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I'm trying to get a feel for some differential graded (dg) structures.

Suppose $C$ is a differential graded coalgebra over a commutative ring $k$, i.e. a graded $k$-module equipped with a coproduct $\Delta : C \to C \otimes C$ and a counit $\varepsilon : C \to k$ satisfying the usual axioms.

I'm interested in some sufficient conditions for the coalgebra structure on $C$ to induce coalgebra structure on the homology $H(C)$ (which is a graded $k$-module).

I guess if $k$ is a field (or a ring for which the relevant $\operatorname{Tor}$'s in the Künneth sequence vanish) then the map $H(C)\otimes H(C) \to H(C \otimes C)$ is an isomorphism, so the inverse can be used to define a coalgebra structure on $H(C)$. What if $k$ is a more complicated ring?

What about conditions "about $C$" instead of conditions "about $k$"?

Is it correct that when dealing with a product-type structure (e.g. a dg algebra or a dg Lie algebra) then no use of the Künneth formula is needed to induce the product-type structure on homology?

Many thanks!

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