Of course, nobody ever uses the singular homology of a scheme, since we actually have sheaf cohomology, and Čech cohomology, which are all much more meaningful. However, my question is, what would happen if we tried to define the singular homology of a scheme, let's say, of $\mathbb{A}_\mathbb{C}^1$, or of $\mathbb{P}_\mathbb{C}^1$?
What is the singular homology of a scheme?
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general-topology
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algebraic-topology
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0It might be useful for you to look up etale cohomology, and in particular the Artin comparison theorem. – 2017-01-23