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Assume that $G,H$ are two sheaves of groups (say in fpqc topology on the scheme $X$) and there is a map $G\to H$ which is representable by a closed immersion. Let us also assume that the quotient is r …

There is a functor from the category of higher displays over $k$ of type $\mu$ to the category of isocrystals over $k$ where $k$ is an algebraically closed field where you forget about the filtration …

if $f:X\to Y$ is a map of small v-stacks, Scholze and Fargues define relative homology as the left adjoint to the $f^{\star}$. They say the left adjoint exist because $f$ is a slice in $v$-site (they …

We can associate two $\mathbb Q_p$ vector spaces to a $p$-divisible group, and I'm a little confused about the relation between these two groups. First of all, I think part of my problem is that when …

Chevalley’s affineness criterion says that if $f: X\to Y$ is surjective and finite, $Y$ is Noetherian and $X$ is affine then $Y$ is also affine. The usual proof uses Serre's criterion and Noetherian i …

First of all $\mathbb{C[[x,y]]}$ is local because it is only depend on $\mathbb{C}[x,y]_{(x,y)}$ because by definition it is $lim \frac{\mathbb{C}[x,y]}{(X,Y)^n}$ and all things outside $(X,Y)$ are in …

for your first question, the idea is basically what MikhailBorovoi said but of course you have to be careful to get a model outside $S$: add all the prime divisors of dominators of the equations defin …

First about the ralation between henselian and formal smoothness property, I think a good idea is to look at what was the first version of Hensel lemma: it says that if $f(\bar{a})=0,f'(\bar{a})\, mo …

Deligne's idea was that Shimura varieties should be understood as moduli space of motives(with extra structures). lot's of Shimura varieties of abelian type can be understood as moduli space of abelia …

do you know how we cunstruct a blow up? for your first question blow up of affine $A$ at the ideal $I$ is covered by $$Spec(A[\frac{x^{r_0}\cdot y^{s_0}}{x^{r_i}\cdot y^{s_i}},...,\frac{x^{r_n}\cdot y …

assume that $X$ is Siegel Shimura variety defined over $\mathbb{Z}_p$, you can take its p-adic formal completion $\mathfrak{X}$,and than take it's adic generic fiber $\mathcal{X}$ and get an adic spa …

assume that $Spf\,A\to Spf\,\mathbb{Z}_p[[t_1,...,t_n]]$ is a closed immersion of flat integral formal schemes over $\mathbb{Z_p}$. I see Kisin several time use that if $Spf\,A(O_K)\subset SPf\,\mathb …

Assume that $X=Spf R$ is p-adic formal scheme over $O_{C_p}$ with generic fiber $X_{\eta}$. I want to know why the nearby cycles map $Ru^\star \mathbb{Z/p}$ is equal to $R\Gamma_{et}(spec R[1/p],\math …

I'm trying to read Scholze's article "Etale cohomology of diamonds" (arXiv link) and both in this article and in Berkeley notes, the diamonds are defined as sheaves on the category of characteristic $ …

If you want to see the relation between differential forms and deformation theory you can look at the part B of Illusie article in FGA explained which shows that why the obstruction to the problem of …