Recent comments—The Stacks project

Comments 1 to 20 out of 4610 in reverse chronological order.

On Feb 24, 2020 procvaustralia left comment #4954 on Lemma 10.38.12 in Commutative Algebra

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On Feb 24, 2020 awllower left comment #4953 on Lemma 29.34.18 in Morphisms of Schemes

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On Feb 23, 2020 SDIGR left comment #4952 on Remark 44.3.7 in Picard Schemes of Curves

Typo!?:
Last sentence:
The divisor $D$ corresponding to a $k$ -rational point of $\underline{\mathrm{Hilb}}_{X/k}^{d+1}$ must be of degree $d+1$ , not of degree $d$ .

On Feb 23, 2020 SDIGR left comment #4951 on Lemma 44.3.5 in Picard Schemes of Curves

Sorry, I mean the other way around.

On Feb 23, 2020 SDIGR left comment #4950 on Lemma 44.3.5 in Picard Schemes of Curves

First sentence of the Proof: $\underline{\mathrm{Hilb}_{X/S}^0}=S\mapsto\underline{\mathrm{Hilb}_{X/S}^d}=S$ .

On Feb 23, 2020 Min left comment #4949 on Section 106.19 in Moduli of Curves

What is T in the functor the dualizing sheaf represents? Should it be U?

On Feb 23, 2020 Min left comment #4948 on Section 106.19 in Moduli of Curves

What is $T$ in the functor the dualizing sheaf represents? Should it be $U$ ?

On Feb 22, 2020 yogesh left comment #4947 on Lemma 10.62.4 in Commutative Algebra

Oops, sorry disregard my previous comment. I got forgot we're talkng about the annihilator as an element of $M$ , not $M/M_{n-1}$ . I was thinking of using the previous result of associated primes of short exact sequences applied to $M_{n-1} \to M \to R/\mathfrak{p}_n$ so $Ass(M) \subseteq Ass(M_{n-1} \cup \{ \mathfrak{p}_n\}$ But the proof you give is basically the proof of that result, whose proof is omitted.

On Feb 22, 2020 yogesh left comment #4946 on Lemma 10.62.4 in Commutative Algebra

Oops, sorry disregard my previous comment. I got forgot we're talkng about the annihilator as an element of $M$ , not $M/M_{n-1}$ .

On Feb 22, 2020 yogesh left comment #4945 on Lemma 10.62.4 in Commutative Algebra

In the proof, I think the inclusion $\mathfrak{p} \subset \mathfrak{p}_n$ is going the wrong way and the rest of the proof is unnecessary. I think any nonzero element of $R/\mathfrak{p}_n$ is annihilated by exactly by $\mathfrak{p}_n$ .

On Feb 22, 2020 yogesh left comment #4944 on Lemma 10.67.10 in Commutative Algebra

In the second to last sentence of the proof, there is no need to appeal to Lemma 07DV (looks like this was necessary before you fixed $f_j^e$ to $f_j$ in 2014)

On Feb 22, 2020 yogesh left comment #4942 on Lemma 10.38.14 in Commutative Algebra

oops, nevermind. disregard my previous comment

On Feb 22, 2020 yogesh left comment #4941 on Lemma 10.38.14 in Commutative Algebra

the exact sequence in the first line of the proof is always exact. maybe it should be N in place of $\ker(\alpha)$

On Feb 22, 2020 Rankeya left comment #4940 on Lemma 10.48.3 in Commutative Algebra

$R$ should be $S$ in this statement.

On Feb 22, 2020 Antoine Chambert-Loir left comment #4939 on Section 29.50 in Morphisms of Schemes

I wonder whether these results do extend when the assumption on the base (locally Noetherian) is replaced by the assumption on the morphism (finite presentation). It looks automatic to me.

I have one example in mind where such a reduction is used — for Lemma 2.4.12 of Maclagan/Sturmfels book, the base is a valuation ring (but they would need some slightly different result, though, because it is not clear a priori that if $I$ is an ideal of $K[T]$ (K, field of fractions of a valuation ring R), then $I \cap R[T]$ is finitely generated ; but they consider a finitely generated subideal of $I\cap R[T]$ which does the job.

On Feb 22, 2020 Laurent Moret-Bailly left comment #4938 on Lemma 10.28.3 in Commutative Algebra

The proof contains the result that the constructible sets of $\mathrm{Spec}(R)$ are the finite unions of sets of the form $D(f)\cap V(g_1,\dots,g_m)$ . This fact is useful by itself, so why not make it a separate statement (or incorporate it in 00F6)? It would make 00F8 immediate, and probably simplify other proofs, too.

On Feb 22, 2020 awllower left comment #4937 on Lemma 10.28.3 in Commutative Algebra

I think the last ring map should be $R_f\rightarrow R_f/(g_1,\ldots,g_m)$ ?

On Feb 20, 2020 Laurent Moret-Bailly left comment #4936 on Definition 9.12.2 in Fields

The comment that the definition is restricted to algebraic extensions is welcome, but why not add a reference to sections 030I and 05DT?

On Feb 19, 2020 Johan left comment #4935 on Lemma 38.32.2 in More on Flatness

You are right! To see this we should appeal to the result stated in Lemma 29.6.8 for the "partial section" $s = (j, f)$ to the projection $\overline{X} \times_S Y \to \overline{X}$ . When I last edited this section I tried to add as many references to Lemma 29.6.8 as I could, but I missed this one. Sorry!

On Feb 19, 2020 awllower left comment #4934 on Lemma 38.32.2 in More on Flatness

Why $(1,f)$ being a closed immersion implies that $X\rightarrow\overline X'$ is an open immersion? I can only tell that $X\times_SY\rightarrow \overline X\times_SY$ is an open immersion, but this does not seem to help.