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Let $R$ be a unital ring. Let $\mathbf{A}_\bullet$ and $\mathbf{C}_\bullet$ be positive chain complexes of $R$-modules. If $\mathbf{A}_\bullet$ consists of flat $R$-modules then there is homology Künneth spectral sequence $$E^2_{p,q}:=\bigoplus_{s+t=q}\mathrm{Tor}_p^R(H_s(\mathbf{A}_\bullet),H_t(\mathbf{C}_\bullet))\Rightarrow H_{p+q}(\mathbf{A}_\bullet\otimes_R\mathbf{C}_\bullet).$$

I am interested in a cohomological version, specifically, is the following true?

Suppose $\mathbf{C}_\bullet$ is a negative complex. If $\mathbf{A}_\bullet$ consists of projective $R$-modules, then there is a cohomology Künneth spectral sequence $$E^2_{p,q}:=\bigoplus_{s+t=q}\mathrm{Ext}^p_R(H_s(\mathbf{A}_\bullet),H_t(\mathbf{C}_\bullet))\Rightarrow H_{p+q}(\mathrm{Hom}_R(\mathbf{A}_\bullet,\mathbf{C}_\bullet)).$$

A version of this appears in Rotman's introduction to homological algebra (first edition) but it does not appear in the second edition and I do not know of another reference.

If the "theorem" is true, what is a reference for it?

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    $\begingroup$ Are you sure you have the degrees right? $\endgroup$
    – Theo Johnson-Freyd
    Jan 24, 2022 at 17:14
  • $\begingroup$ Possibly not, what do you think they should be? $\endgroup$
    – Sam Hughes
    Jan 24, 2022 at 17:24
  • $\begingroup$ This could be found in EKMM Thm 4.6 (or Lurie's Higher Algebra). $\endgroup$
    – Z. M
    Jan 24, 2022 at 19:31
  • $\begingroup$ I didn't actually think through what I would expect the degrees to be, but as a rule of thumb homological and cohomological degrees are minus each other. And Ext and Hom are contravariant in the first entry. So the $(p,s,t)$ summand on the LHS looks to me like it is in total homological degree $-p-s+t$, not $p+s+t$. $\endgroup$
    – Theo Johnson-Freyd
    Jan 25, 2022 at 14:36

1 Answer 1

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This is a special case of the hyper(co)homology spectral sequence from Chapter XVII, Section 2 of Cartan-Eilenberg (1953), for the functor $T(C,A) = \mathrm{Hom}_R(A,C)$. The $E_2$-term is given in equation (4) on page 368, essentially as $$ E_2^{p,q} = \prod_{s+t=q} \mathrm{Ext}^p_R(H_s(\mathbf{A}_\bullet), H^t(\mathbf{C}^\bullet)) \Longrightarrow H^{p+q}(\mathrm{Hom}_R(\mathbf{A}_\bullet, \mathbf{C}^\bullet)) \,, $$ with $p$ as the filtration degree.

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