27 problems

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 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 $R$ 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…

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

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…

Isomorphism conjecture. The q.l.b. are isomorphic to each other. The paper has proved this for the particular pair $E^{1,1}({\hbar})$ and $E^{0,0}({\hbar})$, but states that the ge…

Let $X$ be a symplectic manifold and let $(\Omega(X, {\cal W}), \nabla, R)$ be its topological $Q$-algebra. If a strict deformation quantization of $X$ exists, consider the categor…

Let $G$ be a finite group, let $p$ be a prime, and let $(\mathbb{K},R,\mathbb{F})$ be a splitting $p$-modular system for $G$. For $x\in G$, let $C_G(x)$ denote its centralizer; cal…

Let $G$ be a finite group, let $p$ be a prime, and let $(\mathbb{K},R,\mathbb{F})$ be a splitting $p$-modular system for $G$. For each $p^{\prime}$-element $x\in G$, let…

Let $G$ be a finite group, let $p$ be a prime, and let $(\mathbb{K},R,\mathbb{F})$ be a splitting $p$-modular system for $G$. Write $G_p$ for the set of $p$-elements and…

Serre's conjecture. Every finitely generated projective module over $\mathbb{K}[x_1,\ldots,x_r]$ is free.

Let $R$ be a commutative ring. A finitely generated projective $R$-module $P$ is stably free if there exist $n,m\in\mathbb{N}$ such that $P\oplus R^n\cong R^m$, and $R$ is Hermite…

Let $k\geq1$ and let … Here $\gamma_1$ and $\gamma_2$ are the parameters defined earlier in the paper. The off-diagonal coefficient vanishing conjecture. If…

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 $E$ be a finite graph, let $\mathsf{k}$ be a field, and let $L_\mathsf{k}(E)$ be its associated Leavitt path algebra. For $n\in\mathbb{N}$, write $L_n$ for the classical Leavit…

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…

There is a polynomial ring $A=k[x_1,\cdots,x_n]$ over a field $k$, and a finite projective $A$-module is a finitely generated projective module over $A$. Serre's conjecture. Every…

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 $r+1$…

Let $A$ be a domain of dimension $n$, let $R=A[X_1,\ldots,X_m]$, and let $I$ be the ideal of $R$ generated by $(X_1,\ldots,X_m)$. In this case…

Let $G$ be a torsion-free group. A finitely generated projective $\mathbb{Q}[G]$-module is a finitely generated projective module over the group ring $\mathbb{Q}[G]$. The projectiv…

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

Let $G_{22}$ be the lower-right block of a stable transfer matrix $G(z)$, and let $G_{22}$-stabilizability denote the property defined in the surrounding robust-control framework.…