Bijection between strong torsion classes and maximal support pre-tilting modules

Let (A,C)(A,\mathcal{C}) be an almost-directed dd-homological pair, meaning that AA is an algebra equipped with the subcategory C\mathcal{C}, and suppose that

gl.dim(A)d.\operatorname{gl.dim}(A)\leq d.

A dd-strong torsion class in C\mathcal{C} is a torsion class satisfying the stated dd-strongness condition, and a maximal support pre-dd-tilting AA-module is a support pre-dd-tilting module maximal with respect to the relevant inclusion. The bijection conjecture. There is a bijection between dd-strong torsion classes in C\mathcal{C} and maximal support pre-dd-tilting AA-modules. This conjecture would classify dd-strong torsion classes in C\mathcal{C}, extending the preceding theorem's construction from proper support-dd-tilting modules to all maximal support pre-dd-tilting modules. The supplied text gives no resolution, so the conjecture remains open.

Sources & referencesView supporting material

Primary source

Jordan McMahon, “Higher support tilting I: higher Auslander algebras of linearly oriented type A”, arXiv:1808.05184 (2018).

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