TITLE: Closed monoidal structure on the derived category of sheaves
QUESTION [5 upvotes]: Given a topological space X, i'd like to find Der X - the derived category of sheaves of abelian groups on X - to be a closed monoidal category. Hom should be cohomological and the internal-hom should be triangulated.

Is this possible in full generality? (Unbounded complexes, no restrictions on X)
Consider a sheaf of rings R or equivalently a ring of sheaves. This gives us two things: An abelian category of left R modules that we can derive; let's call this one Der R; A monoid R in Der X whose category of modules we denote dMod R. Is Der R = dMod R'? Ff not: how do they relate?
Given monoids R,S,T in Der X do we get the usual adjunctions in two variables between their categories of bimodules?
Given rings R,S,T in Sh X do we get the usual adjunctions in two variables between their derived categories of bimodules?

Now for the question: What is the right setting to do this? As i understand it, there's no suitable model structure that gives 4 in full generality.

REPLY [6 votes]: The most direct answer to question (2) is no, even in the case of sheaves on a point.  In this case we simply have a ring R and we are asking whether the derived category of R-modules is equivalent to the category of R-module objects in the derived category of abelian groups.  For example, let $R = Z/p^2$.  Then in the derived category of R-modules, we have nonzero Ext groups
$$Ext^n_{Z/p^2}(Z/p,Z/p) = Hom_{D(Z/p^2)}(Z/p,Z/p[n])$$
for all positive degrees $n$.  However, because no object in the derived category of Z-modules has any nontrivial Ext-groups in high degrees, we have for $n$ large
$$Hom_{dMod(Z/p^2)}(Z/p,Z/p[n]) = 0.$$
The problem is that dMod(R) doesn't really remember that R is an actual ring, but instead only remembers that it can be lifted to a chain complex with a chain-homotopy associative multiplication map.  These kinds of ring objects are too weak to do really serious homological algebra and define a proper derived category.
The answer to question 3 is correspondingly no, because we can't really construct proper derived functors of the tensor product given only a monoid in Der(X).