The Stacks project

It seems to me that you also need to choose $t\in K$ so that it is $\kappa(\mathfrak{m})$-linearly idependent from $s+m$, and then also choose $\varphi$ to map $s+m$ to $\mathfrak{m}$. Otherwise, I don't see why the obtained locus doesn't contain $\mathfrak{m}$.

Sorry, but I don't see why the case $d=0$ can't occur. Simply take $M=A/\mathfrak{q}$ for some ideal of definition $\mathfrak{q}$. I think that this lemma is simply wrong in this case, while the case $d=0$ can't occur in the next lemma.

"gysin" is a name, so it should be capitalized everywhere.

The final paragraph of the proof of Lemma 08ZY refers to Lemma 08ZN with the phrase "the intersection of all open and closed subsets of $X$ containing $x$". That may be ambiguously read to mean the intersection of the collection of all open sets and all closed sets containing $x$, which is not the desired meaning. Using the usual term "clopen set" would make things shorter and clearer.

In addition to #9757, the expression birational schemes should be defined.

In the paragraph after Definition 29.50.1, it should be "the existence of a birational morphism".

What is K in this context?

Maybe we should add the lemma that any product of local rings is the product of the localizations at its maximal ideals.

Excellent, I saw other comments about the formula, which is excellent! As a beauty machine manufacturer, Litonlaser is very appreciative.

Some possible typos in the proof of lemma10.149.1:

line -4: $\beta'(g)=\beta(g)+\sum_{i}\delta_{i}\cdot \beta(\frac{\partial g}{\partial x_{i}})$

line -3: $d: J'/J^{2} \rightarrow \Omega_{P/R}\otimes_{P} S, g\mapsto \sum_{i} \pi(\frac{\partial g}{\partial x_{i}})\cdot dx_{i}$, where $\pi: P\rightarrow S$ is the natural quotient map.

We need to replace $P^{\bullet}$ by $P_{\bullet}$.

Lindo

For reference, this is Kashiwara, Schapira, Categories and Sheaves, Proposition 3.2.4.

I think "graduation" should be "gradation". I'm not sure that "graduation" is wrong, many people use it too, but since the page also uses "gradation" I think it's better to stick with that one

How to prove the injectivity of Lemma37.4.2? If we assume $H^{0}(X',G_{m})/n\rightarrow H^{0}(X,G_{m})/n$ is surjective, then it suffices to show $H^{1}(X',\mu_{n})\rightarrow H^{1}(X,\mu_{n})$ is injective by Kummer sequence for etale cohomology.

Since $H^{0}(X,\mathcal{I})[n]$, $H^{0}(X,\mathcal{I})/n$ and $H^{1}(X,\mathcal{I})[n]$ are zero, we have $H^{0}(X',G_{m})[n]=H^{0}(X,G_{m})[n]$ and similarly for etale base change. So we have $i_{*}\mu_{n}=\mu_{n}$ as etale sheaves, where $i:X\rightarrow X'$.

Finally we get the desired injection by the spectral sequence $H^{p}(X',R^{q}i_{\star}\mu_{n}) \Rightarrow H^{p+q}(X,\mu_{n})$

perhaps I am missing something, but Lemma 00JA doesn't seem to prove that R is a product of it's localizations at its maximal ideals, just a product of loval artinian rings?

perhaps I am missing something, but Lemma 00JA doesn't seem to prove that R is a product of it's localizations at its maximal ideals, just a product of loval artinian rings?

Isn't the consequence that all local rings of A are fields equivalent to the other assertions? (It seems to me that it implies (3)). In this case, it might be useful to state it as a fourth equivalent condition.

Apologies if this is wrong, but why is this not just the projection formula. Currently what the stacksproject calls the projection formula uses that $K$ is perfect which is a very strong assumption and not very useful in practice (when one specializes to schemes or algebraic stacks). It might be useful to mention that this formula exists in the projection formula section. Also unless I am mistaken, a similar statement holds for qcqs representable morphisms of algebraic stacks. This is Corollary 4.12 (combined with lemma 2.5) of Hall-Rydh 'Perfect complexes on algebraic stacks'.

Ctrl+F "topoological"