Sheafified Duality, Base Change

Unless otherwise indicated, all schemes—and hence all scheme-maps—are assumed henceforth to he concentrated. All proper and quasi-proper maps are assumed to be finitely presentable. 

As in 4.3, a scheme-map / X — F is called quasi-proper if R/ takes pseudo-coherent Ox-complexes to pseudo-coherent Oy-complexes. For example, when Y is noetherian and / is of finite type and separated then / is quasi-proper iff it is proper, see (4.3.3.3). We will need the nontrivial fact that quasi-properness of maps is preserved under tor-independent base change [LN, Prop. 4.4]. 

Recall the characterizations of independent fiber square (3.10.3), of finite tor-dimension map (2.7.6), and of the dualizing pair f, r) in (4.1.1). We write f for f when / is quasi-proper. 

Corollary 4.4.3 (Base Change). In (4.4.1), the functorial map adjoint to the composition 

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The abstract theory begins with Theorem (4.1) (Global Duality), asserting for any map f X —> T of concentrated schemes the existence of a right adjoint for the functor R/ Dqc(X) —> Dqc( )- In order to sheafify this result, or, more generally, to prove tor-independent base change for —see... 

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