In "Residues and Duality" by Hartshorne, the derived tensor $\otimes$ only defined for the bounded above categories (see Chapter II, section 4, p.93), 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.



Why is the derived tensor product only defined for bounded above derived categories?
I would like to add my 2 euro ¢. In a nutshell: your information is **outdated**. "Residues and Duality" by Hartshorne was the first available text on derived categories of coherent sheaves and Grothedieck duality but it was published in 1966. In 1988, Spaltenstein showed in ("[Resolutions of unbounded complexes](http://www.numdam.org/item?id=CM_1988__65_2_121_0)", Compositio Math. 65 (1988), no. 2, 121–154) that one can use unbounded **homotopically flat** resolutions of sheaves to get a derived tensor product of unbounded complexes. Similarly, you may use **homotopically injective** resolutions of sheaves to derive functors on the right.


A very readable and useful update of duality for coherent sheaves using this developments is Joe Lipman's text "Notes on Derived Functors and Grothendieck Duality" in Springer Lecture Notes, no. **1960** (2009), 1–259.


There is a lot of extra literature on variants for Grotehdieck abelian categories, differential graded algebras and so on.


