Recent comments—The Stacks project

Comments 1 to 20 out of 8130 in reverse chronological order.

On Dec 09, 2023 Anonymous left comment #8745 on Lemma 12.5.10 in Homological Algebra

If " $f$ is left inverse to $g$ " means that $f \circ g$ is the identity (where $f \circ g$ means "apply $g$ first"), then I think "left" and "right" should be switched in this lemma.

On Dec 08, 2023 Matt Broe left comment #8744 on Section 10.126 in Commutative Algebra

Someone pointed out to me that this claim holds in more generality: finitely generated projective modules are dualizable objects in RMod, and tensoring with dualizable objects commutes with limits.

On Dec 07, 2023 Zhenhua Wu left comment #8743 on Lemma 30.8.1 in Cohomology of Schemes

I believe the lemma is wrong in the case $q=n=0,d<0$ . Unwrapping the definition we should have $H^0(\mathbb{P}^0,\mathcal{O}(d))=R\cdot T_0^d$ , which is compatitable with $(\frac{1}{T_0}R[\frac{1}{T_0}])_d\neq 0$ but not compatitable with $(R[T_0])_d=0$ when $d<0$ .

On Dec 05, 2023 Matt Broe left comment #8742 on Section 10.126 in Commutative Algebra

I have found the criterion given in claim 1 of the following link for commuting tensor with an inverse limit of modules useful, but have not seen it in any reference: https://math.stackexchange.com/a/4577954/304290 Perhaps it would be worth mentioning here, either on this tag or wherever else it would fit.

On Dec 04, 2023 Zongzhu Lin left comment #8741 on Section 23.12 in Divided Power Algebra

In the definition of Koszul complex $K_n$ , should the differential be defined by $d(\xi_i)=f^n$ ?

On Dec 02, 2023 pkas left comment #8740 on Section 10.160 in Commutative Algebra

In the second part of the proof for 10.160.8 (the case of positive characteristic), the terminology is slightly confusing. I suggest rewording the condition required of $\varphi_n$ to the commutative square $\pi \varphi_n = f \pi$ , where $\pi$ are the projections $\Lambda/p^n \Lambda \to \Lambda/p\Lambda$ and $R/\mathfrak{m}^n \to R/\mathfrak{m}$ , and $f: \Lambda/p\Lambda \to R/\mathfrak{m}$ is a chosen isomorphism. Moreover, the map to lift using formal smoothness is the composition $\Lambda/p^n\Lambda \rightarrow \Lambda/p^{n-1} \Lambda \overset{\varphi_{n-1}}{\longrightarrow} R/\mathfrak{m}^{n-1}$ .

It could also be helpful for readers to mention that this proof shows that the map $A \mapsto A/pA$ from Cohen rings (up to isomorphism) to fields (up to isomorphism) is injective.

On Dec 01, 2023 Paolo Lammens left comment #8739 on Section 110.9 in Examples

Here what is the ideal $I$ that defines the $I$ -adic topology/completion?

On Nov 29, 2023 Yicheng Zhou left comment #8738 on Lemma 37.46.3 in More on Morphisms

I guess that $V$ in the first sentence of the last paragraph should be only "geometrically connected"?

On Nov 29, 2023 Mudassar Hameed left comment #8737 on Lemma 67.41.2 in Morphisms of Algebraic Spaces

Thanks for sharing this useful information.

On Nov 29, 2023 Alexis left comment #8736 on Section 10.39 in Commutative Algebra

For lemma 0BBY, you might also want that $(I\cap J)M=(I\cap J)\otimes_{R}M$ .

On Nov 28, 2023 Hayama Kazuma left comment #8735 on Lemma 33.37.6 in Varieties

In the proof, "Observe that $R=R_n$ ", here $n$ should be $r$ ?

On Nov 27, 2023 Eiki left comment #8734 on Section 60.13 in Crystalline Cohomology

Some possibly pedantic comments; "two ring maps $p_0, p_1$ " aren't they technically maps of sheaves where on the level of sections they are described as above?

If we are later calling $p_0,p_1$ to be the maps on the corresponding schemes, maybe we want to give it some other name (probably $p_i^{-1}$ rather than $p_i^*$ here)?

On Nov 25, 2023 Hao left comment #8733 on Theorem 57.13.3 in Derived Categories of Varieties

Sorry I just found the answer that by \tag{0FDC} they are the same.

On Nov 25, 2023 Hao left comment #8732 on Theorem 57.13.3 in Derived Categories of Varieties

The original version in Orlov's paper concerns the coherent category, but the statement here is about perfect objects. Does this implies the coherent version? Or Does every fullly faithful embedding preserves perfect objects?

On Nov 25, 2023 Michael left comment #8731 on Section 39.7 in Groupoid Schemes

I am wondering about the small detail mentioned in Lemma 047S. Can someone explain it to me ?

On Nov 23, 2023 Kentaro Inoue left comment #8730 on Lemma 10.99.8 in Commutative Algebra

I think that this lemma can be proved more easily. In the first paragraph, (2) is proved in the case $m=1$ . In the general case, (2) can be reduced to the case $m=1$ by considering the $I$ -adic filtration. Then (1) follows from (2) immediately.

On Nov 22, 2023 Erhard Neher left comment #8729 on Lemma 8.10.6 in Stacks

Line 8 of the proof should read: We have to find a morphism $\varphi : y_1 \to y_2$ restricting to ...

On Nov 22, 2023 Erhard Neher left comment #8728 on Lemma 8.10.1 in Stacks

Proof of (2), line (2): it should read $y_{ij} \to x$ instead of $p(y_{ij}) \to p(x)$ .

On Nov 21, 2023 Jim Davis left comment #8727 on Section 111.1 in Exercises

Exercise 078G. Replace idempotent by central idempotent

On Nov 20, 2023 Theodore Sylvan left comment #8726 on Definition 10.17.1 in Commutative Algebra

This is not unlike nothing. however...