Adjoint Monoidal A-Pseudofunctors

We review next the behavior of derived direct and inverse image functors vis-a-vis a pair of ringed-space maps X Y Z. 

relative to the categories of Ox- Oy-, Oz-) modules we have the functorial isomorphism (in fact equality) 

Given a third map Z W, we have the commutative diagram of functorial isomorphisms (actually equalities) 

From these observations we can derive similar ones involving the corresponding derived functors. 

This allows us to build a diagram analogous to (3.6.3), with Re in place of e for each map e involved. The resulting derived functor diagram still commutes, as can be seen by reduction (via suitable quasi-isomorphisms) to the case of q-injective complexes in D(X), for which the diagram in question is essentially (3.6.3).  

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More explicitly (details in 3.4, 3.5), if / X F is a map of ringed spaces, then the derived categories D( x), D( Ay) have natural structures of symmetric monoidal closed categories, given by [Pg.83]

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