By Richard G. Swan

Idea of sheaves.

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For the other terms, we note that by proposition 7 , p. (l) = 0 for i > 0, since I is injective. Proposition 9? If T is left-exact and ^ i s a resolvent functor over T, then HK is a cohomological 6-functor over T. Proof: If 0 -> A 1 -» A -> Au -» 0 is exact so is 0 -4 A 1) -» Tr{A ) ^p(An) -» 0; taking cohomology, we have an exact 6functor. This is augmented and € is, by (ii), a natural equivalence. If I is injective, H1#^(l) = 0, for i > 0, by (iii), and so H 1 % is effaceable. Corollary: Under the conditions of proposition 9> there is a natural isomorphism of augmented 6-functors HK *%+ ~ R ^ T.

Let F, G, H be stacks. A bilinear map (3) b. f: F + G H is a collection of bilinear maps f^: F(U) + G(u) -> H(U) such that if V CL U, the following diagram is commutative: F(U) + G(U) -> H(U) F(V) + G(V) -> H(V) Since direct limits preserve direct sums and bilinearity, f gives rise to a corresponding bilinear map Ti F + G -> H of protosheaves and an argument exactly the same as before shows that f is consistent with the topology given to F, G, and H. , L preserves bilinearity. So, in fact, does T( ,x) .

The associated sheaf of ¥p ( ,M) we write C^, the Alex­ ander-Spanier sheaf of p-dimensional cochains. , Xp+^). 6 is a map of stacks, and 66 = 0. If ,M) is the constant stack, define e(u): Y ”1(U,M) -> ¥°(U,M) by letting e(U)(m) be the constant map with value m. This is called the augmentation. Note that 6 € = 0. , a sequence m 4 y°( , m) 4 y1 ( , m) 4 y2( , m) -» ... with 62 = 0, 6e = 0. This sequence is in fact exact. We define maps s: Y1+1(U,M) -» ’TL(U,M) (i > 0) and t : ¥°(U,M) -> ’J,-1(U,M).

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