13 problems

Bazzoni–Glaz conjecture. The weak global dimension of a Gaussian ring is $0$, $1$, or $\infty$.

Let $\Gamma$ be a discrete subgroup of a Lie group that is not virtually cyclic. Write $vcd(\Gamma)$ for its virtual cohomological dimension, and let $B_{\mathcal{VC}}\Gamma$ denot…

Let $G$ be a discrete group, let ${\mathcal N}(G)$ be its group von Neumann algebra, and let $\frac{1}{{\mathcal F}{\mathcal I}{\mathcal N}(G)}{\mathbb Z}$ be the additive subgroup…

Let $G$ be a group. It is of type $\Phi_{\mathbb{Z}}$ if an $\mathbb{Z}G$-module has finite projective dimension exactly when its restriction to every finite subgroup has finite pr…

Singular hyperplane conjecture. The algebra $\mathcal{A}_lambda$ has infinite homological dimension if and only if $lambda$ lies in a singular hyperplane.

Vanishing upper Serre dimension conjecture. The upper Serre dimension satisfies

Rationality and nonnegativity conjecture. The two Serre dimensions are rational numbers, and the upper Serre dimension is nonnegative:

For a group $G$, consider Bredon cohomology with respect to the family of finite subgroups. A group is locally free if every finitely generated subgroup is free. Bredon analogue of…

Homological-dimension conjecture. The value $\mathrm{hd}_{k}(G)$ is either infinite or equal to $h(G)$.

Local-freeness conjecture. Every group $G$ satisfying

Filtered-colimit conjecture. Every group $G$ with

Let $G$ be an elementary amenable group and let $k$ be a non-zero commutative ring. The homological dimension–Hirsch length conjecture. If $G$ has no $k$-torsion, then … Equivalent…

Phantom dimension conjecture. The phantom dimension $\operatorname{phan. dim.} HR$ should be the pure global dimension of $R$.