Global Duality

Fix once and for all a universe if [M, p. 22]. Henceforth, any category is understood to have all its arrows and objects in It. Call a set small if it is a member of it. A small category is one whose arrows—and hence objects— form a small set. Every topological space X is understood to be small and any sheaf E on A is understood to be such that for every open U C X, T U, E) is a small set. 

For any scheme X,Ox), Ax is, as before, the abelian category of Ox-modules and their homomorphisms, and the full abehan subcategory 

Theorem 4.1 Let X be a concentrated (= quasircompact, quasi-separated) scheme and f X Y a concentrated scheme-map. Then the A-functor R/ Dqc(A) — D(y) has a bounded-below right A-adjoint. 

Corollary 4.1.2 When restricted to concentrated schemes, the Dqc-f /wefl( pseudofunctor derived direct image (see (3.9.2)) has a pseudofunctorial right / -adjoint (see (3.6.7)(d)). 

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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... 

Remark 4 Note that if we had selected as the starting point the optimal solution that is (1,1,0), then the v2-GBD would have terminated in one iteration. This can be explained in terms of Remark 3. Since v(y) is convex, then the optimal point corresponds to the global minimum and the tangent plane to this minimum provides the tightest lower bound which by strong duality equals the upper bound. This is illustrated in Figure 6.3. 

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