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Let R be the constant sheaf on a manifold.I know that the Cech cohomology of M with values in R is isomorphic to the De Rham cohomology. I want to know if I use compact De Rham cohomology,whether there is a subset of Cech cohomology isomorphic to the compact De Rham cohomology?What is this subset?

Feng yi
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  • The compact De Rham cohomology is not generally a "subset" of the de rham cohomology. For example, on $R^n$, the n dimensional de Rham cohomology is zero, but the n dimensional compact support De Rham cohomology is $\mathbb{R}$. In general the relation between compact cohomology and "usual" cohomology is given by Poincare duality. My understanding is that there is a sheaf that computes compactsupported cohomology, and it is discussed in treatments of Verdier duality. I don't know the details there, so I'm going to refer you to this nice exposition: http://www.math.harvard.edu/~amathew/verd.pdf – Elle Najt Mar 09 '17 at 03:26

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