The Stacks project

Comments 281 to 300 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 left comment #9443 on Lemma 5.8.4 in Topology

In the second paragraph, it seems that “take one of the and expand it to an irreducible component ” needs Lemma 5.8.3, point (3), which requires Zorn's lemma. It can be done without the axiom of choice: the second paragraph may be replaced by

Suppose , where is some irreducible subset of . By the argument in the first paragraph, for some . Since the original union does not have redundant members, . Therefore is maximal in the set of irreducible subsets of , that is, is an irreducible component of .


On Branislav Sobot left comment #9442 on Lemma 29.39.3 in Morphisms of Schemes

I belive that in the first part of the proof we should take in the exponent of number instead of , so that total degree is N.


On Ryo Suzuki left comment #9441 on Lemma 10.32.5 in Commutative Algebra

Oops! I rewrite it.

Let be a ring. Let be an ideal of and be a finitely generated ideal of . Suppose . Then for some .


On Ryo Suzuki left comment #9440 on Lemma 10.32.5 in Commutative Algebra

It can be slightly generalized as follows: Let be a ring. Let be an ideal of and be a finitely generated ideal of . Suppose . Then for some .

Then Lemma 00L6 also can be generalized in the same way.


On left comment #9438 on Lemma 12.4.2 in Homological Algebra

Going to leave as is.


On left comment #9437 on Section 14.34 in Simplicial Methods

Thanks and fixed here.


On left comment #9436 on Definition 78.19.3 in Groupoids in Algebraic Spaces

Thanks and fixed here.


On left comment #9435 on Lemma 13.27.7 in Derived Categories

Thanks and fixed here.


On left comment #9434 on Section 29.5 in Morphisms of Schemes

Sounds good. Going to leave as is.


On left comment #9433 on Section 34.3 in Topologies on Schemes

Thanks and fixed here.


On left comment #9432 on Lemma 34.4.13 in Topologies on Schemes

Thanks and fixed here.


On left comment #9431 on Lemma 3.9.1 in Set Theory

Going to leave as is.


On left comment #9430 on Lemma 10.127.3 in Commutative Algebra

I don't agree with this comment. Maybe I misunderstood?


On left comment #9429 on Lemma 12.26.3 in Homological Algebra

Thanks! Fixed here.


On left comment #9428 on Lemma 10.153.9 in Commutative Algebra

Fixed here.


On left comment #9427 on Section 19.10 in Injectives

Going to leave as is.


On left comment #9426 on Lemma 10.8.8 in Commutative Algebra

Side comment: the result "homology commutes with filtered colimits" holds over any abelian category that is AB5 (as is). I wish I knew a reference in the literature (does anybody know some?). In the meantime, I wrote the proof here, see Corollary 4.


On left comment #9425 on Remark 15.86.10 in More on Algebra

Thanks and fixed here.


On left comment #9424 on Lemma 36.3.4 in Derived Categories of Schemes

Thanks and fixed here.


On left comment #9423 on Lemma 13.7.1 in Derived Categories

Already fixed.