24 problems

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 $\Lambda$ 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

Cluster-category correspondence conjecture. There is a one-to-one correspondence between the cluster variables of $\operatorname{\mathcal A}(H)$ and…

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…

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…

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…

Let $A$ be a finite-dimensional algebra over a field $k$. For $n\in\mathbb{N}\cup\{0\}$, an $n$-quasi-generator is a module admitting the quasi-generator property of level $n$, and…

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.

Let $\mathcal{E}$ be an exact category, and suppose that there exists an integer $n\geq 0$ for which $\mathcal{E}$ has at least one $n$-tilting subcategory. Let…

Let $\mathcal{X}$ and $\mathcal{Y}$ be essentially small idempotent complete additive categories. Write $\operatorname{Mod}-\mathcal{X}$ for the category of modules over…

Happel–Unger conjecture. The graph $\mathcal{G}_t(\mathcal{H})$ is connected if and only if $\mathcal{H}$ contains no nonzero projective objects.

Let $f\in S^n$ be an invertible polynomial, and let $L_f$ be its maximal grading. Let $\operatorname{HMF}^{L_f}_{S^n}(f)$ be the homotopy category of maximally graded matrix factor…

Let $(A,\mathcal{C})$ be an almost-directed $d$-homological pair, meaning that $A$ is an algebra equipped with the subcategory $\mathcal{C}$, and suppose that … A $d$-strong torsio…

Let ${\bf w}\in {\mathbb C}[x_1,\ldots,x_n]$ be an invertible polynomial, let $\Gamma_{\bf w}$ be its group of diagonal symmetries, and let…

Let $\mathbb{X}$ be a Geigle–Lenzing projective space and let $d$ be its dimension parameter. A $d$-tilting bundle on $\mathbb{X}$ is a tilting bundle whose endomorphism algebra ha…

Let $(R,\mathbb{L})$ be a Geigle–Lenzing complete intersection, let $d$ be the dimension parameter, and let $A^{\rm CM}$ be its CM endomorphism algebra. A $d$-tilting object is an…

Let $A$ be a basic finite dimensional $k$-algebra and let $T$ be a finitely generated $A$-module. Call $T$ a Wakamatsu tilting module if $\operatorname{Ext}_A^i(T,T)=0$ for all…