The Stacks project

suppose α:w→x and β:w→y, y should be t here

Why not following the proof in EGA or in Raynaud Anneaux locaux henseliens? It does not need to reduce the statement to the case where R is Noetherian.

Would it be possible to expand on why $A$ being a valuation ring implies that there is no prime of $A'$ lying over $\mathfrak{m}$? After all, $A'$ is not necessarily local, so we cannot immediately apply the definition of a valuation ring.

The statement can be improved to "the affine stratification number of $Y$ is at most the number of distict values of $\int_f w$ minus $1$". Namely, the base case of the induction is when $\int_f w$ attains $1$ value and in that case the argument shows that $Y$ is affine, hence has stratification number $0$.

suggested slogan: representabiltiy can be checked on a presentation.

It looks to me that there is a small left-right discrepancy: $\alpha^n_{n+1}$ maps everything to 0, but $h_n(e_i)(\alpha_{n+1}^n) = e_i$ identically. Thus evaluating at zero gives the identity, whereas evaluating at 1 gives 0.

The completed bivariant group should be a subset of $\prod_{p\geq 0}A^p(X\to Y),$ not $\prod_{p\geq 0} A^p(X).$

It took me so long to parse the final sentence "This implies the lemma." I include here an argument for posterity. Write $X\to Y$ for the map $\operatorname{Spec}S\to\operatorname{Spec}R$, and write $x$ and $y$ for the points in $X$ and $Y$ associated to $\mathfrak{q}$ and $\mathfrak{p}$. Assume $S\otimes_R\kappa(\mathfrak{p})$ is a finite product of separable field extensions of $\kappa(\mathfrak{p})$. In particular, $X_y=\operatorname{Spec}(S\otimes_R\kappa(\mathfrak{p}))$, as a topological space, is a finite discrete one. Let $\mathfrak{q}\subset S$ be a prime lying over $\mathfrak{p}$. This corresponds to a point $x'$ in $X_y$ by Schemes, Lemma 26.17.5. Therefore $\kappa(x')=\mathcal{O}_{X_y,x'}\cong S_\mathfrak{q}/\mathfrak{p}S_\mathfrak{q}$ (see #11295) is a field, so claim (1) follows.

Typos : in the first paragraph "$(A/I)^{\oplus n} \cong J/ (J^2 + IB) \oplus \overline{K} \longrightarrow J'/((J')^2 + IB')$" in the second pragraph in the parenthese explain why the morphism at $C_ {\mathfrak q}$ is an isomorphism "this is because by Nakayama's lemma 10.20.1" (I find that Nakayama's algebra lemma 10.20.1 a little weird)

Maybe for more precision in the statement of (2) (resp., (3)) one can replace "is perfect" by "is a perfect complex of $R_\mathfrak{p}$-modules" (resp., "of $R_\mathfrak{m}$-modules"). (This is what it is meant, right?)

The composition of locally projective morphisms is locally projective. This is immediate from 29.43.14, (or from 29.43.6, or from 29.43.4 and 29.43.7), but is still a nice little observation as it shows that locally projective morphisms are the smallest "reasonable" (a la Vakil) class of morphisms containing the projective (or H-projective) morphisms.

I think there is a typo in the proof of Lemma 10.57.7: $\mathfrak{p} \subset \mathfrak{q}$ should be the other way round.

From my understanding, these spectral sequences constitute the old way to compute the total derived functor $RF$ valued at a complex. They are stated by Grothendieck in [Tohoku, p. 146, eq. (2.4.2); EGA, (II.4.3.1), (II.4.3.2)] and by Weibel in his book An Introduction to Homological Algebra, 5.7.6, 5.7.9.

In the literature these are sometimes known as the "hypercohomology spectral sequences," in case it's worth giving a name to the result.

I think the reference for this section should be [Cartan-Eilenberg, Ch. XVII], right?

thats its very cool breakdown and explanation, now i can understand!

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Maybe this is supposed to be clear, but regarding the comments just before this definition:

It seems to me that some additional argument is being made to compare two K-flat resolutions, $\mathcal{K}^{\bullet}_1 \rightarrow \mathcal{F}^{\bullet}$ and $\mathcal{K}^{\bullet}_2 \rightarrow \mathcal{F}^{\bullet}$, in addition to Lemma 21.17.12. I guess one could argue by dominating both by K-flat resolutions by another, i.e. $\mathcal{K}^{\prime \bullet} \rightarrow \mathcal{K}^{\bullet}_1$ and $\mathcal{K}^{\prime \bullet} \rightarrow \mathcal{K}^{\bullet}_2$, which are compatible up to homotopy with the maps to $\mathcal{F}^{\bullet}$. Or maybe you have something else in mind?

By considering a K-flat resolution of the other factor $\mathcal{G}^{\bullet}$, I claim that we can see that the resulting isomorphism of functors does not depend on the choice of $\mathcal{K}^{\prime \bullet}$.

That is, the previous sentence could read:

By Lemma 21.17.12, the resulting functor (up to canonical isomorphism) does not depend on the choice of the K-flat resolution.

This canonical isomorphism is compatible with composition, from $\mathcal{K}^{\bullet}_1$ to $\mathcal{K}^{\bullet}_2$ to $\mathcal{K}^{\bullet}_3$, if that makes sense.

Sorry to double post, but a better phrasing is:

Let $R$ be a Noetherian ring. Then $R$ has finite global dimension $\leq n$ if and only if for every maximal ideal $\mathfrak{m}$ of $R$, the ring $R_{\mathfrak{m}}$ has global dimension $\leq n$.