Derived invariance by syzygy complexes

Abstract

We study derived invariance through syzygy complexes. In particular, we prove that syzygy-finite algebras and Igusa--Todorov algebras are invariant under derived equivalences. Consequently, we obtain that both classes of algebras are invariant under tilting equivalences. We also prove that derived equivalences preserve AC-algebras and the validity of the finitistic Auslander conjecture.

Igusa-Todorov dimensions and derived dimensions of artin algebras

“…Given two subclasses I1, I2 ⊆ Ob T , we denote I1 * I2 by the full subcategory of all extensions between them, that is, The derived dimension can solve some problems. In [39], Wei give the following Problem Is every syzygy-finite algebra derived to an algebra of finite representation type? The answer is negative.…”

“…Theorem 1.2. (see [39,Theorem 4.5 and Theorem 5.4]) Let Λ and Γ be two derived equivalent algebras. Then (1) Λ is an Igusa-Todorov algebra if and only if Γ is an Igusa-Todorov algebra.…”

Reduction techniques of singular equivalences

“…Examples of such algebras include syzygy-finite algebras, algebras with representation dimension not more than three and algebras with infinite-layer length not more than three [22]. It is known that Igusa-Todorov algebras satisfy the finitistic dimension conjecture, and the invariance of syzygy-finite and Igusa-Todorov under recollements and derived equivalences is discussed in [38,39].…”

“…A-B-bimodules), and Ω D b (A) (−) the syzygy functor on derived category, up to some direct summands of projective modules. We point that Ω A (M) = Ω D b (A) (M) for any M ∈ modA, and we refer to [2,38] for more details on syzygies of complexes.…”

The derived dimensions and syzygy finite type

“…In [13], Wei give the following Problem Is every syzygy-finite algebra derived to an algebra of finite representation type?…”