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Let $X$ be a smooth, projective curve. We let $Coh(X)$ be the stack of coherent sheaves on $X$. Its Grothendieck group is $Pic(X)\times\mathbf{Z}$. Is the map $$ Coh(X)\rightarrow Pic(X)\times \mathbf{Z} $$ sending a coherent sheaf to its class in the Grothendieck group algebraic?

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    $\begingroup$ Yes, there is a 1-morphism of algebraic stacks as you describe. $\endgroup$
    – Jason Starr
    Commented Dec 4, 2020 at 11:36
  • $\begingroup$ Thanks ! Is it something obvious that can be generalized to any category having a moduli stack of objects ? I assume we also have to define a algebraic stack structure for the Grothendieck group. Do you have any reference where I can have a look to get a better understanding of this ? $\endgroup$
    – hennlu
    Commented Dec 4, 2020 at 12:09
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    $\begingroup$ I do not know about the general case, and I doubt that the K-group has a natural structure of algebraic stack (think about the associated graded that is the Chow group). For curves, this is “classical”. One construction using the determinant of a perfect complex is given in Definition 2.11 of my article with de Jong and He. $\endgroup$
    – Jason Starr
    Commented Dec 4, 2020 at 12:45

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