The Stacks project

What is the definition of "locally equidimensional" in the proof?

I am confused about the proof for Lemma 10.57.7. Which states the following:\

Let $S$ be a graded ring and let $p\subset S$ be a prime ideal. Let $q$ be the homogeneous ideal generated by homogeneous elements of $p$. Then $q$ is a prime. \

To check if a homogeneous ideal $q$ is prime, it is enough to check if $fg\in q$ then $f\in q$ or $g\in q$ for homogeneous elements $f,g$. Let $f,g$ be homogeneous and $fg\in q$. By definition of $q$, it means $fg=\Sigma a_if_i$ for $f_i\in p$ homogeneous. But that imply $fg\in p$ and thus $f\in p$ or $g\in p$. Then $f,g$ are homogeneous elements in $p$ and hence generator of $q$. Thus $f\in q$ or $g\in q$. Then we are done, right? Not sure we need to assume $f,g$ non-homogeneous.

In the statement of the lemma, you want to call the modules M^a,...,M^b rather than M_a,...,M_b

@9797 I just realized the converse is also true. That is, if the diagram $$\require{AMScd} \begin{CD} F(A)@>>> F(B) @>>> F(C)@>>> F(A)[1]\\ @VVV@VVV@VVV@VVV\\ G(A)@>>> G(B) @>>> G(C)@>>> G(A)[1] \end{CD}$$ commutes for all distinguished triangles $A\to B\to C\to A[1]$ of $\mathcal{D}$, then $\eta:F\Rightarrow G$ is trinatural. Indeed, just apply the hypothesis to the distinguished triangle $A\to 0\to A[1]\xrightarrow{\operatorname{id}}A[1]$.

Sorry but I didnt really understand the first proof... the flat module M is right exact functor that  be like _*M, and it seems not use M' and M'' are flat.

In the proof, $L$ should be $K$. After the last displayed formula, "this implies that" should be "implies that".

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Lean says no. On further inspection, it looks like a missing d (there's a gamma_i on the last but one line of the displayed calculation, and the definition of gamma_i involves d, but there's no d on the line before). It can be fixed simply by multiplying the first three terms in the displayed calculation by d.

Independent of this, we didn't need to do the case splitting on whether things were <= r or > r in Lean because valuations coming from valuation rings are defined on all of K in Lean and take value +infinity on 0; introducing this convention means that you don't need to do the case split just before the calculation because v(0) makes sense. But probably you're not bothered by this. In fact the truth is that in Lean valuations are multiplicative not additive (i.e. they're norms), and we use a (totally ordered) "group with 0" for the target of the valuation (a concept apparently invented by Artin: it's a monoid which is the disjoint union of a multiplicative group with {0}; fields are good examples), with v(0)=0. Not that I'm suggesting you change it, everything else in the proof seems to be fine.

I think you need to prove that the morphisms in lemma 01PM agree on overlaps, so we can glue them, before taling about the "canonical map" colim HomOX(In,F)⟶Γ(U,F)

By $\delta(s')$, do you mean $\delta'(s')$?

In the proof of Krull-Akizuki the beggining seems to be circular, or not well-written: To begin we may assume that $L$ is the fraction field of $A$ by replacing $L$ by the fraction field of $A$ if necessary.

I think $(B,I,\delta)$ should be an $A$-algebra. In the sentense "...and let $\Omega_{D'}^i$ resp. $\Omega_{D'}^i$ be ...", the former $\Omega_{D'}^i$ should be replaced by $\Omega_D^i$.

In Lemma 50.15.2, it maybe worth mentioning that there's another short exact sequence of complexes: 0 $\rightarrow$ $\Omega_{Y/S}^{\bullet}$ $\rightarrow$ $i^{*}\Omega_{X/S}^{\bullet}(log\ Y)$ $\xrightarrow{Res}$ $\Omega_{Y/S}^{\bullet}[-1]$ $\rightarrow$ 0, where $i: Y\rightarrow X$ is the closed immersion. Moreover, it's locally split (locally Res has a section $\omega\mapsto\omega dlogf$) which is dependent of choices of local equation $f$ of $Y$.

Sorry, I do not notice that your convention is start from 0.

In the 2nd paragraph of the proof, W should be $\text{pr}_2(…)$?

In the phrase "The second map is quasi-compact as it is the base change of $f$...", I believe $f$ should be replaced with $g\circ f$.

There's a typo at the third proof of part (1). You should add one more bracket on the right, it should be $\operatorname{Hom}_R (M, \Gamma(X, \mathcal{Hom}_{\mathscr{O}_X} (\widetilde{N}, \mathcal{F})))$.

There's a typo at the third proof of part (1). You should add one more bracket on the right, it should be $\operatorname{Hom}R (M, \Gamma(X, \mathcal{Hom}{\mathscr{O}_X} (\widetilde{N}, \mathcal{F}))).

There are two typos in the proof where $g \circ f$ is $f \circ g$.

Small grammatical issue, and unclear phrasing. This should probably say something like:

In a preadditive category $\mathcal{A}$, we call an object that is both final and initial (as in Lemma 12.3.2 above) a zero object and denote it by $0$.