15 problems

Let $(A,\mathfrak{m})$ be a two-dimensional regular local ring, let $I$ be an $\mathfrak{m}$-primary ideal, and let $\overline{v}_I$ denote the asymptotic Samuel function of $I$. W…

Let $R\rightarrow S$ be a birational extension of regular local rings of dimension $n\geq 3$, and let $V$ be a valuation ring of the quotient field of $S$ that dominates $R$. A mon…

Let $k$ be an infinite field and $(R,\mathfrak{m})=k[[x_0,\ldots,x_d]]$. For general linear forms $\ell_1,\ldots,\ell_i\in\mathfrak{m}$, set … Let $\stirlingI{d+1}{d+1-i}$ denote t…

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…

Let $(R, \mathfrak m=(x_1,\dots,x_d))$ be a complete $d$-dimensional regular local ring, let $J=(x_1^{a_1},\dots,x_d^{a_d})$ with $a_i>0$, and let $\mathfrak a$ be an $\mathfrak m$…

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 $R$ be a complete intersection of even dimension with an isolated singularity, and let ${\mathbb E}(R)$ denote the real vector space referred to in the source. Dao–Kurano's one…

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 $(R,\mathfrak m,k)$ be a local ring, let $M$ be an $R$-module of nonzero finite length, let $F$ be a minimal resolution of $M$, and let $\boldsymbol{x}$ be a parameter set for…

Let $(T,\eta)$ be a Gorenstein local ring of dimension $d$, let $\{x_1,\ldots,x_d\}\subseteq T$ be any system of parameters, and let $Q=(x_1,\ldots,x_d)$ be the ideal generated by…

Polynomial-type conjecture. We conjecture that $h^0_{I,R}(n)$ is always of polynomial type for general rings.

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