The Stacks project

Comments 1 to 20 out of 5404 in reverse chronological order.

\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 #5810 on Section 33.8 in Varieties

Thanks for the suggestion. I added and proved the lemma along the lines you suggested. Of course there are generalizations of this lemma long the lines of 33.7.13 where you have a "dense family" of morphisms with geometrically irreducible. See changes in this commit.


On left comment #5809 on Section 7.8 in Sites and Sheaves

Thanks for pointing this out. I only found one place where the arrow was in the wrong direction but it was confusing ideed. See changes in this commit.


On left comment #5808 on Lemma 41.7.3 in √Čtale Morphisms of Schemes

Thanks and fixed here.


On left comment #5807 on Section 15.61 in More on Algebra

Thanks and fixed here.


On left comment #5806 on Section 10.27 in Commutative Algebra

Thanks and fixed here.


On left comment #5805 on Section 15.60 in More on Algebra

I think you are right about the bidegree of being but I think you swapped the roles of and . So I think the differential for the first spectral squence of the example on the -page is given by maps which is what it says.


On left comment #5804 on Definition 99.4.1 in Morphisms of Algebraic Stacks

OK, I think this isn't necessary because we mention in the introduction 92.1 to this chapter that we are working without any separation assumptions. Moreover, just the fact that you are referring to Definition 4.1 of their book (which is their definition of algebraic stacks and not the definition of a quasi-separated morphism of algebraic stacks) seems to suggest that things are quite a bit different.


On left comment #5803 on Section 10.39 in Commutative Algebra

True but the proof as given is fine too.


On Giulio Bresciani left comment #5802 on Section 33.8 in Varieties

A simple lemma that for some reason no one points out is the following: if is an irreducible scheme of finite type over and is dense, then is geometrically irreducible. This follows from the similar statement 04KV for geometrically connected schemes by considering where is the image of the pairwise intersections of irreducible components of . Obviously it is possible to relax the finite type hypothesis.


On Bryan Shih left comment #5801 on Section 7.8 in Sites and Sheaves

By the definition of morphism of families, 00VT, seems incompatible with the lemmas later on where seems to be a map from rather than , i.e. 0G1L. Or am I missing something?


On left comment #5800 on Lemma 29.38.7 in Morphisms of Schemes

@#5799: If is defined over a field and is relatively very ample on over , then it follows from the definition that there is a -vector space and a surjective module map . Sure this does not mean that is surjective, but we never use this in the argument. (The argument actually shows you can take to correspond to and then the surjectivity also holds on the base but it doesn't always need to be true.)


On Luke left comment #5799 on Lemma 29.38.7 in Morphisms of Schemes

The last step in proving (4) should be modified, since even in the case when base is a field there are examples in which is not surjective, therefore the second map defined only on an open. Lukely we can chose the open as the complement of , then show that image of map is contained in the chosen open.


On James A. Myer left comment #5795 on Lemma 41.7.3 in √Čtale Morphisms of Schemes

I think "By Conditions (2), (3), and (4) imply that . Algebra, Lemma..." should read instead "Conditions (2), (3), and (4) imply that . By Algebra, Lemma..." -- I guess the "By" got cycled around.


On alexis bouthier left comment #5793 on Section 10.94 in Commutative Algebra

What I meant is that to prove theorem tag. 05A9, one can directly prove that being a direct sum of countably generated modules if fpqc local, so this would shorten the argument. The link to Drinfeld's paper is not the right one, I've sent the right version to stacks.project@gmail.com


On Jinhyun Park left comment #5792 on Section 15.61 in More on Algebra

It appears to me that there is a typo. In the displayed equation in the paragraph just above Lemma 068K, the Tor's subscrip indices of the 2nd and the 3rd term should be n+m, not n.


On left comment #5791 on Section 10.27 in Commutative Algebra

There's a small error in the sketch: it says "Thus if non-empty has maximal elements", but it should be "Thus if the set of ideals not in is non-empty, it has maximal elements".


On Brad Dirks left comment #5790 on Section 15.60 in More on Algebra

Is there a typo in the indices for Example 061Z? Homological spectral sequences have with bidegree , so should go from to . This would be a map . It seems that the page in the same example does have the correct indices.


On Leo left comment #5789 on Definition 99.4.1 in Morphisms of Algebraic Stacks

It may be worth pointing out that Laumon-Moret-Bailly seem to define quasiseparated to have separated diagonal (Definition 4.1). Some other places in the literature follow this convention, for example in Olsson's Log Geometry and Algebraic Stacks Remark 3.17, he explicitly states is not quasiseparated even though his Theorem 3.2 shows that its diagonal is quasicompact and quasiseparated (a consequence of his notion of "locally separated," which is the stacks project's notion + quasiseparated).


On Brad Dirks left comment #5788 on Section 10.39 in Commutative Algebra

I believe the first assertion in 00L3 was already shown in 0BUR.


On bouthier left comment #5787 on Section 12.4 in Homological Algebra

In this section, it would be helpful to add the construction of the Karoubi envelope of a category.