13 problems

Let $S$ be a scheme. A linear group scheme over $S$ is a group scheme $G\rightarrow S$ that, locally in the étale topology of $S$, embeds as a closed subgroup scheme of $GL(E)$ for…

Let $G$ be an abelian affine presentable group sheaf over a scheme $X$. For every finite type artinian scheme $Z\rightarrow X$, let $p:Z\rightarrow \operatorname{Spec}(k)$ be the p…

Etingof's conjecture for $\alpha_p(r,s)$. The statement analogous to the generalized Benson conjecture holds for $\alpha_p(r,s)$ as well.

Let $k$ be a field of positive characteristic and let $G$ be a finite commutative unipotent $k$-group scheme. Write $\mathrm{ed}_k(G)$ for the essential dimension of $G$, and let…

For a scheme, its étale fundamental group is the profinite group encoding its finite étale coverings. Grothendieck's group-scheme enhancement conjecture. The étale fundamental grou…

Let $p$) be a prime, let $q=p^l$, and let $\mathbb{F}_q$ be the finite field with $q$ elements. For a compact group $G$, write $\mathcal{M}_\mathrm{dimirr}(G)$ for the multiset of…

Purity conjecture for flat cohomology. The fppf cohomology with supports vanishes in low degrees:

Let $A$ be the base ring and let $G\in\mathbb{F}_{q}\text{-}\operatorname{Gr}^{\boxplus}_{A}$. Suppose that there is a closed embedding … for some $N\in\mathbb{N}$. Let…

Let $A$ be the base ring and let $G\in\operatorname{Gr}^{+}_{A}$ be a group scheme of additive type. Write $G=\operatorname{Spec}(B_G)$ and let $\operatorname{Prim}(B_G)$ denote th…

Let $S$ be a base scheme and let $G\in\operatorname{Gr}_{S}$ be a group scheme of additive type, meaning that locally on $S$ it admits a closed embedding into $\mathbb{G}_a^N$ for…

Let $A$ be the base ring, let $q$ be a power of the characteristic prime $p$, and let $G\in\mathbb{F}_{q}\text{-}\operatorname{Gr}^{+}_{A}$. Suppose that there is a closed embeddin…

Paley type group schemes are considered in abelian groups, including groups of non-square order. Elementary abelian group conjecture. Such abelian groups must be elementary abelian…

Kummer group scheme classification conjecture. Any model of $\mu_{p^n,K}$ over ${\mathcal O}_K$ is a Kummer group scheme.