7 problems

Let $\underline{\mathcal{G}}$ be the relevant parahoric group scheme and let $\mu$ be the specified cocharacter datum, and write $\widetilde{M}_{\underline{\mathcal{G}},\mu}$ for t…

Let $X$ be a normal integral scheme projective over a complete local ring $(R,\mathfrak{m})$ with residual characteristic $p>0$, and let $B$ be a $\mathbb{Q}$-divisor on $X$. Pertu…

Let $X$ be a normal integral scheme of finite type over a complete local ring $(R,\mathfrak{m})$ such that $R/\mathfrak{m}$ has positive characteristic. Let $B\geq 0$ be a…

Let $(R,pR,k)$ be a discrete valuation ring essentially smooth over $\mathbb{Z}$, with residue field $k$. Gersten's Arithmetic DVR Conjecture. For every $n \geq 0$, the transfer ma…

Let $(R,\operatorname{\mathfrak{m}})$ be a $d$-dimensional complete local domain of mixed characteristic $p$. Assume that $p,x_2,\dots,x_d$ is a system of parameters. Set … and, fo…

Let $R$ be a complete local domain of mixed characteristic. A Q-sequence is a sequence with the quasilength property used in the definition of robust algebras. Mixed-characteristic…

Let $R$ be a finitely generated integral domain over $\mathbb Z$, let $S=\operatorname{Spec}R$, and let $\pi:\mathcal X\to S$ be a smooth proper morphism. Let $\mathcal L$ be an in…