66 problems

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…

Lech's conjecture. Then

Let $(R,\mathfrak{m})$ be a complete local ring. A small Cohen–Macaulay module over $R$ is a nonzero finitely generated $R$-module $S$ such that $\operatorname{depth}(S)=\dim(R)$.…

Polini–Ulrich conjecture. One has

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…

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…

Polynomial dimension-and-frequency conjecture. There exist $k\geq 1$, polynomials

Let $(R,\mathfrak{m})$ be a Noetherian local ring such that its completion $\widehat{R}$ is reduced and satisfies Serre's $S_2$ condition. Let $B$ be a balanced big Cohen–Macaulay…

The naive generator bound. If

Ma's multiplicity conjecture.

Let $(R,\mathfrak m)$ be a local ring and let $I$ be an ideal of $R$. For an ideal $I$, write $\operatorname{cd}(I,R)$ for its cohomological dimension, and let $H^c_I(R)$ denote th…

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 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…

Let $A$ be a one-dimensional local domain essentially of finite type over a field $K$ of characteristic zero, and let $\Omega_{A/K}$ denote its module of Kähler differentials. Berg…

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…

Lusztig's cohomological realization conjecture. Every irreducible representation of $G_r^F$ appears in the virtual representation

Use the notation above: let $d\geq1$, let $p>2$ be prime, define … and let $(A,\mathfrak m,k)$ be a $d$-dimensional unmixed local ring with $k=\overline{\mathbb F_p}$. Higher-dimen…

Let $d\geq1$ be an integer and let $p>2$ be prime. Define … Let $(A,\mathfrak m,k)$ be a $d$-dimensional unmixed local ring with $k=\overline{\mathbb F_p}$. Han–Monsky lower-bound…

Linear Artin Approximation Theorem. Then the system has a true solution $\underline{U}$ such that

Foxby's conjecture. If $M$ is faithful and

Let $d\geq 2$. Set … where … Let $(R,\mathfrak{m},k)$ be an unmixed non-regular local ring of characteristic $p$ and dimension $d$. Here, unmixed means that…

Let $R$ be a regular local ring of dimension $2$, and let $K$ be its field of fractions. The Pythagoras number conjecture for regular local rings asserts that … Here $p(R)$ and…

Onn's weak conjecture. There is an isomorphism of group algebras