Maximal m-almost rigid modules capturing higher Ext dimensions

Determine whether, for a gentle algebra A with quiver Q and S = ⊕_{i∈Q0} S(i) the direct sum of all simple A-modules, there exist classes of A-modules—potentially termed maximal m-almost rigid modules—whose number of indecomposable direct summands captures the dimension of ⋃_{i=0}^m Ext^i_A(S,S), thereby extending the cases m=0 (tilting A-modules with |Q0| summands) and m=1 (maximal almost rigid A-modules with |Q0| + |Q1| summands).

Background

For a gentle algebra A with quiver Q, the number of indecomposable direct summands in a basic tilting A-module equals |Q0|. The paper proves that every maximal almost rigid A-module has exactly |Q0| + |Q1| indecomposable summands, matching dim Hom_A(S,S) + dim Ext1_A(S,S) for S the direct sum of simple modules.

Motivated by these identities for m=0 and m=1, the authors pose whether analogous module classes exist that would align the number of summands with higher extension data up to degree m, thereby generalizing the summand-count interpretations to higher Ext-degrees.

References

Question 5.6. Recall that the cardinality of Q0 is equal to the number of summands of a (basic) tilting A-module. By Corollary 5.5, the cardinality of Q0 ∪ Q1 is the number of summands of a maximal almost rigid module. If we let S = ⊕ i∈Q0S(i) be the sum of all simple modules, we have |Q0| = dimHom(S,S), and |Q1| = dimExt1(S,S). So it is natural to ask if there are other classes of modules, which one might call maximal m-almost rigid modules, that capture the dimension of ∪ i=0m Exti(S,S).

— Maximal almost rigid modules over gentle algebras  (2408.16904 - Barnard et al., 2024) in Question 5.6, Section 5