I am reading a paper concerning the action of monoidal category to another category. Let $k$ be a commutative ring, $R$ is a k-algebra. $A=R-mod$, $B=R^{e}-mod=R\bigotimes _{k}R^{o}-mod$.

Consider the action:

$B\times A\rightarrow A,(M,N)\mapsto M\bigotimes _{R}N$ is an action of monoidal category of $R^{e}-mod=B^{~}=(B,\bigotimes _{R},R)$ on A.

The paper said this action induces the action

$\Phi : D^{-}(B)\times D^{-}(A)\to D^{-}(A)$ of the monoidal derived category $D^{-}(B)$ on $D^{-}(A)$

I know this action should be $(M,N)\mapsto M\bigotimes_{R}^{L}N$.

But I do not know how is this action of monoidal derived category on the other derived category **induced** by the action of monoidal abelian category. Is there a canonical way(A natural transformation)to get this action?

Notice that the action of monoidal abelian category is defined as follows

$\Psi:=(\Phi ,\phi ,\phi _{0})$

$\Phi :B=(B,\bigotimes _{R},R)\rightarrow End(A)$

$\Phi (V)\cdot \Phi (W)\overset{\phi }{\rightarrow}\Phi (V\bigotimes _{R}W)$

The back ground of this question is localization of differential operator in derived category, so I added the tag"algebraic geometry"

This paper is "Differential Calculus in Noncommutative algebraic geometry I" which is available in MPIM

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in the LaTeX code, which does not help readability :) $\endgroup$ – Mariano Suárez-Álvarez Feb 9 '10 at 5:37