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1. • CommentRowNumber1.
   • CommentAuthorRichard Williamson
   • CommentTimeApr 29th 2018
   • PermaLink
   Author: Richard Williamson
   Format: MarkdownItexAdded statement of the fact that any open subset of the underlying topological space of a scheme $X$, together with the sheaf of commutative rings on it obtained by restriction of the structure sheaf on $X$, defines an open subscheme of $X$. Began proof, but not finished yet. Whilst the proof is straightforward, there is something to show; it is not completely trivial. <a href="https://ncatlab.org/nlab/revision/diff/open+subscheme/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/open+subscheme/4">v4</a>, <a href="https://ncatlab.org/nlab/show/open+subscheme">current</a>

   Added statement of the fact that any open subset of the underlying topological space of a scheme $X$, together with the sheaf of commutative rings on it obtained by restriction of the structure sheaf on $X$, defines an open subscheme of $X$. Began proof, but not finished yet. Whilst the proof is straightforward, there is something to show; it is not completely trivial.

   diff, v4, current

2. • CommentRowNumber2.
   • CommentAuthorRichard Williamson
   • CommentTimeApr 29th 2018
   • (edited Apr 29th 2018)
   • PermaLink
   Author: Richard Williamson
   Format: MarkdownItexCompleted the proof, modulo a couple of basic pieces of commutative algebra. Would be great if someone finds time to write up these (on a different page)! I tried to be a little more careful than seems to be typically the case (compare for example the proof in EGA I (Proposition 2.1.3 in Chapter I) or the [stacks project](https://stacks.math.columbia.edu/tag/01II), Lemma 25.9.2), keeping track of the isomorphism between $N$ and $Spec A$, and mentioning some points which one needs to convince oneself are true. <a href="https://ncatlab.org/nlab/revision/diff/open+subscheme/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/open+subscheme/5">v5</a>, <a href="https://ncatlab.org/nlab/show/open+subscheme">current</a>

   Completed the proof, modulo a couple of basic pieces of commutative algebra. Would be great if someone finds time to write up these (on a different page)! I tried to be a little more careful than seems to be typically the case (compare for example the proof in EGA I (Proposition 2.1.3 in Chapter I) or the stacks project, Lemma 25.9.2), keeping track of the isomorphism between $N$ and $Spec A$, and mentioning some points which one needs to convince oneself are true.

   diff, v5, current