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Assume that $G$ is a group and $R$ is a p.i.d. What can we say about the projective dimension of $R[G]$? For example can we say that this dimension is at most $1$ for reductive groups? (I think if $ …

I am trying to understand the proof of lemma 2.2.18 in Lucas Mann's thesis. Its statement is surprising for me, because it talks about general rings which are not necessarily characteristic $p$, and o …

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 …

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 …

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 …

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 …

In the paper "The display of a formal $p$-divisible group" Zink defines some objects and calls them $3n$-display. A $3n$-display over $R$ is a quadruple $P$, $Q$, $F$, $F^1$ such that $P$ is a project …

I'm trying to read Kisin's paper about the Integral model of Shimura varieties. In section five he discusses versal deformation ring of a p-divisible group. Assume that $K$ is a number field with resi …

I'm reading Laurie's note about Fargues-Fontaine Curve and I think he uses a different definition of $B_{\mathrm{cris}}$. Usually when $R$ is a perfect ring of characteristic $p$, $A_{\mathrm{cris}}(R …

Manin proves Mordel's conjecture for function fields in characteristic zero.his proof has a gap but Coleman fill this gap and restate Manin proof in a more modern language.both of them work over chara …