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Hi, I try to understand the orientation sheaves. When searching it in the google, i meet new areas such as local coefficient system and locally constant sheaves. I realize that any system of local coefficients on X is a locally constant sheaves. But what is the relation with orientation sheaves. Which refferences are there to read it? |
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These are purely topological notions and have nothing to do with algebraic geometry in particular. Let $M$ for simplicity be a topological manifold of dimension $n$. Then the orientation sheaf $\mathcal{L}_{or}(M)$ is the sheafification of the presheaf $U\mapsto H_n(M,M-U;\mathbb{Z})$. It's always a locally constant sheaf with stalks equal to $\mathbb{Z}$. One immediately checks that $\mathcal{L}_{or}$ is trivial if and only if $M$ is orientable. This definition can be generalized. As for the references, I'd suggest checking A.Dimca, Sheaves in Topology or B.Iversen, Cohomology of Sheaves. |
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