Let $X$ a topological space, say a complex variety, and $\mathbb{C}_X$ its constant sheaf. $\mathcal{D}(X)$ is the derived category of sheaves of $\mathbb{C}_X$-modules.
Let $F^\bullet\in \mathcal{D}(X)$. We can describe two actions of the cohomology $H^\bullet(\mathbb{C}_X)$ of $X$ on the hypercohomology $H^\bullet(F)$ of $F$.
- We have $H^\bullet(F)=Hom_{\mathcal{D}(X)}(\mathbb{C}_X,F)$, so $H^\bullet(\mathbb{C}_X)=End^\bullet_{\mathcal{D}(X)}(\mathbb{C}_X)$ acts by composition.
- We have $F=F \otimes \mathbb{C}_X$, so $H^\bullet(\mathbb{C}_X)$ acts on $F$, and by functoriality, on $H^\bullet(F)$
The former is left action while the latter is a right action. I want to understand how they are related. For example it should be true that if $X$ is smooth and $H^\bullet(X)$ is commutative as a ring, then they coincide.