Derived Direct and Inverse Image

A basic objective, in the spirit of Grothendieck s philosophy of the six operations, is the categorical formalization of relations among functorial maps involving the four operations R/, L/, and RWom.  

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 g) and RTfom and the adjoint A-functors R/ and L/ respect these structures, as do the conjugate isomorphisms, arising from a second map g Y Z, R gf)t R R/, Lf Lg L gf). We express all this by saying that R— and L— are adjoint monoidal A-pseudofunctors. 

relations among the four operations can be worked with as instances of category-theoretic relations involving adjoint monoidal functors between 

The second and third maps determine each other via L/ -R/ adjunction (3.4.5), as do the third and fourth (3.4.6). When the first map is given, the second and third maps also determine each other via RWom - adjunction. (This is not obvious, see Proposition (3.2.4).) 

any three of the four maps can be deduced category-theoretically from the remaining one. 

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Adjointness of Derived Direct and Inverse Image corresponds via (2.6.1) to the composed map... 

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

Derived Direct and Inverse Image For commutativity, expand the diagram naturally as follows. 

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