Minor additions:
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fixed a couple of typos;
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explicitly stated comparison with l-adic cohomology in Corollary 5.7;
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comparison with Geisser's arithmetic homology in Proposition 7.9;
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remarks about the relation of L^c (X_ét, n) to standard Lichtenbaum's conjectures on finite generation of étale motivic cohomology at the beginning of section 8.
Minor additions:
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note that finiteness of H^i (X_ét, Z^c(n)) for X/F_q already follows from finite generation by duality,
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precise results on boundedness of H^i_W,c (X, Z(n)) in section 7,
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comparison with the construction of Flach and Morin in section 8.
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The new title is "Weil-étale cohomology and duality for arithmetic schemes in negative weights".
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Various improvements in the exposition thanks to suggestions of an anonymous referee, in particular in sec. 3 and 6.
Minor additions:
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note that finiteness of H^i (X_ét, Z^c(n)) for X/F_q already follows from finite generation by duality,
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direct proof via trace formula in section 5.
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The new title is "Weil-étale cohomology and zeta-values of arithmetic schemes at negative integers".
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Minor improvements in the exposition.
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Updated references to 2012.11034