Comments 1 to 20 out of 5160 in reverse chronological order.


On PLACEMENT TRAINING left comment #5545 on Lemma 85.18.10 in Formal Algebraic Spaces

Can’t tell a good article, because it is more than that, thanks keep going... PLACEMENT TRAINING https://www.gobrainwiz.in/pages/crt

On left comment #5544 on Section 4.22 in Categories

But it is enough to suppose that $X$ is a (co-)cone of the diagram, and this observation is actually important in Lemma 05SH, as this lemma is (rightly) formulated without any exactness conditions on $F$. Such conditions are not needed because any functor preserves (co-)cones even if it does not preserve (co-)limits.

On Harry Gindi left comment #5543 on Lemma 58.73.3 in Étale Cohomology

The corresponding lemma in SGA4 (IX.2.14) asserts (1) for a map F into the product rather than the coproduct. Is this a typo?

On Zhenhua Wu left comment #5540 on Lemma 29.40.8 in Morphisms of Schemes

Locally of finite type will suffice as universally closed morphism is quasi-compact, it's best to change this description in order to be consistent with tag 0AH6.

On minsom left comment #5539 on Section 17.12 in Sheaves of Modules

May I ask you something? In the proof of lemma 01BY (3) , how can we show that "There exists an open covering of $U$ such that on each open all the sections $\overline{s_i}$ lift to sections $s_i$ of $G$." ? For sheaf axiom , we need the fact ' for open cover $U_i$ , $U_j$ , each element is same in $U_i \cap U_j$ '

On Michael left comment #5538 on Lemma 29.55.5 in Morphisms of Schemes

It looks like there is a typo here. $(A_{\mathfrak{m}})$ should be $\dim_k(A_{\mathfrak{m}})$ and similiarly for $B$. There is a total of three occurrences.

On David Liu left comment #5537 on Section 30.12 in Cohomology of Schemes

In the proof of lemma 01YH, why does $\mathcal{Q}$ and $\mathcal{Q}'$ have supports that are strictly contained in $Z_0$?

On Zeyn Sahilliogullari left comment #5536 on Lemma 26.11.5 in Schemes

g∈A in the second to last sentence should be g∈B

On left comment #5535 on Lemma 43.22.1 in Intersection Theory

The reference is not precise. I think it is better to write [Chapter V, C), Section 7, formula (10), Serre_algebre_locale]

On left comment #5534 on Lemma 42.25.3 in Chow Homology and Chern Classes

In the statement of the Lemma 02ST, I believe the last line should be $f_*(c_1(f^*L)∩[X])=c_1(L)∩[Y]$.

On left comment #5533 on Lemma 42.25.3 in Chow Homology and Chern Classes

In the statement of the Lemma 02ST, I believe the last line should be $f_*(c_1(f^*L)∩[X])=c_1(L)∩[Y]$.

On Chi Zhang left comment #5532 on Lemma 26.17.5 in Schemes

Maybe $\mathfrak{p}$ is more precisely a maximal ideal? Since it is the kernel of the ring morphism $\kappa(x)\otimes_{\kappa(x)}\kappa(y)\to \kappa(z)$.


On left comment #5531 on Definition 10.87.7 in Commutative Algebra

Yes, but the condition is independent of the choice by part (1) of the proposition.

On Dion Leijnse left comment #5530 on Lemma 48.2.1 in Duality for Schemes

In part (2) of the statement of the Lemma, I think the $K|_U$ should be $K|_{U_i}$.


On left comment #5529 on Lemma 15.63.10 in More on Algebra

Lemma needs "Assume B is flat as an A-module".

On Roman Bezrukavnikov left comment #5528 on Definition 10.87.7 in Commutative Algebra

In Proposition 059E we a given a module M and a directed system having M as a limit, while in this definition we are only given a module. Is the meaning here that there exists a directed system for which equivalent conditions of the Proposition hold?

On Kang Taeyeoup left comment #5527 on Section 26.7 in Schemes

Alternatively, I believe that the proof can be done by much easier and elegant way.

We have a Hom-Tensor adjuntion in the category of $\mathscr{O}_X$-modules.

Using this with the lemma 26.7.1, we get the following bunch of natural isomorphisms.

$\mathrm{Hom} _{\mathcal{O} _X}(\widetilde{M}\otimes _{\mathcal{O} _X} \widetilde{N},F) \cong \mathrm{Hom} _{\mathcal{O} _X}(\widetilde{M},\mathrm{Hom} _{\mathcal{O} _X} (\widetilde{N},F))$ $\cong \mathrm{Hom} _R(M,\Gamma(X,\mathrm{Hom} _{\mathcal{O} _X}(\widetilde{N},F)) = \mathrm{Hom} _R(M,\mathrm{Hom} _R(N,\Gamma(X,F)))$ $\cong \mathrm{Hom} _R(M\otimes _R N,\Gamma(X,F)) \cong \mathrm{Hom} _{\mathcal{O}_X}(\widetilde{M\otimes _R N},F)$

Since this hold for any $F$, we have that $\widetilde{M}\otimes _{\mathcal{O}_X} \widetilde{N} \cong \widetilde{M\otimes _R N}$.

On Kang Taeyeoup left comment #5526 on Section 26.7 in Schemes

I think the first proof of the lemma 01I8 a) is not clear. The proof shows that there is the isomorhpism $M\otimes_R N \to \Gamma(X,\widetilde{M}\otimes_{\mathscr{O}_X} \widetilde{N})$, and this gives the corresponding isomorphism $\widetilde{M\otimes_R N} \to \widetilde{M}\otimes_{\mathscr{O}_X}\widetilde{N}$

However, an adjunction does not guarantee that the isomorphisms in the one part goes to isomorphisms in the other part.

Suppose $F : C \to D$ and $G : D \to C$ are given and $F$ is left adjoint to $G$.

Let $\alpha : c \to Gd$ be an isomorphism, then the corresponding morphism in $Hom_D(Fc,d)$ is given by the composite $Gd \xrightarrow{F\alpha} FGc \xrightarrow{\epsilon_c} c$. Here $\epsilon : FG \to id$ is a counit. In general there is noreason that $\epsilon_c\circ F\alpha$ should be an isomorphism.

On Wen-Wei LI left comment #5525 on Lemma 13.18.7 in Derived Categories

Extra ) in the last displayed formula of this proof.

On Olivier de Gaay Fortman left comment #5524 on Lemma 92.18.2 in Algebraic Stacks

I think you mean by Algebraic-Stacks/S the 2-category of algebraic stacks over $S$, defined using $Sch_{fppf}$, and similarly for Algebraic-Stacks'/S.