12 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 $R$ be a local regular ring with fraction field $K$, let $Q$ be a non-degenerate quadratic form over $R$, and let $c\in R$ be invertible. The equation $Q=c$ means that the quad…

Let $R$ be a local ring of mixed characteristic $(0,p)$ that is excellent, normal, and admits a dualizing complex. Let $\pi\in\mathfrak{m}$ be a non-zero divisor such that…

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…

Skalit's unramified equality conjecture. Equality holds in the lower bound

Hochster–Huneke's mixed-characteristic vanishing conjecture. The corresponding vanishing theorem holds in mixed characteristic. The equal-characteristic theorem gives the model for…

Mixed-characteristic local cohomology conjecture. These three properties should hold for regular local rings of mixed characteristic.

The Monomial Conjecture. For all $t\in\mathbb N$ and all systems of parameters $x_1,\ldots,x_d$ of any local Noetherian ring,