Recent comments—The Stacks project

Comments 1 to 20 out of 4650 in reverse chronological order.

On Mar 28, 2020 songong left comment #5003 on Lemma 10.103.8 in Commutative Algebra

Isn't this just a special case of 10.71.6?

On Mar 28, 2020 Xavier left comment #5002 on Section 26.15 in Schemes

It seems that setting each $V_i$ maximal makes it easier to verify the uniqueness of $g$ , since for another $f$ with $f^* \xi' = \xi$ one automatically have $f^{-1}(U_i) \subset V_i$ .

On Mar 27, 2020 Johan left comment #5001 on Section 61.7 in The Trace Formula

Yes. I should probably move this into the obsolete chapter. But this whole chapter needs to be improved drastically. Also, I think we can improve on the exposition of Section 13.13.

On Mar 27, 2020 Lenny Taelman left comment #5000 on Section 61.7 in The Trace Formula

I presume this was added before 05RX (from the preliminaries part, chapter derived categories), and is now obsolete?

On Mar 26, 2020 Johan left comment #4999 on Section 6.15 in Sheaves on Spaces

The empty limit is a final object.

On Mar 26, 2020 Chris Li left comment #4998 on Section 6.15 in Sheaves on Spaces

Any hint for the existence of final object in 007O (a)? What if we remove the final object from the given category?

On Mar 23, 2020 SDIGR left comment #4997 on Lemma 4.32.10 in Categories

The strongly cartesian morphism in $\mathcal{X}\times_{\mathcal{S}}\mathcal{Y}$ is given by $(a,b)$ not by $(G(a),G(b))$ .

On Mar 19, 2020 Jérôme Poineau left comment #4996 on Lemma 26.20.4 in Schemes

In (1), the morphism Spec $(A) \to S$ should be called $f$ , so that we know what $f$ refers to in (2).

On Mar 18, 2020 Noah Olander left comment #4995 on Proposition 57.89.6 in Étale Cohomology

I think when you write "On the other hand, by Theorem 03Q9 we have $R^2f_*\mathcal{F}_{\overline{\eta}} = H^2 ( \mathbf{A}^1_{\overline{\eta}} , \mathcal{F})$ , you mean $\eta$ is the generic point of $\mathbf{A}^{d-1}$ but you don't say what $\eta$ is.

On Mar 18, 2020 alexis bouthier left comment #4994 on Section 29.50 in Morphisms of Schemes

They do not, take any ring $A$ of finite Krull dimension such that $dim(A[t])\geq dimA +2$

On Mar 18, 2020 Li Yixiao left comment #4993 on Lemma 35.32.4 in Descent

emm I made a trivial mistake...

On Mar 18, 2020 Li Yixiao left comment #4992 on Lemma 35.32.4 in Descent

The maps $V' \times _{V} V' \to V' \to W' \to W$ can be written as $V \times _ X X' \times _ X X' \to V \times _ X X' \to W \times _ X X' \to W$ , which are clearly equal to $V \times _ X (X' \times _X X') \to V \to W$ .

On Mar 17, 2020 Tongmu He left comment #4990 on Lemma 10.97.1 in Commutative Algebra

I would like to give an another proof of lemma 10.97.1.

(1) First, if we only require that $I^n M_n=0$ and $I$ is finitely generated, let's show that $M$ is $I$ -adically complete: We have $M\to M/I^nM\to M_n$ . Taking limit, we get $M\to M^\wedge\to M$ . Hence $M$ is a direct summand of $M^\wedge$ . By lemma 10.95.3, $M^\wedge$ is $I$ -adically complete, so is $M$ .

(2) If moreover $M_ n = M_{n + 1}/I^ nM_{n + 1}$ , let's show that $M/I^nM=M_n$ : Consider short exact sequence $0\to N_n\to M\to M_n\to 0$ . Notice that $\operatorname{Coker}(N_{n+1}\to N_n)=\operatorname{Ker}(M_{n+1}\to M_n)=I^nM_{n+1}$ is killed by $I$ . Therefore, by Nakayama's lemma 10.19.1, $N_{n+1}/(N_{n+1}\cap I^{n+1}M)\to N_{n}/(N_{n}\cap I^{n}M)$ is surjective. On the other hand, (1) shows that $\lim N_{n}/(N_{n}\cap I^{n}M)=0$ . We conclude that $N_{n}/(N_{n}\cap I^{n}M)=0$ and thus $M/I^nM=M_n$ .

On Mar 16, 2020 Noah Olander left comment #4985 on Proposition 57.89.6 in Étale Cohomology

There's a typo right before you write down the E2 page of the spectral sequence: You should write $R^p g_*R^q j_*\mathcal{F}=0$ for $p>2$ instead of $R^pj_* \mathcal{F}$ .

On Mar 15, 2020 Trenday left comment #4984 on Section 15.4 in More on Algebra

which is easily seen to imply that the ideal

On Mar 14, 2020 Noah Olander left comment #4982 on Lemma 57.68.3 in Étale Cohomology

In the second line should you write $\mathrm{Pic}^0 (\overline{X})$ instead of $\mathrm{Pic}^0 (X)$ ?

On Mar 13, 2020 Elyes Boughattas left comment #4981 on Section 91.12 in Algebraic Stacks

Typo in the last line of the proof of lemma 91.12.4: $x\circ f$ should be $f\circ x$ .

On Mar 12, 2020 Kazuki Masugi left comment #4980 on Lemma 42.3.2 in Chow Homology and Chern Classes

Does this lemma hold in general $M$ ? (Is "Cohen-Macaulay module"-ness nessesary?)

On Mar 12, 2020 Kazuki Masugi left comment #4979 on Lemma 42.3.1 in Chow Homology and Chern Classes

What does “In this case we can filter $M$ by powers of $\mathfrak{q}$ " mean? (Should I use lemma 10.61.1(Tag 00L0)?)

On Mar 12, 2020 Elyes Boughattas left comment #4978 on Section 63.4 in Properties of Algebraic Spaces

In the proof of Lemma 63.4.6 : before the first diagram, "which in addition will prove that (2) holds" should be "which in addition will prove that (3) holds". Again, at the paragraph starting before the third diagram, "To finish the proof we prove (1)" should be "To finish the proof we prove (2)".