Stacks project -- Comments

#8745 on tag 010G by Anonymous

Anonymous — Sat, 09 Dec 2023 01:51:50 GMT

If " $f$ is left inverse to $g$ " means that $f \circ g$ is the identity (where $f \circ g$ means "apply $g$ first"), then I think "left" and "right" should be switched in this lemma.

#8744 on tag 05N4 by Matt Broe

Matt Broe — Fri, 08 Dec 2023 01:11:36 GMT

Someone pointed out to me that this claim holds in more generality: finitely generated projective modules are dualizable objects in RMod, and tensoring with dualizable objects commutes with limits.

#8743 on tag 01XT by Zhenhua Wu

Zhenhua Wu — Thu, 07 Dec 2023 04:42:06 GMT

I believe the lemma is wrong in the case $q=n=0,d<0$ . Unwrapping the definition we should have $H^0(\mathbb{P}^0,\mathcal{O}(d))=R\cdot T_0^d$ , which is compatitable with $(\frac{1}{T_0}R[\frac{1}{T_0}])_d\neq 0$ but not compatitable with $(R[T_0])_d=0$ when $d<0$ .

#8742 on tag 05N4 by Matt Broe

Matt Broe — Tue, 05 Dec 2023 05:46:03 GMT

I have found the criterion given in claim 1 of the following link for commuting tensor with an inverse limit of modules useful, but have not seen it in any reference: https://math.stackexchange.com/a/4577954/304290 Perhaps it would be worth mentioning here, either on this tag or wherever else it would fit.

#8741 on tag 0GZ3 by Zongzhu Lin

Zongzhu Lin — Mon, 04 Dec 2023 09:27:42 GMT

In the definition of Koszul complex $K_n$ , should the differential be defined by $d(\xi_i)=f^n$ ?

#8740 on tag 0323 by pkas

pkas — Sat, 02 Dec 2023 04:37:27 GMT

In the second part of the proof for 10.160.8 (the case of positive characteristic), the terminology is slightly confusing. I suggest rewording the condition required of $\varphi_n$ to the commutative square $\pi \varphi_n = f \pi$ , where $\pi$ are the projections $\Lambda/p^n \Lambda \to \Lambda/p\Lambda$ and $R/\mathfrak{m}^n \to R/\mathfrak{m}$ , and $f: \Lambda/p\Lambda \to R/\mathfrak{m}$ is a chosen isomorphism. Moreover, the map to lift using formal smoothness is the composition $\Lambda/p^n\Lambda \rightarrow \Lambda/p^{n-1} \Lambda \overset{\varphi_{n-1}}{\longrightarrow} R/\mathfrak{m}^{n-1}$ .

It could also be helpful for readers to mention that this proof shows that the map $A \mapsto A/pA$ from Cohen rings (up to isomorphism) to fields (up to isomorphism) is injective.

#8739 on tag 05JF by Paolo Lammens

Paolo Lammens — Fri, 01 Dec 2023 07:38:31 GMT

Here what is the ideal $I$ that defines the $I$ -adic topology/completion?

#8738 on tag 0EY5 by Yicheng Zhou

Yicheng Zhou — Wed, 29 Nov 2023 10:42:23 GMT

I guess that $V$ in the first sentence of the last paragraph should be only "geometrically connected"?

#8737 on tag 03K8 by Mudassar Hameed

Mudassar Hameed — Wed, 29 Nov 2023 06:58:07 GMT

Thanks for sharing this useful information.

#8736 on tag 00H9 by Alexis

Alexis — Wed, 29 Nov 2023 03:46:14 GMT

For lemma 0BBY, you might also want that $(I\cap J)M=(I\cap J)\otimes_{R}M$ .