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On chapter III.4 ("Metrized $\mathcal{o}$-modules") of this book on algebraic number theory, Neukirch credits his treatment of the theory of finitely generated $\mathcal{o}$-modules to the course "Arakelov theory and Grothendieck-Riemann-Roch" taught by Günter Tamme. He continues:

There, however, proofs were not given directly, as we will do here, but usually deduced as special cases from the general abstract theory.

In the references, Neukirch only lists Tamme's book on Étale Cohomology and his article on the proceedings "Beilinson's Conjectures on Special Values of L-functions".

Was any of this ever put on writing?

Alternatively, is there a good reference where this things are "deduced as special cases from the general abstract theory"?

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Here is a link of Gunter Tamme's course. I take a brief look and it seems centering around proving Gronthendieck-Riemann-Roch using K-theory machinery. I did not see Arakelov theory anywhere. The course notes at here are incomplete (the page 106 is missing, for example).

Personally I would guess the "general abstract theory" can be found in Fulton's book and SGA6. Neukirch should be credited for all the concrete proofs he presented in his book. I think the recent expository article by Gerald Montplet in the Springer "Progress in Mathematics" book and his papers are great examples of ""deducing special cases from the general abstract theory". This is usually very difficult in Arakelov theory or even in index theory.

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  • $\begingroup$ Thank you very much, I had lot hope of finding it! I'll give it a good look. $\endgroup$
    – Myshkin
    Commented Mar 23, 2018 at 19:23
  • $\begingroup$ @Bombyxmori which book of Fulton are you referring to, he's written several? Also where in SGA 6 are you referring to, it's over 700 pages? I can't find mention of Arakelov anything in SGA 6, Fulton's Intersection Theory, or Fulton and Lang's Riemann-Roch Algebra. The chapter of the CIMPA summer school, PM vol 321 by Gerard Montplet is here, or available from his website. $\endgroup$
    – Tim Campion
    Commented Dec 4, 2018 at 5:14
  • $\begingroup$ @TimCampion: The intersection theory by Fulton. It is heavily used in Lang's book. SGA6 contains a chapter of Deligne pairing, which may be the first time this shows up in literature. $\endgroup$
    – Bombyx mori
    Commented Dec 5, 2018 at 1:18

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