The Stacks project

I have a question about whether an alternative form of this lemma exists: The functor $Rf_!:D^+(X_{\text{ét}},\Lambda)\to D^+(Y_{\text{ét}},\Lambda)$ admit a right adjoint for a possibly non-torsion coefficient ring $\Lambda$. So for example for $\Lambda=\mathbb{Z}$, so that we have $D^+(X_{\text{ét}},\mathbb{Z})=D^+(X_{\text{ét}})$. I tried checking the references and I do not see any point at which this fails, so I thought I might ask.

What does Lemma 032L state about the integral closure of a Noetherian normal domain? Regard IT Telkom

In the last line of the statement, there is a typo "mapsto".

This lemma does imply that faithful flatness can be lifted along nilpotent surjections $R\to S$. Flatness follows from lifting tor-amplitude which is documented elsewhere and faithfullness from the above. I couldn't find a reference elsewhere so perhaps it might be good to add it in this section.

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minor typo, in the proof of Lemma 06CM, line -5, "Namely, $\Omega_{S'/R'}$ is projective as a direct sum of the free module $\Omega_{P'/R'} \otimes_{P'} S'$" $\Omega_{S'/R'}$ is a direct summand of $\Omega_{P'/R'} \otimes_{P'} S'$.

minor typo, in the proof of Lemma 06CM, line -5, "Namely, $\Omega_{S'/R'}$ is projective as a direct sum of the free module $\Omega_{P'/R'} \otimes_{P'} S'$" $\Omega_{S'/R'}$ is a direct summand of $\Omega_{P'/R'} \otimes_{P'} S'$.

some latex errors here?

Agree with the initial comment here. Most undergraduate students with an interest in algebraic geometry have not encountered category theory and/or morphisms

In Proposition 08YN, item (3) it is in fact to take a particular map Y-->Z of finite topological spaces where $Y$ has 4 points, and $Z$ has 3 points. This map happens to be surjective and proper, and the resulting lifting diagram is equivalent to Definition 5.26.1 without assuming anything about $X$.

Would you consider adding this reformulation ? Perhaps this reformulation might clarify Definition 5.26.1 and "explain" use of surjectivity and compactness here to category-theoretically minded. I can spell out some details if you are interested.

Furthermore, one can reformulate Proposition 08YN and Lemma 090D in terms of weak factorisation systems cogenerated by surjective proper maps of finite topological spaces of size at most 7.

Chris: the fibres are stated to be (geometrically) connected, in particular they are non-empty.

Is there any reason the $f'$ is not surjection in the Stein Factorization Theorems? It is not explicitly said, but certainly it must be?

Just a note on typesetting that there's a \PP^1_C in the exact sequence from the first paragraph where the subscript of C should be \mathbb{C} or whatever convention is used for \mathhbb{C} throughout.

Can I suggest to mention whether the property of being regular (all local rings are regular) ascends along regular maps here?

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This remark is a great illustration of how projective resolutions preserve morphism spaces under quasi-isomorphisms. A clear and elegant dual to the injective case—very helpful for understanding derived functors in practice kakahoki

Apologies if this is nonsense, but isn't there a universal differential operator $d^k_{univ,M}\colon M\to P^k_{S/R}(M)$ ? As written the construction doesn't make this clear. Apologies again if I am mistaken or this is too trivial to make explicit.

For part (1), "$I^\bullet$ computes..." should be replaced by "$F(I^\bullet)$ computes...".

In the proof of Lemma 01VS, you write

"Hence they give rise to a morphism $$j\colon X\longrightarrow \mathbf{P}^m_S\text{ with }m=n+\sum n_i"$$

but shortly after that you write $i^{-1}D_+(T_i)=X_{a_i}$, which should be $j^{-1}D_+(T_i)=X_{a_i}$ as you denote the morphism by $j$.