On derived equivalences and homological dimensions
In representation theory, the global dimension of an algebra is considered to be a fundamental invariant, used to define and to characterise classes of algebras. Dominant dimension, is used to describe, and to measure the quality of, double centraliser properties as in the Morita-Tachikawa correspondence and in Schur-Weyl duality, and to formulate Nakayama's conjecture. The interplay of global and dominant dimensions is crucial in Auslander's correspondence, Auslander's definition of representation dimension and Iyama's higher Auslander correspondence. Derived equivalences are used to connect different areas of mathematics and to compare different situations. In representation theory, derived equivalences from one algebra to another induced by tilting complexes share some homological or \(K\)-theoretic invariance. As is known, Hochschild (co)homology and algebraic \(K\)-theory are derived invariant. But homological dimensions appear to be rarely invariant under derived equivalences. For instance, the difference of global dimensions and finitistic dimensions is bounded above by the length of the tilting complex inducing the derived equivalence. For dominant dimensions, derived equivalent algebras may be have different dominant dimensions, and it seems to be unknown whether finiteness of dominant dimension is preserved. In the paper under review, the authors discuss global dimensions and dominant dimensions are preserved by derived equivalences for a class of algebras with anti-automorphisms preserving simples. Cellular algebras and many finite dimensional algebras in algebraic Lie theory have anti-automorphisms. They also consider derived equivalences between algebras with positive \(\nu\)-dominant dimension always restrict to derived equivalences between their associated self-injective algebras, consequently, two self-injective algebras of finite representation type are seen to be derived equivalent if and only if their Auslander algebras are derived equivalent.
scientific article; zbMATH DE number 7291138
scientific article