37 problems

King's conjecture. Every smooth toric variety has a tilting bundle consisting of line bundles.

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…

Happel–Unger conjecture. The graph $\mathcal{G}(\mathcal{T}_{\mathbb{X}})$ is connected.

Let $P$ be a reflexive polytope. Let $\Sigma$ be a simplicial fan whose ray primitive generators are the vertices of $P$, and let $\mathcal{V}$ be the fan of the canonical bundle o…

Self-orthogonal $\tau$-tilting conjecture. If $T$ is a self-orthogonal $\tau$-tilting $A$-module, then $T$ is $1$-tilting.

Let $Q$ be a quiver and let $\theta$ be a stability condition such that the corresponding quiver moduli space carries a universal bundle $\mathcal{U}$. An object is partial tilting…

Let $B=\operatorname{End}_\Lambda(T_M)$, let ${\mathcal T}_B^{\rm cl}$ denote the relevant class of classical tilting $B$-modules, and let ${\mathcal F}(\Delta)$ be the category de…

Let $R$ and $S$ be Artin algebras, let $_RU$ be a generalized tilting module, and set $S={\rm End}(_RU)$. A module is (quasi) Gorenstein when it satisfies the corresponding Gorenst…

Let $\Lambda$ be the selfinjective preprojective algebra under consideration, let $T$ be a basic maximal rigid $\Lambda$-module, and let $\operatorname{End}_\Lambda(T)$ be its endo…

Derived exchange-pair conjecture. The quotient module categories are equivalent:

Exchange-relation conjecture. The exchange relation in $\operatorname{\mathcal A}(H)$ is

Classification conjecture. There are no other such hereditary categories ${\cal C}$.

Let $A$ be a finite-dimensional symmetric algebra. The algebra $A$ is $\tau$-tilting finite if it has only finitely many isomorphism classes of support $\tau$-tilting modules, and…

Ballard–Huang conjecture. Every smooth toric (weak) Fano Deligne–Mumford stack has a tilting bundle consisting of line bundles.

Let $X$, $G$, and $\mathcal{E}$ be as in the surrounding construction, with $G$ finite, and let $\pi:[\mathbb{A}(\mathcal{E})/G]\rightarrow[X/G]$ be the induced projection. Suppose…

Let $\Sigma$ be a complete, simplicial fan such that the toric Deligne–Mumford stack $\mathcal{X}_\Sigma$ has a partial tilting complex $\mathcal{T}$ with finite global dimension e…

Self-orthogonal Wakamatsu-tilting conjecture. If $|T|=|A|$, then $T$ is Wakamatsu-tilting.

Let $A$ be a finite-dimensional algebra, and let $M$ be an almost-tilting module, meaning a partial-tilting module with one fewer indecomposable direct summand than a basic tilting…

Let $A$ be a finite-dimensional algebra, let $M$ be a maximal partial-tilting module, and let $n$ denote the number of isomorphism classes of indecomposable direct summands of a ba…

Maximal-subcategory and mutation conjecture.

Converse Wakamatsu tilting conjecture. If $|T|=|\Lambda|$, then $T$ is Wakamatsu tilting.

Weak maximal self-orthogonal conjecture. Every projectively Wakamatsu tilting module is maximal self-orthogonal.

Maximal self-orthogonal conjecture. Every Wakamatsu tilting module is maximal self-orthogonal.

Modified upward-closedness conjecture. If $T\in\mathcal{X}_U$ and $U\in\mathcal{Y}_T$, and $U$ is tilting, then $T$ is tilting.

Iwanaga–Gorenstein Wakamatsu tilting conjecture. Any Wakamatsu tilting $\Lambda$-module is tilting.