The theory of sheaves by Richard G. Swan
By Richard G. Swan
Concept of sheaves.
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Extra resources for The theory of sheaves
For the present, we note the following pro position and lemma. (CJ) = 0 i > o, all j. Such a complex we call a T-resolution of A. , if 0 -» A -» C, 0 A* -» C 1 are T-resolutions and - 52 - A -> C if. Ca ) The reader will easily see what preservation of 6 means. (Z^) -> R1! ^ ) -» . By assumption R^T(C^) = 0. (ZJ+1). (Z1 ) « ... (Zi"1). iC1-1) = 0, R ' S M Z 1"1) = Coker t . (Z^) = Ker Z 1. Therefore Coker t = H ^ R ^ - d ) ) « H ^ T d ) ) , since T is leftexact. Naturality follows from the naturality of all the isomorphisms considered.
Proof: Split the exact sequence into 0 - > A - > A 1 ~ » Z 2 ^ 0, 0 Z 2 -» A2 -» -» 0, etc. and use the initial argument of the previous proposition. Lemma 7 : Let T be a covariant K-functor and let ITU be a class of objects of Cfc such that: 1 . I injective I e'ftt/. 2. 0 - » M ' - > M - » A - > 0 exact and M 1, M e ® ^ A e TJV. 3- exact sequence of elements of 1VU H O -* T( M 1) -» T(M) -» T(M") -» 0 exact. Then R^-iM) = 0, i > o, and all M € H t . Proof: Let 0 M 4 I be an injective resolution of M.
We leave the reader to prove that the isomorphism of the proposition preserves e and 6. Definition: An element A e fl is called T-acyclic if R ^ T . ( A ) = 0 when i > o. Lemma 6 : Let 0 -» A -» A1 ... An B -> 0 be exact, with A l'” ',An T-acyclic. (A). This is natural with respect to maps of exact sequences of the type considered. Proof: Split the exact sequence into 0 - > A - > A 1 ~ » Z 2 ^ 0, 0 Z 2 -» A2 -» -» 0, etc. and use the initial argument of the previous proposition. Lemma 7 : Let T be a covariant K-functor and let ITU be a class of objects of Cfc such that: 1 .