On universal deformation rings and stable equivalences of Gorenstein-projective modules

Abstract: Let $\mathbf{k}$ be a field and let $Λ$ and $Γ$ finite dimensional $\mathbf{k}$-algebras. Assume that ${<em>Γ}X</em>Λ$ and ${<em>Λ}Y</em>Γ$ are bimodules that define a singular equivalence of Morita type with level (in the sense of Z. Wang) between $Λ$ and $Γ$ and which also induce an equivalence between the stable categories of finitely generated Gorenstein-projective modules $Λ$-$\underline{\text{Gproj}}$ and $Γ$-$\underline{\text{Gproj}}$. We prove that if $V$ is an indecomposable object in $Λ$-$\underline{\text{Gproj}}$ with $\underline{\mathrm{End}}<em>Λ(V)\cong \mathbf{k}$, then $X\otimes</em>ΛV$ is an object in $Γ$-$\underline{\text{Gproj}}$ such that $$\underline{\mathrm{End}}<em>Γ(X\otimes</em>ΛV)\cong \mathbf{k}$$ and the universal deformation rings (in the sense of F.M. Bleher and the second author) $R(Λ,V)$ and $R(Γ, X\otimes_ΛV)$ are isomorphic. This result generalizes the one obtained by the second author assuming that $Λ$ and $Γ$ are Gorenstein $\mathbf{k}$-algebras.

• The paper proves that universal deformation rings of indecomposable Gorenstein-projective modules remain invariant under singular equivalence of Morita type with level.
• It leverages deformation functor techniques and syzygy constructions to establish natural isomorphisms between universal deformation rings.
• The results generalize deformation theory invariance to a broader class of algebras, including non-Gorenstein and Morita context algebras.

Universal Deformation Rings and Stable Equivalences of Gorenstein-Projective Modules

Introduction

This paper addresses the interplay between universal deformation rings of Gorenstein-projective modules and stable equivalences (particularly of Morita type with level) between finite-dimensional algebras. The focus is on extending results concerning the invariance of universal deformation rings—typically established under the assumption of Gorenstein properties for the underlying algebras—to a broader class of singular equivalences that also preserve the triangulated structures of stable categories of Gorenstein-projective modules. The work both generalizes prior results and situates them in the context of modern representation theory, where singular equivalences of Morita type with level play a key role.

Background and Definitions

Let $\Lambda$ and $\Gamma$ be finite-dimensional algebras over a field $\kappa$. The study centers on their categories of finitely generated Gorenstein-projective modules (denoted $\Lambda\textup{-Gproj}$ and $\Gamma\textup{-Gproj}$), which are Frobenius and thus possess triangulated stable categories $\underline{\Lambda\textup{-Gproj}}$ and $\underline{\Gamma\textup{-Gproj}}$.

Singular equivalence of Morita type with level is defined as follows: a pair of bimodules $_\Gamma X_\Lambda$ and $_\Lambda Y_\Gamma$ induces a singular equivalence (of level $\ell \ge 0$) between $\Gamma$0 and $\Gamma$1 if:

• $\Gamma$2 is projective as a left $\Gamma$3-module and right $\Gamma$4-module
• $\Gamma$5 is projective as a left $\Gamma$6-module and right $\Gamma$7-module
• $\Gamma$8 in the stable category
• $\Gamma$9 in the stable category

Such equivalences generalize both stable and singular equivalences of Morita type; bimodules $\kappa$0 and $\kappa$1 induce equivalences of singularity categories and, under additional assumptions, equivalences of stable categories of Gorenstein-projective modules.

A module $\kappa$2 is Gorenstein-projective if it is reflexive, with vanishing Ext groups from $\kappa$3 and its dual to the base algebra in all degrees. Over a Gorenstein algebra, the stable category of Gorenstein-projective modules is equivalent to the singularity category.

Universal deformation rings play a central role: for $\kappa$4 indecomposable Gorenstein-projective with stable endomorphism ring isomorphic to $\kappa$5, following Bleher and Vélez-Marulanda, the deformation functor of $\kappa$6 is pro-representable by a complete local commutative Noetherian $\kappa$7-algebra $\kappa$8. This ring encodes the deformation theory of $\kappa$9 over Artinian $\Lambda\textup{-Gproj}$0-algebras with residue field $\Lambda\textup{-Gproj}$1.

Main Results

Invariance of Universal Deformation Rings

The paper's principal achievement is to demonstrate that, under an equivalence of triangulated stable categories of Gorenstein-projective modules induced by bimodules implementing a singular equivalence of Morita type with level, the universal deformation ring of an indecomposable Gorenstein-projective module with simple stable endomorphism ring is preserved. Precisely, if $\Lambda\textup{-Gproj}$2 is such a module over $\Lambda\textup{-Gproj}$3, then $\Lambda\textup{-Gproj}$4 is indecomposable Gorenstein-projective over $\Lambda\textup{-Gproj}$5 with the same property for the endomorphism ring, and one has an isomorphism of universal deformation rings:

$\Lambda\textup{-Gproj}$6

This generalizes results that required both $\Lambda\textup{-Gproj}$7 and $\Lambda\textup{-Gproj}$8 to be Gorenstein. By instead relying only on the existence of singular equivalences of Morita type with level and the preservation of the stable Gorenstein categories, the result applies in greater generality to non-Gorenstein settings, including Gorenstein-finite and Morita context algebras.

Consequences and Corollaries

Several corollaries expand upon the main theorem:

• The preservation of universal deformation rings under stable equivalence of Morita type, provided neither bimodule has a projective direct summand.
• The analogous invariance for modules over Morita context algebras, connecting deformation-theoretic properties across larger algebraic constructions built from the original algebras and bimodules.

The results further generalize established facts for self-injective and triangular matrix algebras to broader classes of non-self-injective and infinite global dimension finite-dimensional algebras.

Technical Highlights

• The proof leverages the naturality and continuity properties of the deformation functor, the bijection between first-order deformations and $\Lambda\textup{-Gproj}$9, and the structure theory of syzygies and their interaction with singular equivalences.
• The explicit construction of natural transformations between deformation functors induced by tensoring with the bimodule $\Gamma\textup{-Gproj}$0 (and its quasi-inverse $\Gamma\textup{-Gproj}$1) ensures the functorial equivalence at the level of lifted deformations.
• Examples are provided illustrating the main result in concrete settings, such as modules over specific quiver algebras with explicit relations, verifying the isomorphism of universal deformation rings.

Implications and Future Directions

The findings significantly broaden the class of situations where deformation-theoretic invariants are preserved under equivalences stemming from singular or stable equivalence of Morita type. Practically, this deepens the connection between representation theory, deformation theory, and the structure of derived and stable categories, permitting transfer of deformation data through categorical equivalences even in non-Gorenstein environments.

On the theoretical side, the results reinforce the robustness of universal deformation rings as invariants under derived and singular equivalence. This invites further exploration of deformation theory in the representation theory of non-Gorenstein and higher singularity algebras, possibly relating deformation rings with other categorical and homological invariants.

Future developments may examine the behavior of universal deformation rings under broader classes of equivalences—such as perverse equivalences or recollements—and further analyze the structure of the deformation functor for objects with nontrivial endomorphism rings, or under less restrictive homological conditions.

Conclusion

The paper establishes the invariance of universal deformation rings of indecomposable Gorenstein-projective modules under equivalences induced by singular equivalence of Morita type with level, provided these equivalences extend to the triangulated stable categories of Gorenstein-projective modules. This generalizes earlier results restricted to Gorenstein algebras and enhances the understanding of connections between deformation theory and singular equivalence in the modular representation theory of finite-dimensional algebras (2606.06648).