6 problems

Let $R$ be a ring. A ring is pseudo $\pi$-polar if it satisfies the pseudo $\pi$-polar condition, and $\pi$-UU if every unit $u\in U(R)$ has $1-u^n\in J(R)$ for some integer…

Let $R$ be a $p$-complete noetherian regular local ring with perfect residue field. Let $G$ be the group scheme identified in the source with the Hodge–Tate stack presentation…

Let $(A,\fra^t)$ be a pair, where $A$ is the regular ring and $\fra$ is the ideal appearing in the construction of the perfectoid test ideal ${\mathrlap{\mJ}\,\,\uptau}(A,\fra^t)$.…

Let $X=\operatorname{Spec}(R)$ be the spectrum of a regular local ring $R$. For each integer $l$, let $\mathrm{M}_l(X)$ be the category of coherent sheaves on $X$ whose supports ha…

Unit-regularity conjecture. If $R$ is $*$-regular and $\overline{L}(R)$ is proatomic with an orthogonal large partial $n$-frame, then $R$ is unit regular. Every representable ring…

Representable-ring ultraproduct conjecture. Every subdirectly irreducible representable $*$-regular ring can be embedded into a homomorphic image of a regular $*$-subring of an ult…