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?



Künneth spectral sequence for cohomology of chain complexes of $R$-modules
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.


