Let $R$ be any commutative ring, let $V^\bullet$ and $W^\bullet$ be (co)chain complexes of $R$-modules, indexed cohomologically. We can also assume that they have both cohomology in nonpositive degrees. Using K-flat resolutions we can define the derived tensor product:

$$ V^\bullet \overset{\mathbb L}{\otimes}_R W^\bullet. $$ I'm looking for assumptions on $V^\bullet$ or $W^\bullet$ that ensure that the following "Künneth formula" holds: $$ H^*(V^\bullet \overset{\mathbb L}{\otimes}_R W^\bullet) \cong H^*(V^\bullet) \otimes_R H^*(W^\bullet), $$ where the right hand side is the tensor product of graded $R$-modules. Searching the literature, it seems that there should be some spectral sequence involving Tor of the cohomologies, as mentioned for instance in the nLab entry.

• Is there a more precise reference for such a result?
• Looking at these Künneth formulas and the spectral sequence, I suspect that the following is true: $H^*(V^\bullet \overset{\mathbb L}{\otimes}_R W^\bullet) \cong H^*(V^\bullet) \otimes_R H^*(W^\bullet)$ holds if $V^\bullet$ or $W^\bullet$ has flat cohomologies, namely $H^k(V^\bullet)$ (say) is a flat $R$-module for all $k$. Is it correct?

A little more wildly, could one expect that without assumptions one has $$ H^*(V^\bullet \overset{\mathbb L}{\otimes}_R W^\bullet) \cong H^*(V^\bullet) \overset{\mathbb L}{\otimes}_R H^*(W^\bullet), $$ so taking K-flat resolutions of the cohomology graded $R$-module if necessary?