8 problems

A ring is called $*$-regular if it is regular and its involution is proper. It is directly finite if $ab=1$ implies $ba=1$, and unit-regular if for every element $a$ there is a uni…

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 ${\mathrlap{\mJ}\,\,\uptau}(A,\fra^t)$.…

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…

Let $k$ be a field, let $A=k[X_1,\ldots,X_n]$, and let $\mathfrak p\subset A$ be a prime ideal such that $A/\mathfrak p$ is regular. Write $\mu(\mathfrak p)$ for the minimal number…

Huneke's Bass-number finiteness conjecture. These Bass numbers are finite for all integers $i$ and $j$ and all prime ideals $\frak p$ of $R$. In particular, the injective resolutio…