10 problems

Let $R$ be a local complete Noetherian domain with coefficient field $k$, and set $X:=\operatorname{Spec}R$. For every $n\in\mathbb{N}_0$, consider the $n$-th Nash blowup…

Stirling-number Lech conjecture. The first displayed inequality, together with the equivalent reindexing given in the source, holds for every $\mathfrak{m}$-primary monomial ideal…

Let $(R,\mathfrak{m})$ be a normal local domain essentially of finite type over a field of characteristic zero. Suppose that $R$ is lim-stable, for example, semistable. Lim-stabili…

Let $(R,\mathfrak{m})$ be a local ring. Suppose its completion $\widehat{R}$ has an isolated singularity, meaning that $\widehat{R}_P$ is regular for every…

Grothendieck's conjecture. The following statements hold:

Let $R$ be a complete local ring, and let $F$ be a short complex in the perfect-complex category, meaning a short complex in $\operatorname{perf} R$ supported on the maximal ideal.…

Let $(R,\mathfrak m)$ be a local ring, let $e(R)$ denote its Hilbert–Samuel multiplicity, and let $F$ be a short complex supported on the maximal ideal, meaning a non-exact complex…

Let $(R,\mathfrak m)$ be a local ring, let $e(R)$ denote its Hilbert–Samuel multiplicity, and let $\ell_R M$ denote the length of an $R$-module $M$. Minimal length conjecture. For…

Regularity conjecture for amenable d-sequences. If ${\mathbf x}$ is an amenable d-sequence relative to $M$,

Serre's intersection multiplicity conjecture. The following assertions hold: 1. $\operatorname{Kdim}M+\operatorname{Kdim}N\leq\operatorname{Kdim}R$. 2. If…