Relative cohomology dimensions of complexes based on degreewise cotorsion pairs

Let $R$ be an associative ring with identity, and ($\mathcal{A}$, $\mathcal{B}$) a hereditary cotorsion pair generated by a set in $R$-Mod. Then $(\operatorname{dw}\widetilde{\mathcal{A}}, (\operatorname{dw}\widetilde{\mathcal{A}})^{\bot})$ is a complete and hereditary cotorsion pair (we call it the degreewise cotorsion pair) in the category of $R$-complexes, where $\operatorname{dw}\widetilde{\mathcal{A}}$ denotes the class of all complexes $X$ with components $X_{n} \in \mathcal{A}$ for all $n\in \mathbb{Z}$. For any complexes $X$ and $Y$ and any $n \in \mathbb{Z}$, we define the relative cohomology groups $\operatorname{Ext}^{n}_{\operatorname{dw}\widetilde{\mathcal{A}}}(X, Y)$ based on the degreewise cotorsion pair and investigate the vanishing of the relative cohomology groups. Specifically, we introduce the relative cohomology dimension of $X$ related to $\operatorname{dw}\widetilde{\mathcal{A}}$-precovers, and then show that such a dimension of $X$ is equal to the least integer $n$ for which $\operatorname{Ext}^{i}_{\operatorname{dw}\widetilde{\mathcal{A}}}(X, Y) = 0$ for all $i > n$ and all $R$-modules $Y \in \mathcal{B}$, which recovers the result on relative cohomology dimensions (defined by Liu) of complexes related to Gorenstein projective precovers.