Monomial Gorenstein algebras and the stably Calabi–Yau property

Abstract

A celebrated result by Keller–Reiten says that 2-Calabi–Yau tilted algebras are Gorenstein and stably 3-Calabi–Yau. This note shows that the converse holds in the monomial case: a 1-Gorenstein monomial algebra and stably 3-Calabi–Yau is 2-Calabi–Yau tilted, moreover is Jacobian. We study the case of other stably Calabi–Yau Gorenstein monomial algebras.

Appendix A: Auslander–Reiten translations for Gorenstein algebras by Sondre Kvamme and Matthew Pressland

Let Λ be an Iwanaga–Gorenstein algebra. Since GP(Λ) is a functorially finite subcategory of modΛ, it has Auslander–Reiten sequences [30], inducing an Auslander–Reiten translation τ_GP on $\underline {GP}({\Lambda })$, typically different from the Auslander–Reiten translation τ_Λ on $\underline {\textup {mod}}{\Lambda }$. The goal of this appendix is to relate the objects τ_ΛM and τ_GPM of $\underline {\textup {mod}}{\Lambda }$ when M is Gorenstein projective. Indeed, we will show that these objects eventually coincide after repeated application of the syzygy functor. While this fact may not be surprising to experts, and can be deduced quickly from results already in the literature (most easily from [31, Thm. 3.7]), we give here a very direct proof, which even exhibits a natural isomorphism of functors. Moreover, we explain how this result provides a new perspective on results of Garcia Elsener (Corollary 4.6 of the present paper) and Garcia Elsener–Schiffler [32, Thm. 1] characterising Gorenstein projective modules over certain Calabi–Yau tilted algebras Λ, by relating these characterisations directly to a Calabi–Yau property of $\underline {GP}({\Lambda })$.

We denote by ${\Omega }\colon \underline {\textup {mod}}{\Lambda }\to \underline {\textup {mod}}{\Lambda }$ the syzygy functor, taking a module to the kernel of a projective cover, and by ${\Sigma }\colon \underline {\textup {mod}}{\Lambda }\to \underline {\textup {mod}}{\Lambda }$ its left adjoint, taking a module to the cokernel of a right projΛ-approximation. The restrictions of these functors to $\underline {GP}({\Lambda })$ are mutually inverse, the restriction of Σ being the suspension functor on this triangulated category.

Lemma A.1

The functor ${\Sigma }^{d}\colon \underline {\textup {mod}}{\Lambda }\to \underline {\textup {mod}}{\Lambda }$ has essential image in $\underline {GP}({\Lambda })$.

Proof

Since Λ is d-Iwanaga–Gorenstein, we have ${\Omega }^{d}{\Sigma }^{d}M\in \underline {GP}({\Lambda })$ for any Λ-module M. Then ${\Sigma }^{d}{\Omega }^{d}{\Sigma }^{d}M\in \underline {GP}({\Lambda })$ since Σ preserves Gorenstein projectivity. It follows from the triangular identities for the unit and counit of the adjoint pair (Σ^d,Ω^d) that Σ^dM is a summand of Σ^dΩ^dΣ^dM, and hence is itself Gorenstein projective. □

Theorem A.2

Let Λ be a finite-dimensional Iwanaga–Gorenstein algebra of Gorenstein dimension at most d. Then there is a natural isomorphism

$${\Omega}^{d}\tau_{\Lambda}\simeq{\Omega}^{d}\tau_{\textup{GP}}$$

of endofunctors of $\underline {GP}({\Lambda })$.

Proof

Using the Yoneda embedding, it suffices to show that there is a natural isomorphism

$$\underline{\textup{Hom}}_{\Lambda}(-,{\Omega}^{d}\tau_{\textup{GP}}X)\cong\underline{\textup{Hom}}_{\Lambda}(-,{\Omega}^{d}\tau_{\Lambda} X)$$

of functors on $\underline {\textup {mod}}{\Lambda }$, for any X ∈GP(Λ). By adjunction and the Auslander–Reiten formula for GP(Λ), we obtain natural isomorphisms

$$\underline{\textup{Hom}}_{\Lambda}(-,{\Omega}^{d}\tau_{\textup{GP}}X)\cong\underline{\textup{Hom}}_{\Lambda}({\Sigma}^{d}-,\tau_{\textup{GP}}X)\cong\mathrm{D}{\textup{Ext}}^{1}_{\Lambda}(X,{\Sigma}^{d}-),$$

The validity of the second isomorphism depends on Lemma A.1, showing that Σ^d takes values in $\underline {GP}({\Lambda })$. Alternatively, using the Auslander–Reiten formula in modΛ, we get

$$\underline{\textup{Hom}}_{\Lambda}(-,{\Omega}^{d}\tau_{\Lambda} X)\cong\underline{\textup{Hom}}_{\Lambda}({\Sigma}^{d}-,\tau_{\Lambda} X)\cong\mathrm{D}{\textup{Ext}}^{1}_{\Lambda}(X,{\Sigma}^{d}-).$$

The two right-hand sides coincide, completing the proof. □

Theorem A.3 ([32, Thm. 1])

Let Λ be a 2-Calabi–Yau tilted algebra. Then a Λ-module M is Gorenstein projective if and only if M≅Ω²τ_ΛM in $\underline {\textup {mod}}{\Lambda }$.

Proof

By a result of Keller and Reiten [21, Prop. 2.1], a 2-Calabi–Yau tilted algebra is Iwanaga–Gorenstein of Gorenstein dimension at most 1. Thus ${\Omega }^{2} N\in \underline {GP}({\Lambda })$ for any N ∈modΛ, and the ‘if’ part of the theorem follows immediately.

For the ‘only if’ direction, Keller and Reiten also prove [21, Thm. 3.3] that $\underline {GP}({\Lambda })$ is a 3-Calabi–Yau category, meaning that Σ³ is a Serre functor. On the other hand, the Auslander–Reiten formula states that Στ_GP is always a Serre functor on $\underline {GP}({\Lambda })$. Since Serre functors are unique up to natural isomorphism, the 3-Calabi–Yau property is equivalent to there being a natural isomorphism $\tau _{\textup {GP}}\simeq {\Sigma }^{2}$. Now by Theorem A.2, using again the Iwanaga–Gorenstein property of Λ, we have

$${\Omega}^{2}\tau_{\Lambda} M\cong{\Omega}^{2}\tau_{\textup{GP}}M\cong{\Omega}^{2}{\Sigma}^{2} M\cong M.$$

□

We close by observing that characterisations of Gorenstein projective Λ-modules as in Theorem A.3 are closely related to Calabi–Yau properties of $\underline {GP}({\Lambda })$ in wider generality.

Proposition A.4

Let Λ be Iwanaga–Gorenstein of Gorenstein dimension at most m. Then $\underline {GP}({\Lambda })$ is (m + 1)-Calabi–Yau if and only if there is a natural isomorphism ${\Omega }^{m}\tau _{\Lambda }\simeq \text {Id}_{\underline {GP}({\Lambda })}$. In this case, a Λ-module M is Gorenstein projective if and only if M≅Ω^mτ_ΛM in $\underline {\textup {mod}}{\Lambda }$.

Proof

As in the proof of Theorem A.2, $\underline {GP}({\Lambda })$ is (m + 1)-Calabi–Yau if and only if there is a natural isomorphism $\tau _{\textup {GP}}\simeq {\Sigma }^{m}$ on $\underline {GP}({\Lambda })$. Since Σ^m has quasi-inverse Ω^m on this category, this is equivalent to asking that ${\Omega }^{m}\tau _{\textup {GP}}\simeq \text {Id}_{\underline {GP}({\Lambda })}$. But Theorem A.2 tells us that ${\Omega }^{m}\tau _{\textup {GP}}\simeq {\Omega }^{m}\tau _{\Lambda }$ in this setting, and the first part of the statement follows. The second part is then proved as in Theorem A.3, replacing 2 by m. □

Note that the isomorphisms constructed in the proof of Theorem A.3 arise from a natural isomorphism in this way, using the 3-Calabi–Yau property of $\underline {GP}({\Lambda })$ [21, Thm. 3.3]. We also deduce that a similar Calabi–Yau property holds for certain gentle m-cluster-tilted algebras.

Corollary A.5

Let Λ be an m-cluster-tilted algebra of type $\mathbb {A}$ or $\tilde {\mathbb {A}}$. Then $\underline {GP}({\Lambda })$ is (m + 1)-Calabi–Yau.

Proof

By Corollary 4.6, Λ is Iwanaga–Gorenstein of Gorenstein dimension at most m, and M≅Ω^mτ_ΛM for each M ∈GP(Λ). Since Λ is a gentle algebra, meaning that $\underline {GP}({\Lambda })$ is semi-simple [33], there must be a collection of such isomorphisms that is natural in M—for example, by first choosing such an isomorphism for each M in a set of representatives for the isoclasses of indecomposable objects, and then extending to arbitrary objects by choosing direct sum decompositions. (We use here that the ground field k is algebraically closed, so that each indecomposable has endomorphism ring k.) The result then follows by Proposition A.4. □