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**Additional resources for Residues and Duality: Lecture Notes of a Seminar on the Work of A. Grothendieck, Given at Harvard 1963 /64 (Lecture Notes in Mathematics)**

Is a functorial isomorphism Trpf: for G" E D + (Y). quality controls Rf~f~(G" ) N> G" Then there 156 evidence. seeing that to developing X and and D(Qco(X)) : f~ : [II.?. lg] is an equivalence taking derived be aware that Trpf: measurement on says that with this isomorphism. D(Qco(X)) for the reason that f~ is Qco(X). ) an isomorphism ~ > G" G" E D(Qco(Y)). We practice B = Qco(Y), size [I. 7. four] to the types A = Qco(X) and to the functor F = f~. n on Qco(X). sheaves of the shape P satisfies L(Rnfw) Rf~f~(G" ) > D + (Y) quality control and [1. five. 6] which is appropriate actually, we'll build for all > D(Qco(X)). is outlined on all of of finite cohomological the place J> D(Qco(Y)) D(Qco(Y)) of different types, we decrease for G" 6 D+(Qco(Y)), which says that D+(Qco(Y)) functors Rf~ are in the neighborhood noetherian, the same isomorphism Rfw (We use the following Y L permit exists, P ~ Ob Qco(X) of Lemma four. 1. the hypotheses Now f~ and has cohomological be the gathering Then by way of the 2 lemmae, of loc. cit. and we finish and there's an isomorphism that of 157 ~: Rf. ~ > of functors from D(Qco(X)) We observe to L(Rnf. )[-n] to D(Qco(Y)). f~G" for G" 6 K(Qco(Y)), which supplies an i somorphi sm _R_f. f~(G ") -- ~f. (f*(G') ~(Rnf. )(f*(G')| Now each one sheaf is f*(GP)| Rnf. -acyclic, | w[n]) . is in P (since w ~ ~x(-n-l)~ as a result so the expression at the correct is simply Rnf. (f*(G ") | w). yet for every p, the projection Rnf. (f*(GP)| formulation provides us an isomorphism ~ > Rnf. (w) | G p and composing with the isomorphism 7 of Theorem three. four, this turns into G p. Composing some of these isomorphisms now we have the necessary isomorphism Trpf: feedback. i. Rf. f~G " = > G" . keep in mind that the isomorphism simply outlined relies on the isomorphism ~ Trpf we've of Theorem three. four, and so relies it appears at the projective coordinates. 158 To If one doesn't desire to use [II. 7. 19] and possible outline Trpf for G" 6 D~c(Y) proj ectio n formulation [II. five. 6 ]. [I. 7. 4], just by utilizing the the subsequent proposition indicates that the 2 tools of creating the hint map agree while either are outlined. Proposition four. four. proposition, F',G" enable enable f,X,Y F',G" E D~c(Y) , has finite Tor-dimension. be as within the prior and suppose that one among Then the subsequent diagram is commutative: ~" =~ ~f. dG" > R_f. (~*F" ~ f~a" ) Trpf F'~G" \~ Rf. f~(F" @ G') the place the higher horizontal arrow is the projection formulation [II five. 6] and the right-hand vertical arrow is the isomorphism of Proposition 2. 4a. 159 facts. Y affine. The query is neighborhood on Y, so we may possibly suppose Then w e can take a solution of sums of copies of ~y ~ F" by means of direct hence we may match fullyyt with quasi-coherent sheaves, and should turn out the assertion for F~ E D'(Qco(Y)). Then the end result follows simply from the definition of the morphisms concerned, on account that if Cartan-Eilenberg solution of f~G', then Cartan-Eilenberg solution of f~F" | f~G" . C'" f~F'~C" is a is a 160 w The duality theorem for projective house. The duality theorem for projective area now follows simply from what has long past prior to.