HOmA(M,N) is an isomorphism. instance. One can't quasi-coherent of quintessential noetherian non-singular closed, count on a duality sheaves, even schemes. allow theorem for a finite Y ~tale morphism be a __.. , curve over a box ok, a l g e b r a i c a l l y and permit X be a double for non- masking of y i ~ Y. T allow y 6 Y be a closed element, and permit Xl,X 2 be Y the 2 issues mendacity over y. be the functionality the sheaf K(Y), fields consisting of 2 copies injective at x 2. ~X-mOdules. and K(Y) on the element y. indecomposable One sees simply one c o n c e n t r a t e d K(X) of X and Y, respectively. centred (non-quasi-coherent) [II. Z. ll]. permit that of K(X), injective f~(G) enable Then G G is a [y-module is the sheaf on one c o n c e n t r a t e d it's the direct be X at Xl, and sum of 2 indecomposable 172 Now allow F = ~X" Then we have now f, HOm~x(F,f~G ) = 2K(X) HOm~y(f,F,G) = K(X) targeted hence the duality morphism ~f comment preschemes 6. eight. allow at the effects at y at y. can't be an isomorphism. f: X (not inevitably a in the community whole targeted > Y be a closed immersion in the neighborhood noetherian) intersection in Y [w of (rasp. with X Then we will increase of this part as follows. we will outline : D+(Y) (rasp0 f~ : ,> D+(X) D(Y) > D(X) by means of an identical formulation as above, whole intersection, cohomological ) noting that if then the functor X is a neighborhood HOm~v(f~x,-) has finite size.

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