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K ~(Spec k,F') [[Spec K ~ S p e c k]] may be assumed Galois over k for by taking The naturality of the above isomorphisms yields, in the notation of Prop. e. if F' F' :(Cat T) 0 -*Ab is an additive presheaf commutes with finite coproducts of schemes), then composition with G yields an additive functor F : ~Ab and isomorphisms satisfying (*) • lim Notation. As noted prior to Thm. 3. _! primes Let L/k be a Galois field extension with group the Grothendieck topolo~ constructed above. p , Then ~ for - •d .

Let F : ~ ~ Ab be left exact additive functors such that e~uivalent. As usual, Then for all n > 0 , RnF F and and and GS (RnG)S ~e G : ~ ~Ab natural~ are naturall~ equivalent. 7. Proof. A¢ 8 ~Ab PSpec k ") and given by Let ~-module A : C P(F) = F(Spec k) . ~Ab be given by M • I~ are naturally equivalent. 3 provides natural isomorphisms s(spec A) Now setting --~ A = k (~0~S)(Spec A) ~ ($S)*eHGA ~ (lim SGK)*eA • gives (since ek = k) natural isomorphisms - 56- S(Spec k) and ~ (lim SGK) g since k' = g.

E:A*B Any open subgroup g that obtained by composing U of the compact group index, and hence is of the form ~' M with the g is of finite for some finite field extension - Kl of L, then Thus k inside K' L • If K 28 - is the normal closure of is a normal, open subgroup of MU C and, since its subgroups MV as V M is discrete, ~ in g contained in M is the union of ranges over the normal, open subgroups of S . M on objects, the last assertion of Thm. 9 and the usual The final conclusion now follows from the definition of construction of direct limits in A functor objects U • C of Ca F:~ ~ A b will be called torsion if, for all C , F(C) is a torsion abelian group.

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An Introduction To Linear Algebra by Kuttler

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