13 problems

Let $G$ be a group, let $H$ be a subgroup of finite index, and let $R$ be an arbitrary ring. Assume that for every nontrivial element $x$ of $G$, at least one of the following cond…

Let $A$ be a regular ring of finite Krull dimension, and let $n\ge 1$. A finitely generated projective module over $A[x_1,\ldots,x_n]$ is extended from $A$ if it is obtained from a…

Let $R$ be a regular local ring with maximal ideal $m$, and let $f\in m\setminus m^2$. Quillen's question. Every finitely generated projective $R_f$-module is free. This is the mod…

Let $R$ be a smooth affine algebra of Krull dimension $d\geq 3$ over an algebraically closed field. Let $P$ be a projective $R$-module of rank $d-1$. Murthy's splitting conjecture.…

Let $n\in\mathbb{N}^*$, and let $\{\mathfrak{b}_{m,n}\}_{m\geq0}$ be a sequence satisfying … These sequences encode the diagonal elements of $P_0(A^\infty(\mathbb{R}^{2n}_\Theta))$…

Let $P_k(A^\infty(\mathbb{R}^{2n}_\Theta))$ denote the class of projectors of level $k$ in the smooth noncommutative Euclidean algebra, where $k\geq0$, $n\geq1$, and $\Theta$ is th…

Let $\mathbb{G}=(W\varphi^{-1},L)$ be as in the principal-block dimension conjecture, and suppose that $W$ is an $\ell'$-group, $\varphi$ is trivial, and $q\equiv1\pmod\ell$. Let…

Let $R$ be a local ring. Assume either that $R$ is a regular local ring or that $1/r!\in R$. Let $T=(T_1,\ldots,T_n)$ and let $v$ be a unimodular vector over $R[T]$ of length $r+1$…

Homotopy Conjecture. There exists a surjective map $F:P[T]\twoheadrightarrow J$ such that

Let $p,p'\in\mathbb{Z}_{>0}$ with $p,p'\geq 2$ and $p$ and $p'$ relatively prime. Let $\mathcal{W}_{p,p'}$ be the logarithmic vertex operator algebra, and let $\overline{V(p,p')}$…

Let $\mathcal{W}_{3,2}$ be the logarithmic vertex operator algebra associated with the coprime pair $(3,2)$, and let $\overline{V(3,2)}$ and $\overline{MV(3,2)}$ be the logarithmic…

The norm inequality. The inequality

Moore's conjecture. If $M$ is projective over $\mathbb{Z}H$, then it is projective over $\mathbb{Z}\Gamma$. In particular, for torsion-free $\Gamma$, projectivity over…