Cartesian Finite Morphisms

The construction of e is compatible with restrictions. So we may assume that /, = / X — y is an affine morphism of single schemes. Further, the question is local on Y, and hence we may assume that Y = Spec A is affine. As / is affine, X = SpecB is affine. By Lemma 14.3, we may assume that F = FxG for some G L)(Qch(X)). In view of Lemma 14.6, it suffices to show that e g g G — G is an isomorphism if G is a X-injective complex in Qch(X). 

To verify this, it suffices to show that s g g M A4 is an isomorphism for M G Qch(X). By Lemma 7.19, / = i g on Qch(X) respects coproducts and is exact. Since respects coproducts and is faithful exact, gr respects coproducts and is exact. So g j g Qch(X) Qch(X) respects coproducts and is exact. 

Since X is affine, there is an exact sequence of the form 

We say that an C z-module M. is locally quasi-coherent (resp. quasi-coherent, coherent) if is. The corresponding full subcategory of Mod(Z) is denoted by Lqc(Z) (resp. Qch(Z), Coh(Z)). 

Lemma 27.4. Let the notation be as above. Then an Oz-tnodule X is locally quasi-coherent if and only if for any j e ob(7) and any affine open subscheme U ofYj, there exists an exact sequence of Oz)j) u- Tf odules 

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The fourth step is to prove various commutativities related to the well-definedness of the twisted inverse pseudofunctors, Chapters 16, 18, and 19. Among them, the compatibility with restrictions (Proposition 18.14) is the key to our construction. Given a separated G-morphism of finite t3rpe / X Y between noetherian G-schemes, the associated morphism BQ f) Bq X) Bq Y) is cartesian, see (4.2) for the definition. If we could find a compactification... 

Theorem 28.11. Let I be a finite ordered category, and f, X, —> Yt a morphism in Vil. Schl. Assume that Y, is noetherian with flat arrows, and /, is separated cartesian smooth of finite type. Assume that /, has a constant relative dimension d. Then for any F G L)Y (Yt), there is a functorial isomorphism... 

By Lemma 7.17 and Lemma 7.16, the all morphisms in the diagrams are cartesian. As pi is smooth of finite type of relative dimension d, A is a cartesian regular embedding of the constant codimension d. 

Let T (resp. J-m) denote the subcategory of V((A). Sch/S) (resp. subcategory of P(Am. Sch/5)) consisting of noetherian objects with flat arrows and cartesian morphisms separated of finite type. 

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