In "Residues and Duality" by Hartshorne, the derived tensor $\otimes$ only defined for the bounded above categories (see Chapter I, section 4), that is one has

$$\otimes: D^{-}(X) \times D^{-}(X) \to D(X), $$ where $X$ is a variety, and $D^{-}(X), D(X)$ are derived category of **bounded above** complex of $\mathcal{O}_X$-modules and derived category of complex of $\mathcal{O}_X$-modules respectively .

The main reason it is defined for the bounded above complex is that Hartshorne used the fact that every sheaf of module has a flat resolution.

However, why cannot one use the injective resolution for the derived tensor, and defined it for **bounded below** derived cateogry $D^{+}(X)$?

In fact, the later is the case I need (to be precise, I need $F\otimes : D^{+}(X) \to D(X)$, where $F$ is a sheaf of module), and I checked every conditions for the existence of derived functor, it seems that everything is satisfied.

injectiveresolutions for aleftderived functor. – Zhen Lin Mar 6 at 16:20acyclic, in this case such that the functor $P\otimes -\ $ is exact. – abx Mar 6 at 17:17leftderived functor. There is no use making it a right derived functor. – Zhen Lin Mar 7 at 7:59