8 problems

Proper stable module category equivalence. There should be an equivalence of symmetric monoidal stable $\infty$-categories

Cohomological approximation conjecture. If

Finite-cohomology generation conjecture. If $M$ has cohomology finitely generated over $H^*(G,\mathsf{k})$, then $M$ belongs to the thick subcategory generated by the finitely gene…

Type-$\Phi$ conjecture. A group $G$ is of type $\Phi$ over $\mathbb{Z}$ if and only if it admits a finite-dimensional model for $\underline{E}G$.

Let $G$ be a finite group and let $\mathfrak{p}$ be a homogeneous prime ideal of $H^*(BG,k)$, where $k$ is a field of characteristic $p$. In the stable module category of $G$, let…

Let $k$ be a field of characteristic $2$, and let $A_4$ be the alternating group on four letters. The generating number of the group algebra $kA_4$ is the least generating length o…

Let $k$ be a field of characteristic $2$, and let $Q_8$ be the quaternion group of order $8$. The generating number of the group algebra $kQ_8$ is the least generating length occur…

Let $G$ be a finite group, and let $k$ be a field whose characteristic divides the order of $G$. The ghost number finiteness conjecture. The ghost number of $kG$ is finite. The con…