Cellular Noetherian algebras with finite global dimension are split quasi-hereditary (Q6607137)

Two main problems in representation theory of algebras are to classify the simple modules over an algebra and to understand the homological properties of an algebra. Cellular algebras \(B\) are certain algebras characterized by the existence of an involution \(i\) with \(i^2 = \text{id}_B\) and a certain chain of ideals that provide a filtration of the regular module \(B\). T. Cruz proves that cellular Noetherian algebras with finite global dimension are split quasihereditary over a regular commutative Noetherian ring with finite Krull dimension and their quasi-hereditary structure is unique, up to equivalence. The author establishes that a split quasi-hereditary algebra is semi-perfect if and only if the ground ring is a local commutative Noetherian ring. He gives a formula to determine the global dimension of a split quasi-hereditary algebra over a commutative regular Noetherian ring (with finite Krull dimension) in terms of the ground ring and finite-dimensional split quasi-hereditary algebras. For the general case, Cruz gives upper bounds for the finitistic dimension of split quasi-hereditary algebras over arbitrary commutative Noetherian rings.