Question 1: Let $X$ be a scheme. Then generally for the complex $C^{\bullet}$ in $D^b(X)$, we define $R\Gamma(C^{\bullet})\colon$ = Complex obtained by applying $\Gamma$ to the injective resolution $I^{\bullet}$ of $C^{\bullet}$.

Why is this notation so useful, and do people like to employ $R\Gamma(C^{\bullet})$?

Hypercohomology $H^{i}(C^{\bullet}) \colon=$ $i$-th homology of $R\Gamma(C^{\bullet})$ is useful and defined via $R\Gamma(C^{\bullet})$, but I often see $R\Gamma(C^{\bullet})$ itself without taking its homology.

In what purpose do we need $R\Gamma(C^{\bullet})$?

Question 2: For the complex $C, D \in C^{\bullet}$ in $D^b(X)$, we can define $C ⊗^{L} D$ by using the flat resolution of $C$ (or $D$). Why do we need to take flat resolution not just the tensor product $C ⊗ D$ of two complexes?

I am sorry, I am very weak about these concepts. Hopefully please teach me the reasons of these.

Pierre MATSUMI