Stacks project -- Comments

#10983 on tag 0C0X by Anonymous

Anonymous — Wed, 12 Nov 2025 01:50:11 GMT

Apologies if this is already somewhere in the Stacks project, but the cited Lemma 10.163.3 also shows the following.

Let $f \colon X \rightarrow Y$ be a Cohen–Macaulay morphism of locally Noetherian schemes. If $x \in X$ is such that $\mathcal{O}_{Y, f(x) }$ is Cohen–Macaulay,

then $\mathcal{O}_{X,x}$ is also Cohen–Macaulay.

And sorry for this, but: I wonder if it's better to use "Cohen–Macaulay" instead of "Cohen-Macaulay", i.e. an n-dash instead of a hyphen.

The weird formatting above is because the comment box resued to let me typeset the two local rings on the same line...

#10982 on tag 0DIJ by ani

ani — Wed, 12 Nov 2025 01:07:28 GMT

What does "bis" in the title mean? Is this a technical term? - I have not seen this before in the context of algebraic geometry.

#10981 on tag 00Y3 by Alejandro González Nevado

Alejandro González Nevado — Tue, 11 Nov 2025 08:40:34 GMT

@S. Nivodski For references and examples about sites or topoi without points, one can check \ref{here}{https://mathoverflow.net/questions/306483/locales-as-spaces-of-ideal-imaginary-points/}.

#10980 on tag 0H7L by Junyan Xu

Junyan Xu — Tue, 11 Nov 2025 08:23:39 GMT

By $I$ you probably mean $A\cap J$ .

#10979 on tag 03A2 by Alejandro González Nevado

Alejandro González Nevado — Tue, 11 Nov 2025 07:02:24 GMT

SS: A morphism of topoi factors via a special cocontinuos functor and a continuous morphism of sites that commutes with fibre products.

#10978 on tag 0GWK by Alejandro González Nevado

Alejandro González Nevado — Tue, 11 Nov 2025 12:36:41 GMT

SS: Sheaves canonically reconstruct via (are) their absolute glueing data.

#10977 on tag 047B by Manolis C. Tsakiris

Manolis C. Tsakiris — Mon, 10 Nov 2025 11:45:25 GMT

Little typo in the proof: It should be $Y_a$ instead of $Y_{\kappa(a)}$ .

#10976 on tag 0738 by Alejandro González Nevado

Alejandro González Nevado — Mon, 10 Nov 2025 05:35:42 GMT

SS: On a site, the canonical map of colimits: (1) inherits the injectivity of all the transition maps, (2) transforms quasi-compactness of objects into its own injectivty, (3) transforms the combination of the two previous conditions into bijectivity, (4) transforms the existence of a finitely indexed cofinal system of coverings with quasi-compact fibered products into bijectivity.

#10975 on tag 0D06 by Alejandro González Nevado

Alejandro González Nevado — Mon, 10 Nov 2025 05:22:13 GMT

SS: "On a site, quasi-compactness coincides with the existence of finitely indexed subsurjections of sheaves."

#10974 on tag 0D05 by Alejandro González Nevado

Alejandro González Nevado — Mon, 10 Nov 2025 05:16:08 GMT

Wouldn't it be better to put separate names to these properties and introduce them separately as a weaker notions of quasi-compactness on objects of a site? (2) is quasi-compactness by finite refinements and (3) quasi-compactness by finite subcoverings. Thus we can also establish the slogan: "On sites, quasi-compactness implies quasi-compactness by finite refinements, which implies quasi-compactness by finite subcoverings. The reverse implications are not true".