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Let $A$ be a Banach algebra, let $A$-mod be the category of left Banach modules (as defined in Helemskii's "Banach and locally convex algebras"), $A$-mod is an additive category, but not abelian apparently? (is it never abelian or are there cases where it is?), however, as shown in that book, we can define chain complexes and do homology and cohomology in $A$-mod. Let $R$-Mod be the category of $R$-modules, we can build the derived category $D^b(R)$ from $R$-Mod using its category of chain complexes, passing to the homotopy category, localizing, etc.

Can we define some sort of derived category for $A$-mod ? Or for any additive category for that matter?

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    $\begingroup$ addressing the more specific question regarding e.g. Ban - you should look for work of Prosmans, Schneiders, and Buehler. (In general I think there is something old in JPAA looking at the "abelian envelope" of a derived category) $\endgroup$
    – Yemon Choi
    Feb 10 '15 at 22:07
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    $\begingroup$ Prosmans and Schneiders point out in their respective work that if you apply their machinery to Ban, what naturally emerges is the abelian category qBan that was introduced by Noel/Waelbroeck $\endgroup$
    – Yemon Choi
    Feb 10 '15 at 22:08
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    $\begingroup$ You can take the derived category of (say right) modules over any sekeletally small additive category. This answers your second question, but it seems to me that your first and second questions are rather unrelated. $\endgroup$ Feb 10 '15 at 22:58
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    $\begingroup$ @FernandoMuro Presumably the point is that the OP wants the derived category of A-mod to faithfully encode the Tor, Ext and Hochschild cohomology developed by Helemskii's school. My recollection of reading Schneiders numdam.org/item?id=MSMF_1999_2_76__R3_0 many years ago is that for this to work it is important that A-mod is what he calls quasi-abelian $\endgroup$
    – Yemon Choi
    Feb 10 '15 at 23:31
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    $\begingroup$ @YemonChoi sure, that's what I meant when I said that both questions seem unrelated to me. $\endgroup$ Feb 10 '15 at 23:33

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