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Let $X$ be a regular scheme which is flat over $\mathbf{Z}$. The arithmetic Grothendieck group $\hat{K}(X)$ is defined to be the quotient of $\hat{G}(X)$ by $\hat{G}^\prime(X)$. This is actually quite a length definition which I added below for the sake of completeness.

In the classical case, for $X$ any noetherian scheme, the Grothendieck group $K_0(X)$ is defined to be the Grothendieck group of the category of vector bundles on $X$. That is, one applies the notion of a Grothendieck group for an additive subcategory of an abelian category. (In our case the abelian category is the category of coherent sheaves on $X$.) This means just modding out by short exact sequences.

I would like to know if there is a categorical type of definition for this group. Thus, first one needs to decide what kind of categories we're talking about (objects are pairs in some sense) and then the notion of exact sequence should coincide in some sense with the below definition.

Probably there is no such thing. I just ask this question in order to understand the arithmetic Grothendieck group better.

Note. Let me sketch the definition of the arithmetic Grothendieck group as given in Faltings. In the above $\hat{G}(X)$ is the direct sum of "the free abelian group generated by all vector bundles which have a hermitian metric on $X_{\mathbf{C}}$ which is invariant under complex conjugation $F$" and the abelian group $\widetilde{A}^\ast(X)$. Here $\widetilde{A}^\ast(X)$ is generated by all $p$-forms $\alpha^p$ such that $F^\ast \alpha^p = (-1)^p \alpha^p$. Furthermore, $\hat{G}^\prime(X)$ is the subgroup generated by elements of the form $E_2 - E_1-E_3 - \widetilde{ch}(E)$, where $E$ is the short exact sequence $$0\rightarrow E_1 \rightarrow E_2 \rightarrow E_3 \rightarrow 0$$ and $\widetilde{ch}(E)$ is the secondary Chern form.

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Could you say a little more about this "secondary Chern form"? I can't find a good reference on-line. –  some guy on the street Apr 12 '10 at 17:33
    
Basically, all of the above is written down really carefully in Faltings's "Lectures on the arithmetic Riemann-Roch theorem". You can find it on google books. Unfortunately, the pages 19-20 are missing from it. –  Ariyan Javanpeykar Apr 12 '10 at 18:39
    
The "arithmetic curves case" is treated really nicely in Rossler's math.u-psud.fr/~rossler/mypage/pdf-files/rossler-diplom.pdf . If you're still wondering what this secondary Chern form is you should look at pages 16-18. –  Ariyan Javanpeykar Apr 14 '10 at 16:11
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@Ariyan The arithmetic Grothendieck group is devised to approximate the Grothendieck group of a scheme over $\bf Z$, "compactified" at $\infty$; this "compactified" object never appears as such in Arakelov geometry and maybe (probably) doesn't exist. This suggests that there is no natural categorical definition of the arithmetic Grothendieck group. Notice also that since the direct image in $\hat{K}_0$ involves the analytic torsion, which is highly analytic, such a definition would have to include an sort of algebraic interpretation of the latter, which seems to be a tall order. –  Damian Rössler Aug 26 '11 at 14:04

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The classical group $K_0$ can also be thought of as consisting of equivalence classes of chain complexes of vector bundles, such that the exact sequences represent the zero of $K_0$ --- and furthermore every complex is equivalent to a two-term complex, a.k.a. a bundle morphism; the graded tensor product of complexes also gives a ring structure on $K_0$. If you like, classical $K_0$ is a categorical interpolation between the Euler Characteristic and the homology of a chain complex. The present construction looks like a refinement of that idea, where instead of representing a trivial element, an exact sequence is equivalent to a particular (sum of) differential form(s).

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