The Stacks project

Comments 2181 to 2200 out of 9050 in reverse chronological order.

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

On XYETALE left comment #7370 on Section 10.113 in Commutative Algebra

I checked a bit that the notation probably has never defined for a ring, only for fields. Maybe it is a bit clearer to mention what it means.

On 代数几何真难 left comment #7369 on Section 31.24 in Divisors

I think in lemma 31.24.2, in order to define as a section in , one should first lift to a global section . But is not surjective, seems that is not well-defined.

On Alekos Robotis left comment #7368 on Section 37.11 in More on Morphisms

In definition 37.1.11, there is a slight typo : it says "soure" instead of "source."

On Shizhang left comment #7367 on Lemma 13.27.7 in Derived Categories

Maybe the vertical arrows should all go downward?

On David Holmes left comment #7366 on Section 10.34 in Commutative Algebra

Hi Zongzhu Lin, I think 005K might be the reference you are looking for? Alternatively, I think it's not so hard to see the claim from the definition of constructibility.

On left comment #7365 on Section 112.3 in A Guide to the Literature

The page has been archived by the Wayback Machine. At least some parts of the book are still available through it: https://web.archive.org/web/20110707004531/http://www.math.uzh.ch/index.php?pr_vo_det&key1=1287&key2=580&no_cache=1

On Yijin Wang left comment #7364 on Lemma 28.22.11 in Properties of Schemes

Typo in the proof of lemma 28.22.11: the first sentence should be 'A_1，A_2 ⊂A'

On Alex Ivanov left comment #7363 on Lemma 29.35.16 in Morphisms of Schemes

In (2), it should in fact suffice to assume that is locally of finite type over . (This is also consistent with Lemma 02FW, to which the proof refers).

On left comment #7362 on Section 10.34 in Commutative Algebra

@7361: That is a limitation of how things are being displayed. There is a work-around possible, the question is whether there are enough cases of this causing confusion to put in the effort. I'll put it on the possible features list though, thanks for noticing!

On Laurent Moret-Bailly left comment #7361 on Section 10.34 in Commutative Algebra

Strangely, in "tags" mode, parts (1) and (2) are also converted to tags in the proof (but not in the statement).

On Zongzhu Lin left comment #7360 on Section 10.34 in Commutative Algebra

The first part (1) of Theorem 10.34.1 is called by many as Zeriski's Lemma, which is the H_3 in Zriski's 1947 paper "A new proof of Hilbert's Nullstellensatz" in Bulletin of AMS. The proof presented is roughly the same as Zariski's original proof except the case . Zariski's specifically constructed a non-zero element has the property that any nonzero element has the property that is a factor of in for some (thus k[x_n]$ has only finitely irreducible elements etc,) which is impossible. Zariski's proof is much more self-contained with out using the Chevalley's theorem. There are also other ways to prove Zariski's Lemma using Krull's dimension, or Noether's Normalization Theorem.

By the way I am not clear (in the proof of case) why the constructible set containing a point has to contain a standard open . Constructible sets need not be open. A reference would be helpful.

On Zhiyu Z left comment #7358 on Lemma 29.10.3 in Morphisms of Schemes

For Comment #7350-#7351: any injective morphism of schemes is separated, see 26.23.6.

On Yijin Wang left comment #7357 on Lemma 37.53.3 in More on Morphisms

Typo in lemma 37.52.3: In the forth line 'we see that X_s×Spec(κ(s) Spec(k) is disconnected ' should be 'we see that X_s×Spec(κ(s)) Spec(k) is disconnected '

On Hao Peng left comment #7356 on Lemma 10.29.6 in Commutative Algebra

Nevermind, it is a closed embedding...

On Hao Peng left comment #7355 on Lemma 10.29.6 in Commutative Algebra

The proof has a gap that the may not be equal to . But this can be easily solved using explict form in tag0G1P.

On Hao Peng left comment #7354 on Lemma 5.15.15 in Topology

Ah it seems this can be any subset of . But this lemma only makes sense when is the closure of , so it is still slightly more general.

On Hao Peng left comment #7353 on Lemma 5.15.15 in Topology

I think this lemma is true without the hypothesis that is closed. One proof is given in https://arxiv.org/abs/1005.1423v2 Lemma2.1

On Yijin Wang left comment #7352 on Lemma 37.43.1 in More on Morphisms

Typo in the proof of lemma 37.42.1: in the beginning， (b) should be U=f’^｛-1｝（U’）

On Hao Peng left comment #7351 on Lemma 29.10.3 in Morphisms of Schemes

The last comment contains a mistake, surjective should be separated.

On Hao Peng left comment #7350 on Lemma 29.10.3 in Morphisms of Schemes

I believe more general is true that any injective morphism is surjective. I am not sure if this is stated somewhere?


feed