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Your question might be compacted to someting like: Do I need derived categories to study cohomology of sheaves? Of course, this depends on your particular interests. Let me anyeway give you some starting points to help you to make up your mid.

Your question might be compacted to someting like: Do I need derived categories to study cohomology of sheaves? Of course, the answer depends on your particular interests. Let me anyway give you some starting points to help you to make up your mind.

The every time $\mathbf{R}f$ reduces to $F$ you obtain similar formulas than the ones you obtain by the collapse of the spectral sequence. The advantage is that the argument is simpler and you don't have limitations on finiteness or boundedness of the complex involved.

I won't assert they are easier but they are in my opinion clearer and broader. For base-change and Künneth I suggest you to look at [L, Theorem (3.10.3)]. The Künneth formula is more general than any other I've seen in the literature. Its expression via spectral sequences, if possible, it would look extremely complicated.

Thus, every time $\mathbf{R}f$ reduces to $f$ you obtain similar formulas than the ones you obtain by the collapse of the spectral sequence. The advantage is that the argument is simpler and you don't have limitations on finiteness or boundedness of the complex involved.

I won't assert they are easier but they are in my opinion clearer and broader. For base-change and Künneth I suggest you to look at [L, Theorem (3.10.3)]. The Künneth formula is more general than any other I've seen in the literature. Its expression via spectral sequences, if possible, would look extremely complicated.

A few pointers and comments:

Let $f : \mathcal{A} \to \mathcal{B}$ a left exact functor between abelian categories and denote also by $f$ its extension to the corresponding homotopy categories , i.e. $\mathbf{K}(\mathcal{A})$ denotes the category of complexes with maps up to homotopy. Its derived functor $\mathbf{R}f : \mathbf{D}(\mathcal{A}) \to \mathbf{D}(\mathcal{B})$ satisfies a universal property that implies that the collection $\{\mathbf{R}^if\}_{i \in \mathbb{N}}$ is a universal $\delta$-functor such that $\mathbf{R}^0f = f$. Where $\mathbf{R}^if: = \mathrm{H}^i\mathbf{R}f$.

Your question might be compacted to someting like: Do I need derived categories to study cohomology of sheaves? Of course, this depends on your particular interests. Let me anyeway give you some starting points to help you to make up your mid.

Let $f : \mathcal{A} \to \mathcal{B}$ a left exact functor between abelian categories and denote also by $f$ its extension to the corresponding homotopy categories. $\mathbf{K}(\mathcal{A})$ denotes the category of complexes with maps up to homotopy. The derived functor $\mathbf{R}f : \mathbf{D}(\mathcal{A}) \to \mathbf{D}(\mathcal{B})$ satisfies a universal property that implies that the collection $\{\mathbf{R}^if\}_{i \in \mathbb{N}}$ is a universal $\delta$-functor such that $\mathbf{R}^0f = f$. Where $\mathbf{R}^if: = \mathrm{H}^i\mathbf{R}f$. See [L, $\S2.1$].