Stacks project -- Comments

#8475 on tag 08JZ by Et

Et — Fri, 09 Jun 2023 07:20:19 GMT

Is the map $\pi$ a splitting? If so it would be clearer to just deduce injectivity from that.

#8474 on tag 00S2 by Et

Et — Fri, 09 Jun 2023 07:06:06 GMT

Suggestion: since this proposition uses multiple rings, maybe add a subscript for the tensor products indicating over what they are taken

#8473 on tag 0AS8 by Et

Et — Thu, 08 Jun 2023 09:02:28 GMT

Where is the induction here specifically done? Wouldn't it be enough just to replace R by R/I^k+1, M by M/I^k+1 and I by I/I^k+1 and apply the precceding lemma?

#8472 on tag 051C by Et

Et — Thu, 08 Jun 2023 08:24:42 GMT

Maybe note in the almost last paragraph that you are using the snake lemma? Took me a moment to figure out what you were trying to do

#8471 on tag 009F by Paul

Paul — Thu, 08 Jun 2023 05:03:19 GMT

I guess in the proof of (2) and (4) the notation was heavily relaxed, since why should $\varphi_{ii_j}(s) = \varphi_{i'i_j}(s')$ hold, actually it holds for some restriction of s and s'. Also it should be $\varphi_{ii_j}(U)$ or better of some $U_{i_j}$ . I recommend to mention it or rework completly the proof.

#8470 on tag 00KZ by Laurent Moret-Bailly

Laurent Moret-Bailly — Thu, 08 Jun 2023 01:57:56 GMT

In the proof, the minimality of $n$ is not needed, so "a finite set of generators" would be more to the point. Also, it would be worth pointing out that the $M_i$ 's are finite modules, and perhaps that the last quotient $M/M_{n-1}$ is finitely presented if $M$ is.

#8469 on tag 03L7 by Ryo Suzuki

Ryo Suzuki — Wed, 07 Jun 2023 11:05:29 GMT

I think Lemma 03GI is used to deduce $U_j$ is quasi-compact. It might worth to note it explicitly.

#8468 on tag 01KR by Elías Guisado

Elías Guisado — Wed, 07 Jun 2023 09:59:47 GMT

For the sake of having some reference that is actually instructive, here's a neat proof of the cartesianity of the square https://mathoverflow.net/a/80812/101848

#8467 on tag 01KP by Elías Guisado

Elías Guisado — Wed, 07 Jun 2023 07:54:26 GMT

Minor typo: instead of $\operatorname{Spec}((A \otimes_R B)/J)$ in "we see that $\Delta^{-1}(U \times_S V) \to U \times_S V$ can be identified with $\operatorname{Spec}((A \otimes_R B)/J)$ ," one should write $\operatorname{Spec}((A \otimes_R B)/J)\to\operatorname{Spec}(A \otimes_R B)$ .

#8466 on tag 01JY by Elías Guisado

Elías Guisado — Wed, 07 Jun 2023 07:13:33 GMT

I think it could be nice to reference https://stacks.math.columbia.edu/tag/001U#comment-3413 to make explicit the formal nature of the argument (I believe such little hints are quite useful for those recently initiated to algebraic geometry—and often to category theory too—reading this webpage, like the version of myself one year ago).

An alternative reformulation of the proof would be by placing this lemma after remark https://stacks.math.columbia.edu/tag/01JW#comment-8464 and then just invoking the remark plus 26.17.6.