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\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

On left comment #8451 on Lemma 29.44.16 in Morphisms of Schemes

I think my comment was wrong. There is a filtration after base change, and you can use it to prove the result. However, that is not the shortest proof, and I do not know if the filtration comes from a sequence of sections. I need to double-check what is in "N'eron models" -- they have effectively evicted us from our offices while they repair the HVAC system.

On left comment #8450 on Lemma 29.44.16 in Morphisms of Schemes

Thanks Jason!

On left comment #8449 on Lemma 29.53.4 in Morphisms of Schemes

Since we've defined the normalization only for a quasi-compact and quasi-separated morphism, shouldn't one remark at some point during the proof that this is indeed the case for ? (It follows from 26.21.13 and 26.21.14.)

Also, instead of "with a unique morphism . We omit the verification that the diagram commutes," one could write "with a unique morphism over . On the other hand, by , we have ."

On left comment #8448 on Section 27.4 in Constructions of Schemes

I propose to add the following remark to the webpage: Let be a scheme and let be a morphism of quasi-coherent -algebras. As it was explained in #8438 (ii), we obtain a morphism over . This gives a transformation of functors that induce a transformation of functors , where is our old friend and is the functor taking a scheme over to . We claim that at the transformation is given by precomposition by . We begin with a morphism of -algebras that gives a morphism over . In turn, we get a composite . By #8438 (i), this gives a morphism of -algebras that on sections over an open affine reads as , i.e., it equals the composite , which is what we wanted to show.

On left comment #8447 on Lemma 29.44.16 in Morphisms of Schemes

By the discussion in "N'eron models" around finite flat morphisms, after a faithfully flat base change there is a filtration of f_*O_X whose associated graded pieces are f-pushforwards of pushforwards of invertible O_S-modules along sections of f (i.e., multisections are "scheme-theoretic unions" of sections after faithfully flat base change). The composition of those sections with sigma give a corresponding filtration of the structure sheaf of the relative Proj (by ideal sheaves) compatible with the filtration on the pushforward of O_X. You can use this to prove that the morphism from the structure sheaf of the relative Proj to the pushforward of O_X is surjective.

On left comment #8446 on Lemma 10.36.12 in Commutative Algebra

I think instead of "we have and in " it would be more informative to write "there are such that in ." At least in the calculation I derived I end up with different 's (this doesn't affect the rest of the proof).

On Shubhankar Sahai left comment #8445 on Section 47.8 in Dualizing Complexes

I agree with MAO Zhouhang that it's useful to mention here that is the category of complexes with -torsion cohomology.

On left comment #8444 on Lemma 29.44.16 in Morphisms of Schemes

Argh, the reference to Lemma 26.21.11 is still wrong because the Proj is over and not over . The best thing would be for people to work out for themselves why is a closed immersion (by doing a local computation). But we can also see it using theory. Write as where . (Namely, we have and there is a relative version of this.) Then with there is a map of graded -algebras. (Namely, given a ring map there is a graded -algebra map , and there is a relative version of this.) The map is surjective in all sufficiently high degrees. (Namely, in degrees .) Whence the corresponding morphism is everywhere defined and a closed immersion, see Lemma 27.18.3.

On left comment #8443 on Lemma 27.2.2 in Constructions of Schemes

Okay, sorry, the conditions and already follow from the current hypotheses (maybe one could mention this?) .

On left comment #8442 on Lemma 27.2.2 in Constructions of Schemes

In, I think the middle and last arrows should be labelled as and , respectively.

Instead of saying "by covering the intersection by elements of and taking" I think it would be better to write "by considering any such that and defining", since actually using some covering is not enough to later construct .

To give just the minimum amount of hints to avoid the first use of "we omit," one could:

  1. After "condition (d) says exactly that this is compatible in case we have a triple of elements of ," add "i.e., ."
  2. If we used the notation for what the proof currently denotes , then instead of "we omit the verification that these maps do indeed glue to a " one could write "since for with , by Sheaves, 6.33.1, we get a ."

On the other hand, I think that for to follow, in the lemma hypotheses we would need to add "suppose and for ." The proof of may be added after "we get our on " by adding "along with isomorphisms , , such that for . In particular, for with , we have , which is (in the middle equality we used the definition of , plus )."

On Reiser left comment #8441 on Lemma 31.31.5 in Divisors

Potential typo in the statement of the lemma: it is written "[l]et be an open", but condition (3) is that , which only makes sense if .

On left comment #8440 on Lemma 29.11.5 in Morphisms of Schemes

Minor style typo: in the statement perhaps one would want to enclose the name of the categories between curly braces to fit the style in 29.11.6.

2nd proof (maybe worth of mention?): The functor is fully faithful by remark (ii) of , and it is essentially surjective by 29.11.3.

On left comment #8439 on Section 27.4 in Constructions of Schemes

Final remarks on why I am writing all these comments in this section instead of just introducing a pull request on GitHub: In my opinion, it would be better to change this whole section to the study of the representability of instead of , and maybe mentioning the existence of just at the end plus the categorical remark in #8435. I do think so because (a) is easier to remember and more motivated than , see #8435 (I also think it's what #3430 had on his mind), and (b) proofs with are less wordy than those with . However, I was unsure whether such a big edit would be accepted or if only a proper subset of all the edits would pass. In consequence, I deemed it better to point it all out here.

On left comment #8438 on Section 27.4 in Constructions of Schemes

I think it would be nice to add the following two remarks somewhere in this section. I will be adopting the POV explained in #8435 of representability of instead of .

(i) So far, we have an explicit description of the map Namely, it is the one induced by the universal element . However, we don't have yet an explicit description of the inverse of \eqref{map}. Here's one: Given an -linear map , the induced map is the unique morphism over such that for all open affine , the restricted map over equals , where is the map in 27.3.4 and the first arrow comes from Schemes, 26.6.4. Proof: First we note that is a sheaf on (basically because is a sheaf). Then we note that for any open , we have an induced map Namely, it is the one obtained by the universal element of , where is the inclusion and we are using the notation given in . From here, it is not difficult to verify that the diagram commutes. In other words, we have a morphism of sheaves on . Thus, we can characterize the inverse of \eqref{map} locally on and we finish by the proof in

(ii) An explanation of why is a functor: If we have a morphism of -algebras (we've used 27.4.6, (3)) then we obtain a canonical map over . By remark (i), it is the unique morphism of schemes over such that for every open affine , we have that identifies with via the isomorphisms of Lemma 27.3.4. It follows that is a contravariant functor from the category of quasi-coherent -algebras to the category of schemes over . On the other hand, equation \eqref{map} specializes to so the functor is fully faithful.

On Shubhankar Sahai left comment #8437 on Section 52.1 in Algebraic and Formal Geometry

'The second main result is that for a morphism if ringed sites....' right above the displayed equality of derived completions has a typo. The 'if' should be 'of'.

On left comment #8436 on Lemma 27.4.2 in Constructions of Schemes

I would propose writing the whole proof in this way: I'm going to adopt the POV of representability of instead of , see . We have (where just means regarded as an -module, notation introduced in the paragraph before 10.14.3). To show that represents we check that the identity map is a universal element for . Fix . We have a map , induced by . To construct the inverse, for an -linear map (an element of ), we have a map on global sections , which is -linear. Therefore, it induces a morphism over , by Schemes, 26.6.4. On the one hand, it is easy to see that the composite is the identity (use uniqueness in Schemes, 26.6.4). On the other hand, to see that is the identity, we note that all the involved maps are compatible with restrictions to distinguished open subsets of (i.e., given distinguished open, one restricts to and to ). Hence, it suffices to see that commutes. But this is easy.

On left comment #8435 on Section 27.4 in Constructions of Schemes

I think it's also interesing to think about the presheaf on that the object represents. Let be a category, and let be an object of . Let be a presheaf on . Consider the induced functor that sends to . Then is representable if and only if is representable (not difficult to show). In our situation, , , the functor sends to , and is

That is, in Görtz & Wedhorn words (see (11.2) of the 2nd ed.), this construction can be regarded as a globalized version of the natural isomorphim where is an -algebra.

On ZL left comment #8434 on Lemma 65.20.2 in Properties of Algebraic Spaces

I'm a little confused about the statement of the lemma and its proof. In the statement, we see is an algebraic space étale over . But in the proof, after citing lemma 17.5.2, we see that , having an underlying topological space, is a scheme.

I guess that previous lemma 65.20.1 is used to prove the space case. Since the , we may assume . Then consider the sheaf assigning to a singleton if and assigning to otherwise. Then using 65.20.1 we get the open subspace .

There is also a typo for the conclusion : where .

On left comment #8433 on Lemma 27.4.4 in Constructions of Schemes

I think the expression should be instead. Here's an alternative way to phrase the proof that doesn't require invoking : "Let be open affine. Since the functor is represented by , let be the universal element. By the proof of 26.15.4 (we verified the hypotheses in the proof of 27.4.3), it suffices to see that . This follows from the construction of and ."

On Ben Church left comment #8432 on Lemma 37.24.7 in More on Morphisms

In line 4, it should read "such that (X_{K'})_{red} is geometrically reduced"