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Let $R'/R$ be an extension of DVR such that an uniformizing element of $R$ is also an uniformizing element of $R'$ and such that the residue extension is separable (e.g. … So we can reduced to the case of flat morphism of finite type $W\to Y$ between two regular schemes. Let $y\in Y$ and let $w\in W$ be its image by a section $Y\to W$. …

You can see how this phenomenon requires the setting of a non-reduced root system to happen. … At the hyperspecial points, the group would be a quasisplit $SU_3$ over the residue field. …

Presumably you meant for $R$ to be noetherian (max-adic completion would not be appropriate otherwise), and also excellent (or else the completion could fail to be reduced when $R$ is reduced, etc.). … Since the category of finite etale algebras over a henselian local ring is naturally equivalent to that over its residue field, the invariance of etale fundamental group under passage to the completion …

Thus, to prove that $Y_k$ is geometrically integral, it suffices to prove that it is geometrically irreducible and reduced at the generic point. … https://arxiv.org/abs/math/9604227 This has been extended to positive characterstic and mixed characteristic by Jan Gutt (using totally different techniques that do not reduce to Kobayashi-Ochiai as in …

For instance, the recent progress on finding large gaps between primes (see e.g. these two papers) relies on finding somewhat large gaps inside the reduced residue system of a primorial. … Residue class rings ${\bf Z}/q{\bf Z}$ are often referred to by other names, such as cyclic groups or reductions of the integers modulo $q$, and the reduced residue system might be referred to as the group …

This allows us to use Galois descent to reduce the situation to the case of rational singularities. … Let $X_0$ be a reduced projective curve over $k$ with only ordinary double points. …

Thus the number of points of $\pi$ and $\widetilde{\pi}$ are congruent modulo the size of the residue field. QED Fano complete intersections in weighted projective space. … Then, for the remaining polynomials with $m_i\geq 2$, the results of SGA 7 imply that the geometric generic fiber of the linear system is smooth (away from the singularities of the weighted projective …

Indeed, if $j:N' \rightarrow N$ is a closed submodule of a pseudo-compact $R$-module and $\{N_i\}$ is a cofinal system of open submodules of $N$ then $N'_i = N' \cap N_i$ is one for $N'$, and the inclusions … In this way we reduce to the case when $R$ is artinian. Hence, by the formal smoothness hypothesis, the quotient map $A \rightarrow \kappa$ lifts to an $R$-algebra section $s:A \rightarrow R$. …

This is already a well-known phenomenon for $p$-adic fields $K$ (complete discretely valued with perfectly residue field), where letting $C=\widehat{\overline{K}}$ it is the difference between $K$-vector …

requires first deep modularity results of Wiles, Taylor-Wiles, Skinner-Wiles and Kisin on deformations of Galois representations with coefficients in a finite extension of $\mathbb Q_p$ (so with infinite residue …

For $n=1$ it is the reduced special fiber $\mathcal M_1=\mathcal M_{\mathrm{red}}$, thus a smooth scheme over the residue field $\kappa(E)$. … However, among other things, their geometric description implies that the reduced special fiber $\mathcal N_{\mathrm{red}}$ is in fact... not smooth! …

Over $\mathbb{P}^n_{K_v}$, a distance can be given (once a system de coordinates is fixed) by $$ d((x_0,\dots, x_n), \ (y_0, \dots, x_n))= \dfrac{\max_{i, j} \lbrace |x_iy_j-x_jy_i|_v \rbrace}{(\max_i … One can describe this distance as following: there is a canonical reduction map $\pi: \mathbb P^n(K_v) \to \mathbb P^n(\mathbb k_v)$ where $k_v$ is the residue field of $K_v$. …

As a math amateur, I am finding the study reduced residue systems relative a primorial a very interesting way to understand the distribution of primes. … Is it that research in reduced residue systems relative primorial is now focused on highly technical topics? …

Let $\Phi$ be the root system of $G$ (which is $A_r$), and $\Phi^+$ be the positive roots. Let $W\colon G \to \mathbf{C}$ be the spherical Whittaker function on $(\pi_z, V_z)$. … Why does the limit of the spherical Whittaker functions, as the cardinality of the residue field goes to infinity, converge to the character of the corresponding highest weight representation of the dual …

only definition which comes to mind at a good place (namely, cohomology over $\mathfrak{o}_v$ with coefficients in the abelian scheme Neron model at $v$) vanishes due to Lang's theorem over the finite residue … Using this geometric interpretation and a concrete description of finite etale $R$-algebras (and finiteness of the residue fields of the local factor rings $\mathscr{O}_v$), the vanishing is proved on …