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We are saying a > Y within the type of preschemes over S is embeddable (or S-embeddable), if there exists a gentle prescheme P over S and a finite morphism i: X f = P2i. > Py = P• Y such that until differently exact, embeddable will frequently suggest over Spec ~. Examples. A projective morphism f: X > Y the place Y is quasi- compact and admits an abundant sheaf is embeddable (for any S). certainly, f could be factored via a few ~ [EGA II five. five. four (ii)]. Any finite morphism is embeddable, through taking P = S. Any morphism of finite form of affine schemes is embeddable in a few affine area. notice that any composition of embeddable morphisms is embeddable (:) and that embeddable morphisms are sturdy below base extension. 19o ! Theorem 8.? (f" for embeddable morphisms). We repair a base prescheme S, and think about the class Lno(S) of in the community noetherian ! preschemes over S. morphisms Then there exists a thought of f" for embeddable in Lno(S) consisting of the knowledge i) to five) less than, to the stipulations VAR i - VAR 6. moreover targeted within the experience that if 1')-5') topic this concept is is one other set of such facts fulfilling VAR 1 - VAR 6, then there's an isomorphism of the functors i) and i') appropriate with the isomorphisms 2)-5) and 2')-5'). i) for each embeddable morphism f: X > Y in Lno(S), a functor f:" D+qc(Y) ~ 2) D~c(X) for each composition morphisms, X f > I I (gf)" > Z of embeddable f: nine I > f'g" . for each finite morphism df: four) g an isomorphism of functors Cf,g: three) Y f, an isomorphism > f~ for each gentle embeddable morphism el. nine f"' > f~ . f, an isomorphism 191 five) for each embeddable mcrphism u: Y' each flat base extension (where v g and VAR i). > Y, an isomorphism >g'u v'f" u,f: > Y, and for ! ! b f: X are the 2 projections of X' = X ~ Y ' ) . cf,id = Cid, f = I, and there's a commutative diagram of 4 c's for a composition of 3 embeddable morphisms. VAR 2). For a composition compatibility of through d g. df and VAR 3). Proposition 2. 2, ef and in Corollary of Cf,g 6. 2 of Proposition of delicate morphisms, sq. of embeddable 6. four, compatibility utilizing morphisms of that isomorphism with c as v~f through df, dg, ecu and ev. For a composition the hypotheses with VAR 6). embeddable f,g, e . g For a Cartesian VAR 5). enjoyable with the isomorphism f,g Ditto for a composition VAR 4). and Cg,u c of 2 finite morphisms @f,g of Proposition compatibility 2. 1 or 6. three. of morphisms f,g eight. 2 or eight. four, compatibility through the best For a flat base extension morphism, of Proposition of 2 embeddable d's and e's. of a finite or soft b u2f with the isomorphism 192 facts. extra we are going to provide just a comic strip, due to the fact that the same yet tough theorem is proved in a few aspect in bankruptcy VI. ! To outline f" one chooses an f embedding i: X > Py, and ! defines f" = i~p~. The manufactured from embeddings is back one, ! so one indicates that f" is self sufficient utilizing Propositions eight. 2 and eight.

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