Stacks project -- Comments

#11101 on tag 01P7 by yj

yj — Sat, 17 Jan 2026 11:47:20 GMT

FYI, a counterexample lives in this Math StackExchange answer (ﾉ◕ω<)ﾉ*:･ﾟ✧ Example of quasi-compact, non-quasi-separated scheme where qcqs fails? (Answer 1 gives a counterexample to injectivity when qc is dropped, while Answer 2 gives a counterexample to surjectivity under qc but without qs.)

#11100 on tag 01PW by Laurent Moret-Bailly

Laurent Moret-Bailly — Sat, 17 Jan 2026 12:14:01 GMT

@#1198 Quasi-separatedness is treated in the Springer edition of EGA I. For the present result, see section 6.8 there.

#11099 on tag 01NS by Dhruv Goel

Dhruv Goel — Fri, 16 Jan 2026 03:30:55 GMT

In the proof of Lemma 01NU, why can we think of $\psi$ as a map to $\bigoplus_n f_*\mathcal{L}^{\otimes n}$ ? Doesn't $f_*$ not commute with direct sums in general? Similarly, how can we identify $\Gamma(S, \bigoplus_n f_*\mathcal{L}^{\otimes n})$ with $\Gamma_*(T, \mathcal{L})$ ?

#11098 on tag 01PW by Elías Guisado

Elías Guisado — Fri, 16 Jan 2026 08:17:17 GMT

Another reference for this result is Görtz, Wedhorn, Algebraic Geometry I, Theorem 7.22, which in turn is the generalization to quasi-separated schemes of [EGA I, Theorem 9.3.1], the latter being stated for a $X$ a separated quasi-compact scheme or such that its underlying space is Noetherian (in both cases such a $X$ is quasi-separated, but quasi-separatedness is not developed until EGA IV).

#11097 on tag 0B5K by Elías Guisado

Elías Guisado — Fri, 16 Jan 2026 05:00:42 GMT

Alternative description of 28.17.1.1: restriction to $X_s$ induces a map $\Gamma_*(X,\mathcal{L},\mathcal{F})\to\Gamma_*(X_s,\mathcal{L}_{|X_s},\mathcal{F}_{|X_s})$ of graded $\Gamma_*(X,\mathcal{L})$ -modules. The section $s$ acting by scalar multiplication on $\Gamma_*(X_s,\mathcal{L}_{|X_s},\mathcal{F}_{|X_s})$ is an isomorphism by Modules, Lemma 17.25.10, so we get an induced map $\Gamma_*(X,\mathcal{L},\mathcal{F})_s\to\Gamma_*(X_s,\mathcal{L}_{|X_s},\mathcal{F}_{|X_s})$ of graded $\Gamma_*(X,\mathcal{L})_s$ -modules whose degree zero component gives 28.17.1.1.

#11096 on tag 0B5I by Elías Guisado

Elías Guisado — Fri, 16 Jan 2026 04:21:30 GMT

Here is an alternative construction of the map $M_{(f)}\to\Gamma(D_+(f),\mathcal{F})$ from the proof: We have a map $\bigoplus_{n\in\mathbb{Z}}\Gamma(X,\mathcal{F}(n))\to\bigoplus_{n\in\mathbb{Z}}\Gamma(D_+(f),\mathcal{F}(n))$ which is given componentwise by restriction to $D_+(f)$ . This is a map of modules over the graded ring $\bigoplus_{n\in\mathbb{Z}}\Gamma(X,\mathcal{O}_X(n))$ and hence a map of $S$ -modules via the map 27.10.1.3. Now, the action of $f\in S$ on the module $\bigoplus_{n\in\mathbb{Z}}\Gamma(D_+(f),\mathcal{F}(n))$ is an isomorphism. This follows from the fact that $D_+(f)=X_{\alpha_d(f)}$ , where $\alpha$ is the map 27.10.1.3 (see e.g. Lemma 27.10.5 or [EGA II, 2.6.3]) and Modules, Lemma 17.25.10. Thus we get an induced map $\left(\bigoplus_{n\in\mathbb{Z}}\Gamma(X,\mathcal{F}(n))\right)_f\to\bigoplus_{n\in\mathbb{Z}}\Gamma(D_+(f),\mathcal{F}(n))$ of graded $S_f$ -modules whose degree $0$ component is the desired $S_{(f)}$ -module homomorphism $M_{(f)}\to\Gamma(D_+(f),\mathcal{F})$ .

#11095 on tag 00TX by Manolis C. Tsakiris

Manolis C. Tsakiris — Fri, 16 Jan 2026 03:42:27 GMT

Nakayama's lemma is not needed to argue that $df$ generates a direct summand in $\Omega_{R/k}$ . Indeed, as $df$ is not in $m\Omega_{R/k}$ , and $\Omega_{R/k}$ is free on, say, $e_1,\dots,e_n$ , not all coefficients of the expression of $df$ as a linear combination of the $e_i$ 's are in $m$ . As $R$ is local, this immediately implies that $\Omega_{R/k}/(df)$ is finite free of rank $n-1$ .

#11094 on tag 00TU by Manolis C. Tsakiris

Manolis C. Tsakiris — Thu, 15 Jan 2026 10:11:17 GMT

I suggest replacing

"Since $P'(y)\delta = P'(\bar{y})\delta$ we see that we can..."

"Since $(P')^{\varphi}(y) \in R$ is a unit, we see that we can..."

#11093 on tag 0EBF by John Nolan

John Nolan — Wed, 14 Jan 2026 04:34:07 GMT

I believe the "algebraic analogue" here should be Lemma 0EB9 instead.

#11092 on tag 01N8 by Stacks Project

Stacks Project — Thu, 08 Jan 2026 11:47:29 GMT

Thanks! I fixed the typos you noted here. As I said elsewhere, this chapter is a beast and needs to be reworked in some way to avoid all the notational clutter (but I'm not sure that's even possible).