Boundedness assumption for isomorphism between some derived functors in Kashiwara and Schapira

Question

Let $\mathscr{R}$ be a sheaf of ring on a topological space, and denote $\mathbf{D}^+(\mathscr{R})$ (resp. $\mathbf{D}^-(\mathscr{R})$, $\mathbf{D}^b(\mathscr{R})$) to be the bounded below (resp. bounded above, bounded) derived category of sheaves of $\mathscr{R}$-modules.
In the book Sheaves on Manifolds of Kashiwara and Schapira, they define the derived sheaf How, $R \mathscr{H}om(\cdot,\cdot): \mathbf{D}^- (\mathscr{R})^\circ \times \mathbf{D}^+ (\mathscr{R}) \rightarrow \mathbf{D}^+ (\mathscr{R})$ by replacing the second component by quasi-isomorphic injective sheaves. And they define the derived global section functor, $R\Gamma:\mathbf{D}^+ (\mathscr{R}) \rightarrow \mathbf{D}^+ (\mathscr{R})$ and the derived Hom, $R Hom(\cdot,\cdot): \mathbf{D}^- (\mathscr{R})^\circ \times \mathbf{D}^+ (\mathscr{R}) \rightarrow \mathbf{D}^+ (Ab)$ similarly.

But to get the isomorphism $RHom(F,G) \cong R\Gamma(X;R\mathscr{H}om(F,G))$, they assume without explanation that $F$ is in $\mathbf{D}^b(\mathscr{R})$. The same assumption is made when discussing relation between $R\mathscr{H}om$ and $Rf_\ast$. Can someone verify if this assumption is redundant or not? If this assumption is essential, what's the reason?

Thank you

Answer 1

When Kashiwara and Schapira wrote "Sheaves on Manifolds", the theory of unbounded categories was not known (or very little known, I do not know for certain) and that is why the framework of the book is the bounded derived category $\textbf{D}^b$ or the bounded from below $\textbf{D}^+$ derived category. For the applications they had in mind, $\textbf{D}^b$ was enough and most of the book is written in this framework, which does not mean that some results do not extend to $\textbf{D}^+$. Consider the following example.

But to get the isomorphism $RHom(F,G) \cong R\Gamma(X;R\mathscr{H}om(F,G))$, they assume without explanation that $F$ is in $\mathbf{D}^b(\mathscr{R})$.

In the above, Kashiwara and Schapira could have assumed $F \in \textbf{D}^-$, but in practice they always have $F \in \textbf{D}^b$.

The same assumption is made when discussing relation between $R\mathscr{H}om$ and $Rf_\ast$. Can someone verify if this assumption is redundant or not? If this assumption is essential, what's the reason?

These isomorphisms follow from using the Grothendieck spectral sequence, right? If I remember correctly, the issue is that the Grothendieck spectral sequence needs complexes to be bounded below. The relevant proposition in Kashiwara–Schapira is Prop. 1.10.9.

That's what I'm saying. The condition for Prop. 1.10.9, in Kashiwara-Schapira is bounded below for both component. By when they apply this proposition to, for example, Equation (2.6.3), Equation (2.6.24)-(2.6.26), Prop. 3.1.10, etc., they need to assume the first component to be bounded on both side. Not only bounded below. And that's the place I got confused.

I do not understand your objection. In Proposition 1.10.9, the result is stated in $\textbf{D}^+$ and this is the situation of (2.6.3), etc.

A final remark. In the book, derived functors are treated in the unbounded case.

Sorry, (2.6.3) is a typo. What I wanted to mention was (2.6.4) and the result is stated in $D^b \times D^+$. For other equation, they also assume this $D^b \times D^+$ condition and that's what confused me because they assumed $D^+ \times D^+$ in Proposition 1.10.9.

Since $\text{Hom}$ is contravariant with respect to the first variable, perhaps the result can be extended to $\textbf{D}^- \times \textbf{D}^+$ but certainly not $\textbf{D}^+\times \textbf{D}^+$ (unless we use other tools such as those of "Categories and Sheaves" GL Springer 2006, see here).

Kashiwara, Masaki; Schapira, Pierre. Categories and sheaves. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 332. Springer-Verlag, Berlin, 2006. x+497 pp.