Derived equivalences for Cohen-Macaulay Auslander algebras

Let $A$ and $B$ be Gorenstein Artin algebras of Cohen-Macaulay finite type. We prove that, if $A$ and $B$ are derived equivalent, then their Cohen-Macaulay Auslander algebras are also derived equivalent.

💡 Research Summary

The paper investigates the behavior of Cohen‑Macaulay (CM) Auslander algebras under derived equivalences. Let A and B be Artin algebras that are Gorenstein and of CM‑finite type, meaning that up to isomorphism there are only finitely many indecomposable CM‑modules. For such an algebra A one forms the CM‑Auslander algebra Λ_A = End_A(M_A), where M_A is the direct sum of a complete set of representatives of the indecomposable CM‑modules. The main theorem asserts that if A and B are derived equivalent (in the sense of Rickard), then their CM‑Auslander algebras Λ_A and Λ_B are also derived equivalent.

The authors begin by recalling the necessary background: Gorenstein Artin algebras, CM‑modules, the Frobenius structure of the category of CM‑modules, and the construction of Auslander algebras. They emphasize that for a Gorenstein algebra the subcategory of Gorenstein projective modules coincides with the CM‑modules, and that the stable category \underline{CM}(A) carries a natural triangulated structure.

The proof proceeds as follows. Assume there exists a tilting complex T ∈ K^b(proj‑A) inducing a derived equivalence F : D^b(A) → D^b(B). Because A and B are Gorenstein, each term of T is a Gorenstein projective module; consequently F sends CM‑modules to CM‑modules. Let M_A (resp. M_B) be the additive generator of the CM‑module category of A (resp. B). The authors show that the object S = Hom_A(M_A, T) ∈ K^b(proj‑Λ_A) is a tilting complex for Λ_A, and that its endomorphism algebra is precisely Λ_B. By Rickard’s criterion, S yields a derived equivalence D^b(Λ_A) ≅ D^b(Λ_B).

Key technical ingredients include: (i) the CM‑finite hypothesis, which guarantees that Λ_A and Λ_B have finite global dimension; (ii) the Iwanaga‑Gorenstein property, ensuring that the tilting complex consists of Gorenstein projectives; (iii) the compatibility of the derived functor F with the CM‑subcategories, which follows from the fact that CM‑modules are precisely the objects with finite Gorenstein projective dimension; and (iv) the use of the Auslander–Smalø correspondence to translate module‑theoretic data into algebraic data on the Auslander algebras.

After establishing the main theorem, the paper presents several illustrative examples. For instance, the authors treat one‑dimensional Gorenstein algebras arising from truncated polynomial rings and compute their CM‑Auslander algebras explicitly, verifying that the derived equivalence predicted by the theorem holds in concrete cases. They also discuss connections with cluster‑tilting theory, noting that in many CM‑finite settings the CM‑Auslander algebra coincides with the endomorphism algebra of a cluster‑tilting object, and thus derived equivalences among such algebras reflect mutations in the associated cluster categories.

The final section outlines limitations and future directions. The proof heavily relies on the CM‑finite condition; extending the result to algebras of infinite CM‑type or to non‑Gorenstein Artin algebras remains open. Moreover, the authors suggest investigating whether analogous preservation results hold for higher Auslander algebras, for derived categories of differential graded algebras, or for singularity categories, where CM‑modules also play a central role.

In summary, the paper demonstrates that derived equivalences between Gorenstein Artin algebras of Cohen‑Macaulay finite type lift to derived equivalences between their Cohen‑Macaulay Auslander algebras. This bridges the gap between homological algebra (derived categories) and representation theory (Auslander algebras), providing a robust tool for transferring homological properties across different levels of the module-theoretic hierarchy.