11 problems

Let $R$ be a regular local ring with maximal ideal $m$, and let $f\in m\setminus m^2$. Quillen's question. Every finitely generated projective $R_f$-module is free. This is the mod…

Let $R$ be a regular local ring of dimension $d\geq 1$, let $m$ be its maximal ideal, and let $u\in m\setminus m^2$. Let $G$ be a strictly isotropic reductive group over $R$. Nisne…

Gersten's conjecture. The canonical inclusion $\mathcal{M}^{p+1}(A) \hookrightarrow \mathcal{M}^p(A)$ induces the zero map on $K$-groups

Vanishing conjecture. The local cohomology groups satisfy

Vanishing conjecture. The higher cohomology groups vanish:

Let $(R,\mathfrak{m})$ be a regular local ring of dimension $n$. Let $I$ be an ideal generated by an $R$-regular sequence of length $k\leq n-1$, and let $J$ be an ideal generated b…

Integral-closedness conjecture. If $\mathcal{R}'(I)$ is nearly Gorenstein, then $I$ is integrally closed.

Let $(A,\mathfrak{m}_A,L)$ be a regular local ring, and let $\operatorname{CH}^p(A)$ denote its codimension-$p$ Chow group. Claborn–Fossum Conjecture. … The conjecture extends the…

Gersten's conjecture. If $A$ is a regular local ring, then this map is injective for all $n\geq 0$. This is a theorem when $A$ contains a field; the paper studies analogues of this…

Hübl's conjecture. Then

Let $A$ be a commutative noetherian ring with unit and let $0\leq p\leq\dim A$. Denote by $\operatorname{\mathcal{M}}^p_A$ the category of finitely generated $A$-modules whose supp…