The Stacks project

Comments 321 to 340 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 #9402 on Section 10.84 in Commutative Algebra

Thanks to both of you. Fix is here.


On left comment #9401 on Section 19.10 in Injectives

I just realized about this quick proof of AB5AB4:

If our category is AB3 (i.e., all coproducts exist; thus it is cocomplete), then we can write [ref] where is the set of finite subsets of , which is a directed set. On the one hand, since finite coproducts are also products in a preadditive category (Homology, Lemma 12.3.4), they are exact (i.e., they commute with all finite limits and colimits). Thus, if our category is AB5, then the expression \eqref{1} is exact in , i.e., exact for all implies exact.


On left comment #9400 on Lemma 21.37.2 in Cohomology on Sites

Thanks. I agree with the first change, but then for the -page we only are left with nonzero entries , I think. CHange is here.


On left comment #9399 on Lemma 36.22.6 in Derived Categories of Schemes

Thanks and fixed here.


On left comment #9398 on Lemma 15.12.1 in More on Algebra

Thanks and fixed here.


On left comment #9397 on Section 96.14 in Sheaves on Algebraic Stacks

Fixed here.


On left comment #9396 on Lemma 15.11.6 in More on Algebra

Thanks and fixed here.


On left comment #9395 on Section 35.34 in Descent

Thanks and fixed here.


On left comment #9394 on Section 15.52 in More on Algebra

Yes. It turns out we haven't yet needed to define or use (quasi-)excellent schemes, so this isn't relevant yet.


On left comment #9393 on Section 43.18 in Intersection Theory

Thanks and fixed here.


On left comment #9392 on Section 55.8 in Semistable Reduction

Yes, for example is a curve over with definitions as in the Stacks project, but is not equal to .


On left comment #9391 on Lemma 50.24.2 in de Rham Cohomology

Thanks and fixed here.


On left comment #9390 on Lemma 7.12.4 in Sites and Sheaves

Yes, if the two coproducts are for presheaves (on the left) and for sheaves (on the right).


On left comment #9389 on Lemma 21.34.6 in Cohomology on Sites

Thanks and fixed here.


On left comment #9388 on Section 21.4 in Cohomology on Sites

Going to leave as is.


On left comment #9387 on Section 6.29 in Sheaves on Spaces

Thanks and fixed here.


On left comment #9386 on Lemma 3.12.1 in Set Theory

OK, thanks for the comment on this. I have added the condition that we want to be exact in the statement of the lemma. See here.


On left comment #9385 on Proposition 35.3.9 in Descent

I think this means you agree now?


On left comment #9384 on Definition 20.47.1 in Cohomology of Sheaves

Thanks and fixed here.


On left comment #9383 on Lemma 96.17.1 in Sheaves on Algebraic Stacks

Nope: see Remark 96.6.3.