118 problems

Auslander–Reiten conjecture. If

Let $(R,\mathfrak m,k)$ be a Noetherian local ring with a canonical module $K_R$. Tachikawa's conjecture. If … for all $i>0$, then $R$ is Gorenstein. This is the commutative versio…

Let $R$ be the local ring under consideration and let $M$ be an $R$-module. Generalized Auslander–Reiten conjecture. If … for all $n\gg 1$, then $M$ has finite projective dimension…

Lech's conjecture. Then

Schreyer's conjecture. The ring $R$ has finite Cohen–Macaulay representation type if and only if its completion $\widehat{R}$ has finite Cohen–Macaulay representation type.

Sally's conjecture. If $R$ has almost minimal multiplicity, then $\operatorname{gr}_{\mathfrak m}(R)$ is almost Cohen–Macaulay.

Let $S/R$ be an extension of commutative rings, and let … When $S=\overline{R}$, the integral closure of $R$ in its total ring of fractions $\operatorname{Q}(R)$, write $R^*=R_S^*$…

Let $A\to B$ be a local homomorphism of artinian local rings, and let $\operatorname{edim} A$ and $\operatorname{edim} B$ denote their embedding dimensions. de Smit's conjecture. I…

Let $R$ be a local ring, and let $M$ be a nonzero $R$-module of finite length and finite projective dimension. Let $\ell_R(M)$ denote its length and let $e(R)$ denote the Hilbert–S…

Let $(R,\mathfrak{m},k)$ be a $d$-dimensional Noetherian local ring, and let $M$ be a finitely generated nonzero $R$-module. Suppose that $M$ has finite length and finite projectiv…

Rossi's conjecture. The Hilbert function of a one-dimensional Gorenstein local ring should be non-decreasing.

Polini–Ulrich conjecture. One has

Let $(R,\mathfrak m,k)$ be a Noetherian local ring, and let $M$ be a finitely generated $R$-module. Write $\beta_i^R(M)$ for the $i$-th Betti number of $M$. Avramov's conjecture. T…

Let $(R,\operatorname{\mathfrak{m}})$ be a Noetherian local ring, let $\widehat{R}$ denote its $\operatorname{\mathfrak{m}}$-adic completion, and consider … where $I$ ranges over t…

Let $(R,\mathfrak m)$ be a Noetherian local complete intersection ring. Say that $R$ is a UFD in codimension at most three if every localization $R_{\mathfrak p}$ at a prime ideal…

A small Cohen–Macaulay module over a local ring is a finitely generated module whose depth equals the dimension of the ring. The small Cohen–Macaulay conjecture. Every complete loc…

Asymptotic Lech inequality. In this case, the asymptotic supremum of the normalized multiplicities $e(J)/d!$ over $\mu$-primary ideals of length tending to infinity should satisfy

Let $M$ be a module over a local ring, and let its Betti numbers be the ranks of the free modules in a minimal free resolution of $M$. A sequence of Betti numbers is bounded if it…

Let $k$ be the field appearing in the power-series congruence subgroup, let $n,m$ be positive integers, let $p$ be a prime, and write $C_{n,m}(k[[t_1,t_2,\dots,t_m]])$ for the corr…

Let $R$ be a local ring and let $M$ be an $R$-module of finite projective dimension. For every $R$-module $N$ such that $l(M\otimes_RN)<\infty$, finite-projective-dimension decency…

Let $R$ be a local, Noetherian, commutative ring and let $M$ be a finitely generated $R$-module. Say that $M$ is decent if, for every $R$-module $N$ with $l(M\otimes_RN)<\infty$, o…

Let $R$ be a non-Gorenstein local ring. A finitely generated $R$-module is totally reflexive if it has G-dimension zero. Infinitely many indecomposable totally reflexive modules co…

Let $A$ be an almost commutative separated local ring. Denote by $\bar{A}$ the intersection of all local Henselian rings $H$ in the completion $\hat{A}$ such that…

Berger's conjecture. One always has

Let $A$ be a local one-dimensional domain with field of fractions $F$. Geller's conjecture. The ring $A$ is regular if and only if the map … is injective. The conjecture gives a cr…