Stacks project -- Comments

#9536 on tag 09FB by Joe Lamond

Joe Lamond — Fri, 26 Jul 2024 08:56:32 GMT

Ah, I see that in the definition of "field" given in the next section does explicitly rule out $1=0$ . Apologies for not spotting this.

#9535 on tag 09FB by Joe Lamond

Joe Lamond — Fri, 26 Jul 2024 08:48:46 GMT

The definition of a field appears to have a minor error in it. A field is a (commutative) ring in which $1\neq0$ and all nonzero elements are invertible. The definition given here allows for $1=0$ . (One can avoid this by defining a field to be a ring whose nonzero elements form a group under multiplication, or by defining a field as a ring with exactly two ideals.)

#9534 on tag 08JA by nkym

nkym — Thu, 25 Jul 2024 04:11:02 GMT

The first $K,L$ should be $L,M$ .

#9533 on tag 0G6R by nkym

nkym — Thu, 25 Jul 2024 03:27:32 GMT

The second $\mathcal{C}'$ should be $\mathcal{C}$ .

#9532 on tag 08FC by nkym

nkym — Thu, 25 Jul 2024 02:53:32 GMT

In the proof of (3) from (2), * for the case $m=0$ , $B_i$ should rather be the identity of $\mathcal{O}_{U}^n$ and $t$ should be the same as $s$ , and * "but the first column of" should be "but the last column of".

#9531 on tag 09GZ by Laurent Moret-Bailly

Laurent Moret-Bailly — Wed, 24 Jul 2024 09:49:02 GMT

I believe it would help the reader to change the section title to "Separable algebraic extensions".

#9530 on tag 09HC by Anonymous

Anonymous — Wed, 24 Jul 2024 09:18:52 GMT

Maybe this is pedantic, but do you need some argument on why $k(\alpha)$ are separable over $k$ (i.e. that every element of $k(\alpha)$ is separable over $k$ ), and similarly for $k(\beta)$ ?

For example, one could use Tags 9.12.11 and 9.12.4.

Alternative way to say it (but maybe worse exposition): if there were an element $x \in k(\alpha)$ which is not separable over $k$ , we would get an injection $k(x) \otimes_k \overline{k} \rightarrow k(\alpha) \otimes_k \overline{k}$ from a non-reduced ring into a reduced ring, hence a contradiction.

#9529 on tag 0G7B by nkym

nkym — Tue, 23 Jul 2024 08:54:27 GMT

In the proof the last two $\mathcal{K}_1$ should be $\mathcal{K}_2$ and $\mathcal{K}_3$

#9528 on tag 0GMX by nkym

nkym — Tue, 23 Jul 2024 08:33:25 GMT

$\mathcal{G}$ after the second last $\to$ should be $f^*\mathcal{G}$ instead.

#9527 on tag 0656 by Shubhankar

Shubhankar — Tue, 23 Jul 2024 04:32:50 GMT

Here's a possibly useful lemma which I thought was already on here. I also looked in the tor dimension section but couldn't find it. Apologies if I missed this or if there is a mistake below.

Let $P$ be perfect of tor-amplitude in $[a,b]$ . Then $P^\vee$ has tor-amplitude in $[-b,-a]$ . Indeed, if $M$ is an $A$ -module then $P^\vee\otimes^L M=R\mathrm{Hom}(P,M)$ and the RHS can be represented by a complex $E^\bullet$ with $E^i$ non-zero for $i\in [a,b]$ . Relabelling we see that $\mathrm{Ext}^i(P,M)$ is non-zero for $i\in [-b,-a]$ and so we conclude.