New results and more detailed proofs #168
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- Changes in categories.tex: · Added lemma-pasting-law-pullbacks, https://stacks.math.columbia.edu/tag/001U#comment-3413 · Added lemma-3x3-pullback, from which the newly added lemma-magic-square and lemma-base-change-diagonal-1-cat easily follow (as well as one direction in the proof of lemma-pasting-law-pullbacks). - Changes in schemes.tex · Added changes proposed in https://stacks.math.columbia.edu/tag/01JW#comment-8464 and https://stacks.math.columbia.edu/tag/01JY#comment-8466 · Rewrote proof of lemma-fibre-product-after-map by referencing lemma-magic-square. · Simplified proof of (3) and (4) in lemma-separated-permanence by referencing lemma-base-change-diagonal-1-cat.
The aim of this commit is to incorporate the result "(quasi-)separatedness is local on the target" and propositions (1.2.6) and (1.2.7) of EGA IV, première partie (they are in pp. 227-228). (All of them into the section Separation axioms on the Schemes chapter.) The first one is quasi-separated-local-target. The proof introduces a lot of single-use definitions regarding properties of morphisms of schemes, and actually shows a general result for these definitions (P is LOCT implies Δ_P is LOCT). Such a phrasing for a proof might look weird. I would have written a new section instead on the Schemes chapter with the new definitions plus the mentioned result, but there's been objections to this: https://stacks.math.columbia.edu/tag/01KH#comment-7560 I also changed slightly the statement of lemma-closed-local-target and the statement and proof of lemma-quasi-compact-affine to have the results "«being quasi-compact» is LOCT" and "«being a closed immersion» is LOCT" fully spelled. Propositions (1.2.6) and (1.2.7) of EGA IV correspond to lemma-characterize-quasi-separated-scheme and lemma-quasi-separatedness-is-topological. In their statements, I've introduced the modern terminology of "quasi-separated topological space" and "quasi-separated map of topological spaces" (EGA doesn't use them), defined in a newly-added section in Topology. There's been hesitations to the introduction of such a section before (see comments in tag 0067). Nonetheless, I thought that giving the concept a name and using it makes more manifest the topological nature of quasi-separatedness. If you think this is not a sufficient reason to include the new section in Topology (I understand the reasons in your comment in tag 0067), one could instead fully spell out the concept each time it appears in lemma-characterize-quasi-separated-scheme and lemma-quasi-separatedness-is-topological. (Note: the statement of Proposition (1.2.7) of EGA IV contains an erratum, see https://math.stackexchange.com/q/4716228/394668 )
Added lemma-immersion, which provides an easy characterization of immersions of schemes.
Changes in: - schemes.tex: Added proof of the converse of lemma-where-are-they-equal - morphisms.tex: Given detailed proofs of lemma-integral-local, lemma-finite-local and lemma-base-change-finite - algebra.tex: Added converse in lemma-integral-local (for it is used in the proof of the homonymous lemma in schemes.tex)
Changes in: - topology.tex: · added the new lemma-correspondence-sober-generic-points-open. I've noticed that this result is used implicitly in the sections "Relative normalization" and "Normalization" of the Morphisms chapter. It took me a while to find out about it, so I thought it's worth to be added. · Added the new lemma-correspondence-closed-irreducible-open, used in the proof of the former. - schemes.tex: Added a new result, lemma-quasi-separated-coproduct, (and the auxiliary lemma-fibre-product-commutes-coproduct) that was implicitly used in morphisms.tex to show that 035M is quasi-separated (it's now invoked there). - morphisms.tex: · given more detail in the proof of lemma-normalization-generic with the aid of the afore-mentioned newly added result in topology.tex · Given more detail in the final step of the proof of lemma-description-normalization.
…omponents I regarded the characterization interesting for these schemes are used (among other places) in morphisms.tex, section-normalization and divisors.tex, section-meromorphic-functions and section-meromorphic-reduced
The purpose of this commit is to add the previously omitted argument in the last sentence of the proof of lemma-normal-normalization (Chapter on morphisms). For this, I defined in the topology chapter what does it mean for a continuous map of spaces to "induce a bijection on irreducible components." This notion is used undefined for instance in 037A, and in lemma-normal-normalization itself (although its definition is somehow acknowledged in https://stacks.math.columbia.edu/tag/04KX#comment-7268). I also added the obligatory characterization of the notion for sober spaces, lemma-characterize-bijection-irreducible-components-sober. (We also add the reference to "by a topological argument which we omit" in the proof of lemma-normal-normalization.)
Added detail in a step of a proof of lemma-normalization-normal in the Morphisms chapter.
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As always, a detailed account of the changes is explained inside each commit. Summing up, the types of changes I've introduced are (i) added previously omitted proofs, (ii) added more detail to already existing ones and (iii) added all the missing characterizations of quasi-separated and separated (morphism of) schemes I am aware of.
As I said in #167: if there is anything you think should be written/structured in any other way (or not written at all), feel free to change it. Also, I welcome feedback for (a) if you want that I change something of this edit before you accept it and (b) to improve the style of any future contributions.