Comments 1 to 20 out of 4017 in reverse chronological order.


On DS left comment #4260 on Lemma 15.42.6 in More on Algebra

Can I suggest adding to the lemma: (4) If \hat{A} is normal, then so is A (with a reference to 033G as the proof)?

On DS left comment #4259 on Lemma 15.41.2 in More on Algebra

typo "reduced" for "normal" in the proof.

On Antoine Chambert-Loir left comment #4258 on Section 10.151 in Commutative Algebra

In the definition of $S_k$, the convention that the dimension of the zero module can be made useless by requiring that the prime ideal $\mathfrak p$ belongs to the support of the module.

On awllower left comment #4257 on Section 53.3 in Fundamental Groups of Schemes

I am so sorry. I didn't think through enough and didn't realize that $X^n$ is just the product of $n$ copies of $X$, with the obvious action of $S_n$. I am sorry for any inconvenience thus caused. Also feel free to delete the comments to make the site cleaner if that is desired. Thanks in any case for this great site and effort.

On awllower left comment #4256 on Section 53.3 in Fundamental Groups of Schemes

I do not understand what is the object $X^n$ in the proof of lemma 0BN2. It seems to come out of nowhere, endowed with a mysterious action of $S_n$. Could someone explain what is it? Thanks.

On awllower left comment #4255 on Section 53.3 in Fundamental Groups of Schemes

I do not understand what is the object $X^n$ in the proof of lemma 0BN2. It seems to come out of nowhere, endowed with a mysterious action of $S_n$. Could someone explain what is it? Thanks.

On Laurent Moret-Bailly left comment #4254 on Lemma 60.14.2 in Properties of Algebraic Spaces

Another case for (3): $G\to S$ is radicial.

On Remy left comment #4253 on Lemma 60.14.2 in Properties of Algebraic Spaces

The proof also shows that $X \to X/G$ is a $G$-torsor.

On Dario Weißmann left comment #4247 on Lemma 51.3.2 in Resolution of Surfaces

the local equation for $E$ has the same name as the morphism $f:X\to S$. Maybe call it $g$ instead?

On Dario Weißmann left comment #4246 on Lemma 51.3.1 in Resolution of Surfaces

One could mention in the statement that $E$ is the exeptional divisor of the blowup

On Dario Weißmann left comment #4245 on Lemma 51.2.2 in Resolution of Surfaces

couple typos: $H^1(L_{S/R})$ should be $H_1(L_{S/R})$

$\Omega_{R/\lambda}$ should be $\Omega_{R/\Lambda}$

On Kazuki Masugi left comment #4244 on Lemma 10.29.3 in Commutative Algebra

"Hence (3) $\Rightarrow$ (2)" should be "Hence (4) $\Rightarrow$ (2)"

On Kazuki Masugi left comment #4243 on Lemma 10.29.1 in Commutative Algebra

"Write $g$" should be "Write $h$".

7th sentense in the proof.

On Si Yu How left comment #4241 on Section 4.26 in Categories

Why isn't there a mention of the (optional) locally small condition of a (left, right) multiplicative system as in Weibel's homological algebra book? Without this condition $S^{-1}\mathcal{C}$ may not be locally small even though $\mathcal{C}$ is.

On Si Yu How left comment #4238 on Section 29.3 in Cohomology of Schemes

In the above comment, I mean $\cup$ whenever I wrote $\cap$.

On Si Yu How left comment #4237 on Section 29.3 in Cohomology of Schemes

In the proof of Lemma 01XF, instead of working with a closed point $P \in U$, we can work with an open set $V$ where $U \cap V = X$. Then following the exact same proof we can find $f \in \Gamma(X, \mathcal{O}_ X)$ such that $X_f$ is a principal open subscheme of $U$ and $X_f \cap V = X$.

Since $X$ is quasi-compact, it has a finite affine covering $U_1, \dots , U_n$. Since $U_1 \cap (U_2 \cap \dots \cap U_n) = X$, there exists $f_1$ such that $X_{f_1}$ is affine open and $X_{f_1} \cap (U_2 \cap \dots \cap U_n) = X$. Similarly, since $U_2 \cap (X_{f_1} \cap U_3 \cap \dots \cap U_n) = X$, there exists $f_2$ such that $X_{f_2}$ is affine open and $X_{f_2} \cap (X_{f_1} \cap U_3 \cap \dots \cap U_n) = X$. By iterating such steps we can get $X_{f_1} \cap \dots X_{f_n} = X$ and that each $X_{f_i}$ is affine open.

This approach has the advantage that it avoids Zorn's lemma.

On Aolong left comment #4236 on Section 10.19 in Commutative Algebra

In the statement (7) and (8) of Nakayama lemma, it would be better to point out $x_1,\cdots,x_n$ generate $M/IM$ as an $R$-module, since $M/IM$ can be naturally $R/I$-module as well.

On Zhenhua Wu left comment #4235 on Section 30.26 in Divisors

In 30.26.4, the argument will be more transparent if we pick $U$ to be the domain of definition of $f$. A more detailed proof is following:

There exists a largest nonempty open subscheme $U\subset X$ such that $f$ corresponds to a section of $\Gamma(U,\mathcal{O} _ X^ * )$ . Hence any prime divisor $Z$ with generic point $\xi$ s.t. $f\notin\mathcal{O} _ {X,\xi}$ must have $\xi\in X\backslash U$. Since $U$ is irreducible, we have $U\cap Z=\emptyset$. Since $Z$ is of codimension 1, it is easy to see it is an irreducible component of $X\backslash U$. Hence Lemma 30.26.1 gives the desired result.

If $\mathop{\mathrm{ord}} _ Z (f)\neq 0$, equivalently $\mathop{\mathrm{ord}}Z(f)<0$ or $\mathop{\mathrm{ord}} _ Z(f^{-1})<0$. Hence we have either $f\notin \mathcal{O} _ {X,\xi}$ or $f^{-1}\notin \mathcal{O} _ {X,\xi}$. Apply previous argument then we are done.

On Kazuki Masugi left comment #4234 on Lemma 5.13.7 in Topology

"Then we see that $V_z=U_{z,z′1}\cap …\cap U_{z,z′r}\cap U_i$ " should be "Then we see that $V_z=V_{z,z′1}\cap …\cap V_{z,z′r}\cap U_i$".

On left comment #4233 on Theorem 54.66.10 in Étale Cohomology

Thanks for pointing this out! I've recorded it on https://github.com/gerby-project/gerby-website/issues/140 and it will be dealt with soon I hope.