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Extra info for Residues and Duality: Lecture Notes of a Seminar on the Work of A. Grothendieck, Given at Harvard 1963 /64 (Lecture Notes in Mathematics)
1. five] and [T. three. 6. 5]). we'll frequently make statements in just this sort of contexts, and should permit the reader make the required adjustments of for the opposite. ~f~ as an alternative ~+f~, whilst no confusion can come up. Proposition 2. 1. compact, then Rf~ If takes = X we'll additionally write f: X i~y D + (X) quality controls is separated and quasi- into D + (Y). quality controls is noetherian of finite Krull size, into =Rf~ If, in addition, takes Dqc(X ) Dqc(Y ) . facts. by means of [EGA III then the entire Rif~(F) different phrases, ~f~(F) 1. four. 1o], if F is quasi-coherent on Xj are quasi-coherent E D_+c(Y ). q yet so the end result follows from [I.?. three] nine Rf~ sheaves on Y. In is a way-out functor, 89 Proposition 2. 2. and If f: X >Y is in the neighborhood noetherian, then Y Dc+(Y). Rf~ sends D+(X) into If, additionally, X is noetherian of finite Krull measurement, Rf~ Proo f. If is a formal morphism, X takes Dc(X ) into Dc(Y)o related as earlier proposition, utilizing [EGA III three. 2. 1]. is a prescheme, the worldwide part functor F has a correct derived functor R+r: D+(X) > D(Ab) . As above, we will be able to expand the area of definition of D(X) if X is noetherian of finite Krull size. R~ to 90 w The derived functor of Horn'. permit outline X be a prescheme, Hom'(F',G') and permit E K(Mod(X)) Then we F',G" 6 K(Mod(X)). as in [I. 6] via l Homn(F',G ") = ~ pew Hom(FP,G p+n) and d = ~ Then Hom" + (-l)n+Id G. is a bi-~-functor Hom': K(Mod(X))~ • K(Mod(X)) Lemma three. 1. allow F" be a posh of be a posh of injective both F" evidence. that or G" ~X-mOdules, ~X-mOdules, bounded under. is acyclic. Then Hom'(F',G') imagine that is acyclic. it truly is adequate to teach for every open subset F(U, Hom'(F',G')) = Hom'(F'IU , G'IU ) and allow G" is acyclic. U of X yet this follows from [I. 6. 2] utilized to the abelian type of ~U-mOdules, because if is an inJective G is an inJective ~X-mOdule, then GIU ~U-mOdule. utilizing this Lemma, we will be able to stick to the tactic of [1. 6] and procure a derived functor 91 R Horn': D(X)" x D+(X) (more accurately ~I~II Hom'). workout. believe that that each coherent X is in the community noetherian, and ~X-mOdule is a quotient of a in the neighborhood unfastened ~X-mOdule of finite rank (e. g. convey D(X) > X quasi-projective over a field). then that RIIRI Hom': Dc(X )" X D(X) > D(X) exists. challenge. sheaves on X with no the hypotheses of the workout, research that are acyclic for Hom within the first variable. discover no matter if there are adequate of them, and for that reason no matter if ~iHom" exists. Definition. If F" 6 D(X) and G" 6 D+(X), we outline the neighborhood hyperext Exti(F ",G') = Hi(R Hom'(F',G')) Lemma three. 2. enable F allow be a coherent X . be a in the neighborhood noetherian prescheme. ~x-mOdule, and allow G be a coherent 92 (resp. quasi-coherent) ~X-mOdule. Exti(F,G) is coherent (resp. facts. Then for all i ~ O quasi-coherent). The coherent case is [EGA Oiii12. three. three] , and the quasi-coherent case is proved equally. Proposition three. three. prescheme. enable allow F" 6 Dc(X) X be a in the community noetherian and G" E Dc+(X) (resp.