Recent comments—The Stacks project

Comments 2061 to 2080 out of 9050 in reverse chronological order.

On Jul 21, 2022 Hao Peng left comment #7501 on Section 1.1 in Introduction

I am sorry to comment here but I want to know how to cite tags or comment tags that is clickable in the comments?

On Jul 21, 2022 Hao Peng left comment #7500 on Lemma 48.12.7 in Duality for Schemes

It should follow from tag 0G8I that $Rf_\ast\omega_{X/Y}$ has zero-th cohomology sheaf isomorphic to $\mathcal Hom(f_\ast\mathcal O_X, \mathcal O_Y)$ if $f$ is proper smooth of relative dimension $n$ , does this holds for more general situations?

On Jul 21, 2022 Hao Peng left comment #7499 on Lemma 48.12.7 in Duality for Schemes

In thr proof of (1) $Rf_*\mathbb O_Y$ should be replaced by $Rf_*\mathbb O_X$ .

On Jul 19, 2022 tao left comment #7498 on Definition 10.45.1 in Commutative Algebra

Shoud it be finite extension or algebraic extension here?

On Jul 19, 2022 Johan left comment #7497 on Section 85.3 in Simplicial Spaces

@#7493 Yes. With the formulation as given one gets the cleanest presentation of the material, I think. Now if one has a category $\mathcal{C}$ fibred over $\Delta$ , then there is a replacement $\mathcal{C}'$ of it where the pullback functors commute on the nose, see Lemma 4.36.4. If each of the fibre categories of $\mathcal{C}$ over $\Delta$ is endowed with the structure of a site and the pullback functors of $\mathcal{C}$ define morphisms of sites, then I think it is easy to see that each fibre category of $\mathcal{C}'$ is endowed with the structure of a site so that we end up in Case A. The topos of the site $\mathcal{C}'_{total}$ one gets from this doesn't depend on the choices made by Lemma 85.3.4.

Some of this does not belong here. Instead we need to upgrade the discussion in Section 7.18. See also my comment on Tag 59.51.2 in response to a (related) comment of Tongmu He. There are also various additional descent questions one can ask when $\Delta$ is replaced by an arbitrary category $D$ and one has a discussion of $D$ -topoi. Etc.

On Jul 19, 2022 Hao Peng left comment #7495 on Lemma 33.44.6 in Varieties

i noticed that proof of tag01YV works for a natural extension of statements to proper schemes of dimensione $\le2$ , and then this proposition will follow as a direct consequence. The proof will be much shorter, because the current proof is essentially doing devissage for coherent sheaves again.

On Jul 19, 2022 Xiaolong Liu left comment #7494 on Lemma 28.25.3 in Properties of Schemes

Before the last sentence, one should be $W_{jj'k}$ instead of $W_{jj'l}$ .

On Jul 18, 2022 Bogdan left comment #7493 on Section 85.3 in Simplicial Spaces

Case A. Let C be a simplicial object in the category whose objects are sites and whose morphisms are morphisms of sites.

It seems like the definition suggests that the category of sites is considered as a usual category (and not a $2$ -category). In particular, the relation on $u_\varphi$ is $u_\psi \circ u_{\varphi}=u_{\psi \circ \varphi}$ (as opposed to a ''coherent'' system of isomorphisms).

But it seems that this assumption is too strong to hold in practice. Usually, $u_{\varphi}$ is given by a pullback functor, which is defined up to a "canonical isomorphism".

On Jul 18, 2022 WhatJiaranEatsTonight left comment #7492 on Lemma 33.43.2 in Varieties

I think we need $X$ to be separated here. Otherwise \the projecitve line with double original point is a counterexample.

On Jul 18, 2022 Xiaolong Liu left comment #7491 on Lemma 36.8.1 in Derived Categories of Schemes

$\mathcal{I}\subset X$ should be $\mathcal{I}\subset \mathcal{O}_X$ here.

On Jul 18, 2022 Diego left comment #7490 on Section 23.8 in Divided Power Algebra

Line -8 in the proof of Proposition 23.8.4: the left-hand side should read $dim(S/IS)-dim(S/J)$

On Jul 17, 2022 Hao Peng left comment #7489 on Lemma 53.3.2 in Algebraic Curves

$\mathcal O_X^r$ should be replaced by $\mathcal O_X^{r+1}$

On Jul 16, 2022 Louis Carlin left comment #7488 on Lemma 4.2.18 in Categories

I think there is a typo in the second sentence. It is probably intended to be "Suppose for every $X \in Ob(\mathcal{B})$ we are given an object..." or something similar.

On Jul 15, 2022 Hao Peng left comment #7487 on Section 21.36 in Cohomology on Sites

The appearence of X is weird. Should be $\mathcal C$

On Jul 14, 2022 Hao Peng left comment #7486 on Lemma 49.12.6 in Discriminants and Differents

Borrowing drom differential geometry, would it be better to use $df^*$ instead if $df$ ?

On Jul 14, 2022 Hao Peng left comment #7485 on Lemma 49.12.6 in Discriminants and Differents

The central term of the exact sequence should by $\Omega_{Y/S}$

On Jul 14, 2022 Hao Peng left comment #7484 on Definition 49.9.1 in Discriminants and Differents

in fact in tag0BWJ, $f: Y\to X$ may not by Noetherian

On Jul 14, 2022 Hao Peng left comment #7483 on Definition 49.9.1 in Discriminants and Differents

Should it be that all the propositions are true for Noetherian repalce by locally Noetherian? I don't see why we need Noetherian.

On Jul 13, 2022 Hao Peng left comment #7482 on Example 53.13.6 in Algebraic Curves

Shouldn't the map to P^1 be projection to x coordinates instead of y cooediantes?

On Jul 12, 2022 Nicolás left comment #7481 on Section 7.16 in Sites and Sheaves

A small nitpick comment: in the first paragraph, the letter $u$ is used both to reference the functor $u\colon \mathcal{T}_H \to \mathcal{T}_G$ and some element $u \in U$ .