The Stacks project

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

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.

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?

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.

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 $U_i \cap U_j$ '

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.

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

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

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

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

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

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

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

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

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

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?

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 $\widetilde{M}\otimes _{\mathcal{O}_X} \widetilde{N} \cong \widetilde{M\otimes _R N}$.

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.

Extra ) in the last displayed formula of this proof.

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.