Stacks project -- Comments

#11705 on tag 02YT by I. Vanni

I. Vanni — Thu, 06 Aug 2026 01:43:25 GMT

Typo after Definition 02YY: "Note that if T is a schemes".

#11704 on tag 0BI7 by David Grimm

David Grimm — Wed, 05 Aug 2026 07:13:40 GMT

The hypothesis "in the situation above" has unfortunately quite a few possible interpretations. To author it is probably quite logical which ones are meant. Does it mean for example that one already assumes the possibility to resolve singularities of Y (absolutely) as a prerequisite to have embedded resolution ? I think with a more precisely stated hypothesis, this Lemma could become more usefull as a source for reliable citation.

#11703 on tag 0BI7 by David Grimm

David Grimm — Wed, 05 Aug 2026 07:13:16 GMT

#11702 on tag 0DS3 by Nick

Nick — Tue, 04 Aug 2026 01:48:03 GMT

Equation (5.21.1) does not have proper line breaks in the PDF file because the {equation} environment does not support line breaks. Please use the {align} environment instead.

#11701 on tag 09WR by thesnakefromthelemma

thesnakefromthelemma — Tue, 04 Aug 2026 03:08:30 GMT

The issue mentioned by the above commenter is a good example of the too simple to be simple phenomenon; a cute way to correct the statement is to modify the hypothesis to that for any finite (possibly empty!) family $( x _ i ) _ { i \in N }$ of objects of $\mathcal{I}$ , there exists an object $y \in \text{Ob} ( \mathcal{I} )$ and family of morphisms $( f _ i : x _ i \to y ) _ { i \in N }$ . (But, alas, it is perhaps more idiomatic to just stipulate additionally that $\mathcal{I}$ is nonempty. What can one do?)

It's also worth noting that (IINM) this claim holds not just in $\textit{Ab}$ but more generally in the category of models of any algebraic theory (in the sense of universal algebra)!

#11700 on tag 025N by Alex Scheffelin

Alex Scheffelin — Sat, 01 Aug 2026 08:11:29 GMT

It should day $\mathfrak{m}_B/\mathfrak{m}_B^2$ , currently the 2 is missing.

#11699 on tag 02JC by David Holmes

David Holmes — Fri, 31 Jul 2026 11:05:50 GMT

Trivial off by 1 error: in (1), should be "of dimension $2^{i-1}$ ". For example, if $i=1$ then $\mathfrak p_i = (x_1)$ and localising there gives a DVR. This does not affect anything anywhere, of course.

Disclosure: this was pointed out to me by an LLM when I was having it do a some surveying for https://adjectivesproject.org. Not sure how you feel about LLM-inspired contributions...

#11698 on tag 04VM by Elías Guisado

Elías Guisado — Fri, 31 Jul 2026 08:55:32 GMT

For the record, a characterization on ring epimorphisms is given in Corollary 4. I definitely could not find this in the existing literature on ring epis.

#11697 on tag 01L1 by Elías Guisado

Elías Guisado — Fri, 31 Jul 2026 07:58:40 GMT

This result could fit to this section (although I don't know if it serves any purpose):

Monomorphisms are local on source and target in the following sence: Let $f:X\to Y$ be a morphism of schemes and let $X=\bigcup_{i\in I}U_i$ and $Y=\bigcup_{i\in I}V_i$ be open covers such that $f(U_i)\subseteq V_i$ . Then $f$ is monic if and only if $f_{|U_i}:U_i\to V_i$ is a monomorphism for every $i$ .

The proof is not difficult.

#11696 on tag 01L6 by Elías Guisado

Elías Guisado — Fri, 31 Jul 2026 07:09:12 GMT

Hypothesis (2) may be relaxed to

(2') for any $x\in X$ the ring map $j_x^\sharp:\mathcal{O}_{Y,f(x)}\to\mathcal{O}_{X,x}$ is a ring epimorphism.

Moreover, it turns out monicity is equivalent to (1)+(2'), even in the category of ringed spaces, see here.