Derived equivalences, restriction to self-injective subalgebras and invariance of homological dimensions

Abstract

Derived equivalences between finite dimensional algebras do, in general, not pass to centraliser (or other) subalgebras, nor do they preserve homological invariants of the algebras, such as global or dominant dimension. We show that, however, they do so for large classes of algebras described in this article. Algebras $A$ of $\nu$-dominant dimension at least one have unique largest non-trivial self-injective centraliser subalgebras $H_A$. A derived restriction theorem is proved: A derived equivalence between $A$ and $B$ implies a derived equivalence between $H_A$ and $H_B$. Two methods are developed to show that global and dominant dimension are preserved by derived equivalences between algebras of $\nu$-dominant dimension at least one with anti-automorphisms preserving simples, and also between almost self-injective algebras. One method is based on identifying particular derived equivalences preserving homological dimensions, while the other method identifies homological dimensions inside certain derived categories. In particular, derived equivalent cellular algebras have the same global dimension. As an application, the global and dominant dimensions of blocks of quantised Schur algebras with $n \geq r$ are completely determined.