Comments 1 to 20 out of 7360 in reverse chronological order.


On Mingchen left comment #7895 on Lemma 10.50.3 in Commutative Algebra

This is not a big deal, but strictly speaking, one should say for any non-zero $x\in K$, blabla as you talk about $x^{-1}$. The same issue persists in the next lemma.

On Torsten Wedhorn left comment #7894 on Lemma 15.13.1 in More on Algebra

One could more precisely show that $P \mapsto P/IP$ induces a bijection between isomorphism classes of finite projective $R$-modules and isomorphism classes of finite projective $R/I$-modules: The current formulation shows the surjectivity of the map. To show the injectivity, let $P$, $P'$ be finite projective $R$-modules such that $P/IP \cong P'/IP'$. Hence we find an $R$-linear map $P \to P'/IP'$. As $P$ is projective, it lifts to an $R$-linear map $u\colon P \to P'$. As $u$ is an isomorphism modulo $I$, it is surjective by Nakayama's lemma. The ranks of the finite projective $R$-modules $P$ and $P'$ are equal in every open neighborhood of $V(I)$. As $I$ is contained in the radical of $R$, the only open neighborhood of $V(I)$ is $\Spec A$. Hence $u$ is a surjective map of finite projective modules of the same rank. Hence it is an isomorphism.

On Anonymous left comment #7893 on Section 43.9 in Intersection Theory

In the definition of order, there is $\mathcal{O}_{X,z}$ which should be $\mathcal{O}_{X,Z}.$

On left comment #7892 on Section 115.11 in GNU Free Documentation License

On Laurent Berger left comment #7891 on Section 10.69 in Commutative Algebra

In the first sentence "There is a notion of regular sequence which is slightly weaker than that of a regular sequence" the word "quasi" is presumably missing.

On left comment #7890 on Section 115.11 in GNU Free Documentation License

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On left comment #7889 on Section 115.11 in GNU Free Documentation License

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On Matthieu Romagny left comment #7888 on Lemma 59.28.1 in Étale Cohomology

No, the assumption $n\in \mathcal{O}_S^*$ is the correct one: if $n$ is not invertible in the local rings of $S$ then the sequence is not exact on the right (étale-local surjectivity fails). Think about what kind of extension it needs to extract $p$-roots in characteristic $p$.

On Anon left comment #7887 on Lemma 59.28.1 in Étale Cohomology

Typo in assumption, should be $n \in \mathbf{N}$.

On کاشی کاری مشهد left comment #7886 on Lemma 66.4.5 in Morphisms of Algebraic Spaces

شکل ممکن نشان دهد. پروفایل کاشی کاری مشهد در راکت زمان و تاریخ کاشی کاری مشهد

کاشی کاری ۱۷ شهریور مشهد

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کاشی کار مشهد

کاشی کاران مشهد

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کاشی کار حرفه ای مشهد

اوستا کاشی و سرامیک کار مشهد

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کاشی کار مشهد

کاشی کار فلسطین مشهد

کاشی کار مشهد

کاشی کاری خادمی نژاد مشهد

On کاشی کاری مشهد left comment #7885 on Definition 26.9.1 in Schemes

okay great I know now

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On کاشی کاری مشهد left comment #7884 on Definition 26.9.1 in Schemes

okay great I know now

\ref{h}

On left comment #7883 on Section 15.11 in More on Algebra

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On left comment #7882 on Section 15.11 in More on Algebra

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On Elías Guisado left comment #7881 on Lemma 17.12.4 in Sheaves of Modules

I've found this in Serre's Faisceaux algébriques cohérents. Properties (3) and (4) are, respectively, theorems 13.2 and 13.1. Here is an English translation: http://achinger.impan.pl/fac/fac.pdf On this pdf the theorems are on pp. 18-19

On Mingchen left comment #7880 on Lemma 15.27.3 in More on Algebra

$R^{\oplus t}$ should be $A^{\oplus t}$

On parsec left comment #7879 on Section 3.3 in Set Theory

@verisimidude

(I am not a graduate student; tell me if this is inaccurate)

It's not true that everything is a set. The axioms of ZFC are actually very specific rules for how a set can be constructed. This is because of stuff like Russel's Paradox (does the "set" of all sets that do not contain themselves contain itself). A class is a way to talk about some sets without placing them in a set: by definition, classes can't contain themselves, which avoids Russels's paradox. The "set" of all proper classes would actually not be a class or a set: it would be the next step up in the hierarchy.

On Shogōki left comment #7878 on Lemma 33.36.6 in Varieties

While homeomorphisms have "2 out of 3" property, it's unclear if universal homeomorphism also has the property. (In fact, the one being used in your second sentence is the only unclear one, because one needs to look at base changes that doesn't come from the "most downstairs base"? I tried to see this, and easily reduced to the case where the composition is identity, but then got stuck. Maybe this is already discussed somewhere in the SP?)

In any case, it is still true that F_{X/S} is a universal homeomorphism: WLOG one may assume both X and S are affine, then one can check condition (2) in [0CNF] holds true directly.

On qyk left comment #7877 on Lemma 30.21.3 in Cohomology of Schemes

In the proof (2nd paragraph), there is an "abd", which I think should be an "and".

On qyk left comment #7876 on Lemma 36.32.3 in Derived Categories of Schemes

Is the condition that $f$ is flat redundant? We can use (2) in \href{https://stacks.math.columbia.edu/tag/0B91}{Lemma 0B91}, which does not need flatness of $f$.