k$ be the blowing up of the image of $i$. Using the formula for K-theory of a blowing up, $K^0(Y)\to K_0(Y)$ is not a monomorphism, but now $Y$ is reduced. $\endgroup$ – Jason Starr Apr 20 '15 at 14:28
  • $\begingroup$ Thank you! Where can I find the formula for K-theory of a blowing up? Is it also in Manin's book? $\endgroup$ – Zhaoting Wei Apr 20 '15 at 21:11

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