The Stacks project

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".

gerçek kişilerle ihaleli veya gömmeli batak oyunu aynı zamanda normal okey oyunu

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 $0$.

Should the $Y'$ in Lemma 0ABP be actually $X'$? Also, should one interchage the roles of X and Y in Lemma 0BB0?

Is there a typo on line -3 of the example? I believe $P\to X^{\gamma}$ should be $P\to X^g$.

The definition of a spectral map seems to be exactly the same as that of quasi-compact map from 005A.

Isn't it easier to use that M is a direct summand of a finite free module?