The Stacks project

Comments 1101 to 1120 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 David Holmes left comment #8574 on Lemma 53.15.4 in Algebraic Curves

Trivial typo in first line of statement: 'finite morphism OF algebraic".


On Alejandro González Nevado left comment #8573 on Lemma 7.10.8 in Sites and Sheaves

SS: The canonical map between a presheaf of sets on a site and its corresponding zeroth Čech cohomology admits, for every object and every section of the zeroth Čech cohomology over that object, a covering such that each restriction of the section corresponding to that covering is in the image of this canonical map when applied over the object generating such restriction.


On Alejandro González Nevado left comment #8572 on Remark 7.10.7 in Sites and Sheaves

SS: A pair of covergins such that each refines the other induces a canonical identification of the corresponding zeroth Čech cohomologies of any presheaf of sets on the site with respect to these two coverings.


On Alejandro González Nevado left comment #8571 on Lemma 7.10.6 in Sites and Sheaves

SS: Two morphisms between the same pair of covergins of a site inducing (covariantly) the same morphism on the objects of the site covered by these coverings induce (contravariantly) the same map on the zeroth Čech cohomologies of any presheaf of sets on the site with respect to these two coverings.


On Alejandro González Nevado left comment #8570 on Lemma 7.10.5 in Sites and Sheaves

SS: For any given pair of coverings of an object of a site, there exists a covering which is a common refinement of these two coverings.


On Alejandro González Nevado left comment #8569 on Lemma 7.10.4 in Sites and Sheaves

SS: The association sending a presheaf to the map of presheaves sending this presheaf to its zeroth Čech cohomology is a functor.


On Alejandro González Nevado left comment #8568 on Lemma 7.10.3 in Sites and Sheaves

SS: The colimit, when the coverings vary over the opposite of the category of all coverings of a fixed object on a site, of the zeroth Čech cohomology of a fixed presheaf of sets with respect to these coverings of the fixed object on the site defines a presheaf (the zeroth Čech cohomology of the fixed presheaf of sets over the fixed object on the site) and a canonical map of presheaves from the original sheaf to the one produced here via the colimit.


On Alejandro González Nevado left comment #8567 on Lemma 7.10.1 in Sites and Sheaves

SS: Limit of diagram of sheaves exists and coincides with limit as presheaves.


On ZL left comment #8566 on Lemma 96.10.1 in Sheaves on Algebraic Stacks

A minor typo: The third line from bottom "is a coequalizer diagram in " should be "is a coequalizer diagram in "

Also a small question: The conclusion does it include that the composition of structure ring maps is isomorphic to identity?


On left comment #8565 on Lemma 17.28.11 in Sheaves of Modules

Instead of this lemma, one could prove the following more general result and obtain Lemma 17.28.11 as a Corollary:

Lemma. Let be a topological space, let be a sheaf of rings over and let be sheaves of -algebras. Suppose is an ideal sheaf with . Then is naturally a -module. Let be a morphism of -algebras ( becomes an -module via ). Denote to the set of -algebra homomorphisms that lift to . Then is a -torsor via

Proof. The map is well-defined: It is clear that is -linear, that it preserves the multiplicative unit and that as a map of sheaves of abelian groups, it's a lifting of to . It is left to show that is multiplicative. For this, we need first to prove the following: Let , and let be local sections respectively of and of , over the same open subset . Then we claim that , i.e., that . It suffices to verify the equality on germs at , and we leave this as an exercise to the reader (use the natural -module structure on ). Now, one has Thus, and the map from the statement constitutes a group action of on . It is clear that the action is free. To show transitivity, let . We have to show that . On the one hand, it is clear that is -linear. On the other hand, using the same facts as in the last computation, we get which shows Leibniz's rule for .


On left comment #8564 on Lemma 17.28.7 in Sheaves of Modules

In the statement of the theorem, I think one could write "then we have an isomorphism compatible with universal derivations." (I.e., that , where is the isomorphism.) The proof doesn't change, for Lemma 17.28.6 guarantees this on its statement.


On left comment #8563 on Lemma 17.28.6 in Sheaves of Modules

In case anyone needs it, here's a very detailed proof of the last three equalities:

Along the proof we will use the adjunction pointed out in the second paragraph of this comment.

Lemma 1. Let , be sheaves over , respectively, of rings and of sets, and let be a point. There is an isomorphism of -modules , that is natural on .

Proof. Consider the unit of the adjunction and take map on stalks . This induces an -linear map (natural on , by naturality of the unit). Namely, it maps to . Conversely, for each open neighborhood of , there is an obvious map . Since is an -module, this induces a map . Since the latter is compatible with restrictions to open neighborhoods of , we obtain a canonical map . But it maps to , so the two maps are mutually inverse.

Lemma 2. Same notation as in Lemma 1, and suppose that is a sheaf of -modules. Then the following square commutes: where the left arrow is obtained from the induced map on stalks plus the isomorphism of Lemma 1.

We leave the proof to the reader. In the diagram, the horizontal maps are \eqref{1} and the universal property of the free module over a set.

Lemma 3. Notations as in Lemma 1. Let be a continuous map of topological spaces. Then .

Proof. Pulling back the unit of the adjunction gives a map . Note that is a -module. By the adjunction \eqref{1}, we get a map . On stalks, this map equals where we have used Lemma 1. By Lemma 2, \eqref{2} is adjoint to , i.e., to . In other words, the map \eqref{2} equals the canonical morphism of Lemma 1, which is an isomorphism.


On left comment #8562 on Lemma 10.153.11 in Commutative Algebra

A few comments:

  1. is used both in the proof and the statement in different roles.
  2. Statement (3): Strictly speaking, a map mod becomes which is not of the form . So, perhaps, make this statement more precise.
  3. Because this statement is an important fact, I suggest making the proof a bit more transparent. For instance, more explanation can be added in the proof as to why reduces to , possibly highlighting the role of the uniqueness of the splitting.

On left comment #8561 on Lemma 10.153.3 in Commutative Algebra

Statement (8): As far as I understand, the section is unique. This follows from Lemma 06RR.


On left comment #8560 on Lemma 17.28.2 in Sheaves of Modules

How does one argue uniqueness of ? I guess it is because plus facts (i) is -linear and (ii) the image (pre)sheaf of generates as an -module (in the sense of Definition 17.4.5). Does this sound right?

Also, I think it might be interesting to point out that is the left adjoint of the forgetful functor , as it is easy to show (this is actually how one obtains from in the third to last sentence of the proof).


On left comment #8559 on Lemma 10.131.10 in Commutative Algebra

The readers might find interesting to know the following: In Matsumura's Commutative Algebra (not to be confused with Commutative Ring Theory), Ch. 10, (26.I), in Theorem 58 it is stated a sufficient and equivalent condition for the map from Lemma 10.131.9 to have an -linear retraction (namely, that there is an -algebra map which is a right inverse to the canonical map ). It is immediate to verify that "there exists an -algebra map which is a right inverse to " implies the condition between parentheses.


On Jinyong An left comment #8558 on Lemma 28.26.14 in Properties of Schemes

In the proof of the Lemma 28.26.14, where can I find ( associated ) proof of ?


On left comment #8555 on Lemma 10.143.4 in Commutative Algebra

Two very minor comments now:

  1. Perhaps write to indicate the class of mod , instead of .

  2. Last sentence of second paragraph of the proof: "extensions" instead of "extension".


On Elizabeth Henning left comment #8554 on Section 10.18 in Commutative Algebra

+1 for the example suggestion #8512 above


On left comment #8553 on Lemma 12.8.1 in Homological Algebra

In the statement, one could change "canonical" by "unique." Here's the proof of the uniqueness:

Suppose is an additive functor of preadditive categories. Let be objects of and be morphisms in . According to Categories, Lemma 4.27.5, we may represent these by pairs with common denominator . Then Hence, the preadditive structure in is determined by that of .

The reason I find interesting to add the uniqueness to this lemma is to guarantee the uniqueness in the statement of Derived Categories, Lemma 13.5.6.