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I work with the category $A-{\rm Mod}$ of left modules over a unital ring $A$, but I could ask the same question for any abelian category with enough projectives. Let $M$ and $N$ be two $A$-modules and take projective resolutions $P_\star \longrightarrow M$, $Q_\star \longrightarrow N$ of $M$ and $N$ respectively. These data give rise to a bicomplex $E_{p,q}$, where $E_{p,q} = {\rm Hom}_A\, (P_p ,Q_q )$. My question is the following:

Is $(E_{p,q})$ a page of a known spectral sequence ?

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  • $\begingroup$ Provided it converges (for example, if M has finite projective dimension, but it may converge always), it converges to Ext_A(M,N) in two steps. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Oct 22, 2014 at 17:09
  • $\begingroup$ @MarianoSuárez-Alvarez Do you have any reference ? $\endgroup$
    – Paul Broussous
    Commented Oct 22, 2014 at 17:19
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    $\begingroup$ Not really. If M has finite projective dimension, then convergence of the spectral sequence coming from the filtration by columns is immediate; the first oage, which you get by taking homology wrt the differential of Q is easy to get, because each P_i is projective, and the next one is just the complex you usebto compute the ext from P. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Oct 22, 2014 at 17:22
  • $\begingroup$ This is a second quadrant spectral sequence, so the convergence question gets rather tricky in general. I think Theorem 10.1 of Boardman's "preprint" on conditionally convergent spectral sequences hopf.math.purdue.edu/Boardman/ccspseq.pdf apply here. $\endgroup$
    – user43326
    Commented Oct 22, 2014 at 17:25

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