Recent comments—The Stacks project

Comments 1 to 20 out of 10540 in reverse chronological order.

On May 31, 2026 Vihaan Dheer left comment #11547 on Section 13.21 in Derived Categories

You mention on this page that Cartan-Eilenberg resolutions need not use injectives: what kind of generalization can we make? I'm having trouble writing down a better statement because Lemma 013T seems to rely heavily on the injective assumption in two different ways. Do you have something specific in mind or a reference?

On May 30, 2026 Sharang Thimmaiah left comment #11546 on Section 7.6 in Sites and Sheaves

This may be a small nitpick, but based on remark 00VI and example 00VJ, I suppose that definition 00VH must explicitly mention that for each object $U \in \operatorname{Ob}(\mathcal{C})$ there is atleast one family of morphisms with fixed target $U$ .

On May 29, 2026 Jiamin Pan left comment #11545 on Lemma 10.38.5 in Commutative Algebra

In the proof of (2), we see that both two polynomials $x^n+a_{n-1}x^{n-1}+\dots+a_0$ and $x^m+b_{m-1}x^{m-1}+\dots+b_0$ are now polynomials in $R[x]$ by hypothesis. Since they are monic, the coefficients $c_i$ 's in the quotient are necessarily elements in $R$ by the uniqueness of quotients and remainders in Long Division over $R$ (as well as over $K$ ). Thus, it seems unnecessary to pass everything to the ring $R'$ instead.

On May 29, 2026 Juan Pablo left comment #11544 on Definition 4.43.1 in Categories

I think this is used silently often. Before this definition: "en equivalent definition would be an object 1 and an isomorphism $1:1\otimes 1\to 1$ " seems to be use that left unitor= right unitor.

tag 0FFQ, I believe the omitted proof that the maps are inverse of each other uses the two coherences for monoid categories. The omitted proof is that $Z'\to Z'\otimes X\otimes Y\to Z\otimes Y\otimes X\otimes Y\to Z\otimes Y$ equals $Z'\to Z\otimes Y$ . The first two arrows equal (1) $Z'\to Z\otimes Y\to Z\otimes Y\otimes X\otimes Y$ by naturality, and (2) $Z\otimes Y\to Z\otimes Y\otimes X\otimes Y\to Z\otimes Y$ equals the identity by the definition of dual. The issue is that being a bit careful with the defining arrows one sees that the arrow $Z\otimes Y\to Z\otimes Y\otimes X\otimes Y$ in (1) is really: $Z\otimes Y\to (Z\otimes Y)\otimes 1\to (Z\otimes Y)\otimes (X\otimes Y)$ while the arrow in (2) is $Z\otimes Y\to Z\otimes (Y\otimes 1)\to Z\otimes (Y\otimes (X\otimes Y))$ , so here you need the coherence $Z\otimes Y\to (Z\otimes Y)\otimes 1$ equal $Z\otimes Y\to Z\otimes (Y\otimes 1)\to (Z\otimes Y)\otimes 1$

The omitted proof of tag 0FN8 I believe uses the coherence I mentioned for symmetric categories.

On May 29, 2026 Stacks Project left comment #11543 on Lemma 13.7.1 in Derived Categories

Leaving as is.

On May 29, 2026 Stacks Project left comment #11542 on Lemma 7.21.5 in Sites and Sheaves

Fixed here.

On May 29, 2026 Stacks Project left comment #11541 on Definition 4.43.1 in Categories

OK, I don't want to change the definitions if that is not necessary (and it sound like it isn't). We'll add the additional properties (in the Stacks project) if we use them. If that has happened silently, then someone will point it out and we'll add it here.

On May 29, 2026 Stacks Project left comment #11540 on Section 14.23 in Simplicial Methods

Leaving as is.

On May 29, 2026 Stacks Project left comment #11539 on Lemma 10.78.2 in Commutative Algebra

I do think that part (7) of Nakayama's lemma gives exactly the statement we're looking for. Leaving as is.

On May 29, 2026 Stacks Project left comment #11538 on Lemma 76.51.4 in More on Morphisms of Spaces

Thanks. Fixed here. (I removed the comments with the parse errors.)

On May 29, 2026 Stacks Project left comment #11537 on Lemma 94.6.1 in Algebraic Stacks

Leaving as is.

On May 29, 2026 Stacks Project left comment #11536 on Section 4.32 in Categories

Leaving as is.

On May 29, 2026 Stacks Project left comment #11535 on Section 4.40 in Categories

Leaving as is. Instead, I have edited the discussion in the first paragraph of Section 94.6; see this commit.

On May 29, 2026 Stacks Project left comment #11534 on Section 4.33 in Categories

Leaving as is.

On May 29, 2026 Stacks Project left comment #11533 on Section 4.41 in Categories

Leaving as is.

On May 29, 2026 Stacks Project left comment #11532 on Definition 4.40.1 in Categories

Fixed here.

On May 29, 2026 Stacks Project left comment #11531 on Lemma 4.42.5 in Categories

Leaving as is.

On May 29, 2026 Stacks Project left comment #11530 on Lemma 12.7.1 in Homological Algebra

Ha, very good! Fixed here.

On May 29, 2026 Stacks Project left comment #11529 on Lemma 4.42.4 in Categories

Leaving as is.

On May 29, 2026 Stacks Project left comment #11528 on Section 4.31 in Categories

Leaving as is.