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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
started a stub for projective space
Added the first part of the proof that a $\mathbb{Z}$-grading on a commutative ring $R$ is the same as a $\mathbb{G}_m$-action on $Spec R$. (All a bit rough for the time being.)
added to projective space the remainder of the proof that $\mathbb{G}$-actions on $Spec R$ are equivalent to $\mathbb{Z}$-gradings of $R$.
(I guess I should move that elsewhere.)
I have spelled out how real and projective space become topological manifolds and smooth manifolds, here.
I have filled in full details in the proof of the CW-structure at complex projective space. Then I copied this over also to Projective space – Examples – Real and complex projective space, so that the discussion there is now a self-contained proof of the manifold structure on real/complex projective space.
Among the list of elementary facts, the statement that $S^n \to \mathbb{R}P^n$ is locally trivial had been missing. For completeness, have now included statement and proof here.
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