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By Pogorelov, A.V.

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Therefore, ˇ 1 (X, S) := lim H ˇ 1 (U, S) ∼ H = −→ U ˇ 1 (U, S) H U (disjoint union). For details we refer to Mallios [62, Vol. I, p. 183]. ˇ q (X, S), for all q ∈ Z+ , is the The collection of the A(X)-modules H 0 ˇ Cech cohomology of X with coeﬃcients in the sheaf (A ( -module) ˇ ∗ (X, S). 21) ˇ ∗ (X, S) satisﬁes If X is a (Hausdorﬀ ) paracompact space, then H the axioms of a cohomology theory. The assumption that X is a (Hausdorﬀ) paracompact space is a suﬃcient condition ensuring the existence of the long cohomology sequence, derived from a short exact sequence of sheaves.

Sheaves and all that 32 is not necessarily exact. 3) leads to the exact sequence of S(A)-modules S(φi−1 ) S(φi ) Si ) −−−−−→ S(S Si+1 ) −→ · · · · · · −→ S(S Si−1 ) −−−−−−−→ S(S since the inductive limits preserve the exactness. 4) ψ φ 0 −→ R −−→ S −−→ T −→ 0, leads to the corresponding short exact sequence of S(A)-modules S(φ) S(ψ) 0 −→ S(R) −−−−→ S(S) −−−−→ S(T ) −→ 0. 11], Wells [142, p. 52]). 5) from being exact at Γ(T ). 6. Sheaf cohomology Sheaf cohomology can be approached from various points of view.

Hence, in all the expressions involving restrictions of sections, we use the restriction morphisms of the form ρU V. Chapter 1. 14), respectively. ˇ Accordingly, the Cech cohomology of X with coeﬃcients in the presheaf ((A-module) S is the collection of A(X)-modules ˇ q (X, S) ˇ ∗ (X, S) = H H q∈Z+ 0 . 21)): ˇ ∗ (X, S) satisﬁes all the axioms of a cohomology theory, without any H restriction on the topology of X. 1). To be more speciﬁc, regarding the previous comment, assume ﬁrst that U φ ≡ (φU ) : S ≡ S(U ), ρU V −→ T ≡ T (U ), τV is an A-morphism of A-presheaves over any topological space X.