The equivalence conjecture for CM finiteness, stable tilting, and derived dimension

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 object of $\underline{\operatorname{CM}}^{\mathbb{L}}R$ whose endomorphism algebra has global dimension at most $d$. The CM–tilting equivalence conjecture. The following conditions are equivalent:

• $(R,\mathbb{L})$ is Fano;
• $(R,\mathbb{L})$ is $d$-CM finite;
• $\underline{\operatorname{CM}}^{\mathbb{L}}R$ has a $d$-tilting object;
• $A^{\rm CM}$ is derived equivalent to an algebra of global dimension at most $d$.

The diagram preceding the conjecture records several implications among these conditions, including implications proved in the paper. The conjecture asks whether all four conditions coincide; the general equivalence remains open.