By David E. Dobbs (auth.)
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Extra resources for Cech Cohomological Dimensions for Commutative Rings
Hm) . K with The discussion preceding Thm. 15 h~ a(y) = ( y , . ,hY) for y c FK 1 to - 5S - However each element of and ~ is fixed by § K' • and hence in the image of image of 7 is a restriction of some element of Thus x is in the equalizer of FK - F ~ • and the isomorphisms It follows that ($) G x K' and ~ , is in the have been constructed. It remains to check their naturality. Remark. 7 : FK ~ M K Sheaf properties were Just used to prove the map is an isomorphism. for any additive F, The argument shows that 7 exists but it is not clear what else (possibly an analogue of Cor.
P(g) in terms of two cohomology theories defined by Grothendieck ~ ] . The arguments will require a deeper study of the above construction of the functor M It is with this study that we close the section. 14. some nonzero al~ebra Proof. assume K Since and A A , 0 then (M*@)f for some field K and is a monomor2hism. preserves injective morphisms, we may are objects of of projections from prove that each l_~f f c A(K,A) B . If [pi } is the collection A to its simple components, it suffices to M (pi f) is a monomorphism.
Spec As in §4, provides a categorical equivalence G : A ~ (Cat T) 0 m If H is an inverse of may check that G and M M*eH : (Cat T) 0 ~ A b a discrete ~-module, then one is a sheaf by the preceding theorem. Let Cg be the category of discrete g-modules (and g-module maps) - and $ 48 the category of sheaves on For any f e Cg(M,N) and object - T B (and naturaltransformations). of B, composition with f gives a group homomorphism fB : 9 " s e t ( k ' a l g ( B ' L ) ' M ) since f object U ~ g-set(k-alg(B,L),N) is itself, in particular, a group homomorphism.