543 problems

Auslander–Reiten conjecture. If

Let $\Gamma$ be a symmetric numerical semigroup, let $\Bbbk$ be a field, and let $R=\Bbbk[\Gamma]$ be its semigroup algebra. Let $M$ be a 2-generated ideal of $R$, viewed as an $R$…

Nakayama's conjecture. If an Artin algebra has infinite dominant dimension, then it is self-injective.

Let $R$ be a commutative Noetherian ring such that $\operatorname{Spec}(R)$ is connected, and let $M$ be a non-zero, finitely generated $R$-module of finite projective dimension. F…

Let $k$ be a field and let $\Lambda$ be a finite-dimensional $k$-algebra with Jacobson radical $\mathfrak{r}$ such that $E=\Lambda/\mathfrak{r}$ is separable. Write…

Let $d$ be a nonnegative integer, and let $X$ be a nonempty finite free $(\mathbb{Z}/2)^d$-CW complex. Its homology is taken with coefficients in $\mathbb{F}_2$. Carlsson's conject…

Let $R$ and $S$ be artin algebras, and let ${}_RC_S$ be the relevant Wakamatsu tilting bimodule. Write $\mathop{\rm pd}\nolimits_RC$ and $\mathop{\rm pd}\nolimits_{S^{op}}C$ for th…

Odd-primary transfer conjecture. The transfer homomorphism $Tr_h^{\mathscr A_p}(\mathbb F_p)$ is one-to-one for any odd prime $p$ and $1\leq h\leq 4$. This extends the known low-ra…

Atiyah's conjecture. The rank gradient $\beta_0^G(M)$ is an integer. This is a pro-$p$ analogue of the Atiyah conjecture concerning integrality of $L^2$-Betti numbers. The supplied…

Let $A$ be an Artin algebra. Its finitistic dimension is the supremum of the projective dimensions of finitely generated $A$-modules having finite projective dimension. Finitistic…

Auslander's conjecture. Finite-dimensional modules over a finite-dimensional $k$-algebra satisfy (AC).

Let $A$ be an algebra of finite global dimension, and let $C_A$ denote its Cartan matrix. Cartan determinant conjecture. The determinant of the Cartan matrix is equal to one: … Thi…

Bass's finitistic dimension conjecture. For every artin algebra $R$,

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…

Strong Nakayama conjecture. If

Let $\Lambda$ be an Artin algebra, and write $\operatorname{id}_\Lambda\Lambda$ for its left self-injective dimension and $\operatorname{id}\Lambda_\Lambda$ for its right self-inje…

Let $R$ be an Artin algebra. The Finitistic Dimension Conjecture. The supremum of the projective dimension of all finitely generated left $R$-modules with finite projective dimensi…

Let $A$ be the algebra of functions on an affine space, let $V_A$ be the graded Lie algebra of polyvector fields on that affine space, and let $C^{\bullet}(A)$ be the differential…

Let $\mathcal{A}$ be an abelian category with enough injectives, and let $A\in\mathcal{A}$. Consider the contravariant functor $\operatorname{Ext}_{\mathcal{A}}^{1}(-,A)$. Auslande…

Let $S$ be a polynomial ring and let $I\subseteq S$ be a monomial ideal with linear powers, meaning that $I^k$ has a linear resolution for every $k\geq 1$. Let $\alpha(I)$ denote t…

Let $H$ be a noetherian Hopf algebra over a field. Brown–Goodearl conjecture. Every noetherian Hopf algebra is Artin–Schelter Gorenstein. This conjecture concerns the homological r…

Avramov–Buchweitz–Iyengar's total rank conjecture. For every projective flag $D$, the inequality above should hold. The conjecture is false: Iyengar–Walker provided counterexamples…

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…

Let $A$ be a finite-dimensional algebra over a field $K$, and let $\mathcal{N}$ be the ideal of $\operatorname{HH}^*(A)$ generated by all homogeneous nilpotent elements. Hochschild…

Let $R$ be a commutative local Noetherian ring, and let $M$ be a finitely generated $R$-module whose Betti numbers are bounded. Eisenbud's periodicity conjecture. The minimal proje…