A note on deformations of Gorenstein-projective modules over ﬁnite dimensional algebras

. In this note, we present a survey of results concerning universal deformation rings of ﬁnitely generated Gorenstein-projective modules over ﬁnite dimensional algebras.


Introduction
Let k be a field of arbitrary characteristic. Let Λ be a finite dimensional kalgebra and let V be a left Λ-module of finite dimension over k. F. M. Bleher and the author proved in [8,Prop. 2.1] that V has a well-defined versal deformation ring R(Λ, V ), which is a complete local commutative Noetherian k-algebra with residue field k. Moreover, R(Λ, V ) is universal provided that End Λ (V ) = k. The aim of this note is to serve as an introductory article to the deformation theory of finitely generated Gorenstein-projective modules over finite dimensional algebras (in the sense of [8]) as well as a survey of some of the concerning results available in the literature. The main motivation of this work is that the representation theory of finite dimensional algebras provides many sophisticated tools such as stable equivalences and combinatorial description of modules that can be used in order to arrive at a deeper understanding of these universal deformation rings.
This note is organized as follows. In §2, we review the definition of lifts, deformations, and (uni)versal deformation rings in the sense of [8]. We also review the definition of Gorenstein-projective modules in the sense of [17], and review with more detail the known results concerning universal deformation rings of finitely generated Gorenstein-projective modules over finite dimensional algebras. In §3, we review some results concerning universal deformation rings of modules over algebras of dihedral type (as introduced by K. Erdmann in [18]) and over monomial algebras in which there is no an overlap (as introduced by X. W. Chen et al. in [14]). We also introduced a class of finitely generated modules that we call semi-rigid stable bricks, and prove that a finitely generated Gorenstein-projective module that is also a semi-rigid stable brick has a universal deformation ring which is isomorphic either to k or to k[[t]]/(t 2 ).
We refer the reader to look at [4] and [31] in order to review basic notions from the representation theory of algebras and from the homological algebra used in this note. We also refer the reader to [14] and its references for basic properties of finitely generated Gorenstein-projective modules.
This article is an alternative version of the author's 20-minutes talk at the XXII Coloquio Latinoamericano deÁlgebra, which was held in Quito, Ecuador, during August 2017, where the result [5, Thm. 5.2] was presented. The author would like to express his gratitude to the organizers of this event as well as to the coordinators of the session in Representation Theory of Algebras, Professors Raymundo Bautista and Claudia Chaio, for giving the author the opportunity of participating in this event and for inviting him to write this note.

Preliminaries
In this section, we assume that k is a fixed field of arbitrary characteristic. We denote byĈ the category of all complete local commutative Noetherian kalgebras with residue field k. In particular, the morphisms inĈ are continuous k-algebra homomorphisms that induce the identity map on k. Let Λ be a fixed (but arbitrary) finite dimensional k-algebra, and let R ∈ Ob(Ĉ). We denote by RΛ the tensor product of k-algebras R ⊗ k Λ. We assume that all our modules are finitely generated. We denote by Λ-mod the abelian category of left modules over Λ, and by Λ-mod its stable category, i.e. the objects of Λ-mod are the same as those of Λ-mod, and for all objects V and W in Λ-mod, Hom Λ-mod (V, W ) = Hom Λ (V, W ) is the k-vector space which is the quotient of Hom Λ (V, W ) by PHom Λ (V, W ), which is the k-vector space of Λ-module homomorphisms from V to W that factor through a projective Λ-module. In particular, when V = W , we set End Λ (V ) = Hom Λ (V, V ) (resp. End Λ (V ) = Hom Λ (V, V )) and call it the endomorphism (resp. stable endomorphism) ring of V . We also denote by ΩV the first syzygy of V , i.e. ΩV is the kernel of a projective cover P (V ) → V of V over Λ, which is unique up to isomorphism.
Let V be a fixed left Λ-module.

Liftings and deformations of modules
is a lift of V over R, we denote by [M, φ] its isomorphism class and say that [M, φ] is a deformation of V over R. We denote by Def Λ (V, R) the set of all deformations of V over R. The deformation functor corresponding to V is the covariant functorF V :Ĉ → Sets defined as follows: for all objects R ∈ Ob(Ĉ), defineF V (R) = Def Λ (V, R), and for all morphisms α : is unique for all R ∈ Ob(Ĉ) and lifts (M, φ) of V over R, then R(Λ, V ) and [U (Λ, V ), φ U (Λ,V ) ] are respectively called the universal deformation ring and the universal deformation of V . In other words, the universal deformation ring R(Λ, V ) represents the deformation functorF V in the sense thatF V is naturally isomorphic to the Hom functor HomĈ(R(Λ, V ), −). Using Schlessinger's criteria [25,Thm. 2.11] and using methods similar to those in [22], it is straightforward to prove that the deformation functorF V is continuous (see [22, §14] for the definition), that every finitely generated Λ-module V has a versal deformation ring, and that this versal deformation is universal provided that the endomorphism ring of V is isomorphic to k (see [

Gorenstein algebras, Gorenstein-projective modules
Following [17], we say that V is Gorenstein-projective provided that there exists an acyclic sequence of projective Λ-modules such that Hom Λ (P • , Λ) is also acyclic and V = coker f 0 . Note that every projective Λ-module is also Gorenstein-projective, and if V is a Gorenstein-projective Λ-module, then Ext i Λ (V, Λ) = 0 for all i > 0. It is important to mention that Gorenstein-projective modules are also known under many names in the literature (see e.g. at the introduction of [14] for more details). Following [2], we say that Λ is a Gorenstein k-algebra provided that Λ has finite injective dimension as a left and right Λ-module. By [32], these dimensions coincide and their common value is called the virtual Gorenstein dimension of Λ. Note that if Λ is either self-injective (i.e. the regular Λ-module Λ Λ is injective) or of finite global dimension, then Λ is Gorenstein. In particular, self-injective k-algebras have virtual Gorenstein dimension zero. We denote by Λ-Gproj the full subcategory of Λ-mod whose objects are Gorenstein-projective Λ-modules, and by Λ-Gproj the corresponding full subcategory of Λ-mod. Note that Λ is self-injective if and only if the categories Λ-mod and Λ-Gproj coincide. Following [13], we say that Λ is CM-finite if there are at most a finite number of isomorphism classes of indecomposable Gorenstein projective Λ-modules, and that Λ is CM-free if every Gorenstein-projective Λ-module is projective. Note in particular that every algebra of finite global dimension is CM-free. We invite the reader to look at [16,Prop. 3.14] to obtain examples of finite dimensional algebras that are CM-free but not Gorenstein.

Singular equivalences of Morita type
The following definition was introduced by X. W. Chen and L. G. Sun in [15], which was further studied by G. Zhou and A. Zimmermann in [33], as a way of generalizing the concept of stable equivalence of Morita type introduced by M. Broué in [11].
Definition 2.1. Let Λ and Γ be finite dimensional k-algebras, and let X be a Γ-Λ-bimodule and Y a Λ-Γ-bimodule. We say that X and Y induce a singular equivalence of Morita type between Λ and Γ (and that Λ and Γ are singularly equivalent of Morita type) if the following conditions are satisfied: (i) X is finitely generated and projective as a left Γ-module and as a right Λ-module.
(ii) Y is finitely generated and projective as a left Λ-module and as a right Γ-module.
The concept of singular equivalence of Morita type was further generalized by Z. Wang in [30], where the concept of singular equivalence of Morita type with level is introduced. It was proved by Ø. Skartsaeterhagen are equivalences of triangulated categories that are quasi-inverses of each other.

Results concerning (uni)versal deformation rings of Gorensteinprojective modules
Recall that Λ is a Frobenius k-algebra provided that the left Λ-modules Λ Λ and Hom k (Λ Λ , k) are isomorphic. In particular, Frobenius k-algebras are also selfinjective (see e.g. The following result summarizes some of the known results concerning versal deformation rings of Gorenstein-projective modules over finite dimensional kalgebras.
Theorem 2.2. Let Λ be a finite dimensional k-algebra, and let V be a Gorensteinprojective Λ-module.
(i) If V is projective, then R(Λ, V ) is universal and isomorphic to k.
(iv) If Λ is further Frobenius and V is non-projective, then the versal deformation rings R(Λ, V ) and R(Λ, ΩV ) are isomorphic inĈ.
(v) If Λ is Gorenstein and Γ is another finite dimensional Gorenstein kalgebra such that there exist bimodules Γ X Λ , Λ Y Γ inducing a singular equivalence of Morita type (as in Definition 2.1) between Λ and Γ, then X ⊗ Λ V is a finitely generated Gorenstein-projective left Γ-module, and the versal deformation rings R(Λ, V ) and R(Γ, X ⊗ Λ V ) are isomorphic inĈ. (ii) Let Λ be an arbitrary finite dimensional Gorenstein k-algebra and let V is a Gorenstein-projective left Λ-module whose stable endomorphism ring is k. Note that by [3, Prop. 3.1 (c)] we also have that ΩV is also Gorenstein-projective with stable endomorphism ring also isomorphic to k. In particular, the versal deformation ring R(Λ, ΩV ) is also universal.
In [5,Remark 5.5], the following question was given: Are R(Λ, V ) and R(Λ, ΩV ) isomorphic inĈ? Currently, this is still open for when A is a non-self-injective algebra.

Examples
In this section, we assume that k is an algebraically closed field. Recall that a quiver Q is a directed graph with a set of vertices Q 0 , a set of arrows Q 1 and two functions s, t : Q 1 → Q 0 , where for all α ∈ Q 1 , sα (resp. tα) denotes the vertex where α starts (resp. ends). A path in Q is either an ordered sequence of arrows p = α n · · · α 1 with tα j = sα j+1 for 1 ≤ j < n (in this situation we say that p has length n), or for each i ∈ Q 0 , the symbol e i such that se i = i = te i . We call the symbols e i the trivial paths, which have length zero. For a nontrivial path p = α n · · · α 1 we define sp = sα 1 and tp = tα n . A non-trivial path p in Q is said to be an oriented cycle provided that sp = tp. The path algebra kQ of a quiver Q is the k-vector space whose basis consists in all the paths in Q, and for two paths p and q, their multiplication is given by the concatenation pq provided that sp = tq, or zero otherwise. Let J be the two-sided ideal of kQ generated by all the arrows in Q. We say that an ideal I of kQ is admissible if there exists d ≥ 2 such that J d ⊆ I ⊆ J 2 . In this situation, the quotient kQ/I is a finite dimensional k-algebra. If p is a path in Q, we denote also by p is equivalence class a call it a path in kQ/I. In particular, a path p in kQ/I is zero if and only if p belongs to I. It is well-known that every finite dimensional k-algebra is Morita equivalent to an algebra of the form kQ/I, where Q is a finite quiver and I is an admissible ideal of kQ (see e.g. [4, §III.1]). Since versal deformation rings of finitely generated modules over finite dimensional algebras are invariants under Morita equivalence (see [8,Prop. 2.5]), it is enough to assume that all our finite dimensional k-algebras are all basic and of the form Λ = kQ/I, where Q and I are as above.

Algebras of dihedral type
Consider the k-algebras Λ of dihedral type (as introduced in [18]) in Figure 2. It follows in particular that these algebras are all symmetric, i.e., the functors Hom Λ (−, Λ) and Hom k (−, k) from Λ-mod to Λ op -mod are natural equivalent, where Λ op is the opposite algebra of Λ. This in particular implies that all these algebras are also self-injective (see e.g. [4, Prop. IV.3.8]). Moreover, it follows from [20,Thm. 3.4] that these algebras are also derived equivalent. Since derived equivalences between self-injective k-algebras induces stable equivalences of Morita type (see [23,Cor. 5.5]), and thus singular equivalences of Morita type as in Definition 2.1, it follows that the algebras in Figure 2 are all singularly equivalent of Morita type. Let Λ 0 = D(3R) 1,2,2,2 . In [8, §3], all left Λ 0 -modules V with End Λ0 (V ) = k were completely classified. Since self-injective algebras are also Gorenstein, it  [7] concerning the algebras of dihedral type in Figure 2.
(ii) In [29], the author studied the universal deformation rings of modules over more general cases of algebras of dihedral type of the class D(3R).

Monomial algebras in which there is no an overlap
Although the study of finitely generated Gorenstein-projective modules goes back to [1], explicit descriptions of indecomposable Gorenstein-projective modules have been found for only a few classes of not necessarily self-injective k-algebras (see e.g. [12,14,21,24]). In the following, we recall such description given in [14] for monomial algebras. Recall that an admissible ideal I of kQ is said to be monomial if it is generated by paths of length at least two. In this situation we say that the quotient kQ/I is a monomial algebra. Let Λ = kQ/I be a monomial algebra. Following [14], we say that a pair (p, q) of non-zero paths in Λ is a perfect pair provided that the following conditions are satisfied: (P1) both p and q are non-trivial with sp = tq and pq ∈ I; (P2) if pq ∈ I for a non-zero path q with tq = sp, then q = qq for some path q in Λ; (P3) if p q ∈ I for a non-zero path p with tq = sp , then p = p p for some path p in Λ.
A non-zero path p in Λ is perfect, provided that there exists a sequence p = p 1 , p 2 , . . . , p n , p n+1 = p such that for all 1 ≤ i ≤ n, the pair (p i , p i+1 ) is a perfect pair. It follows from [14,Thm. 4.1] that a finitely generated indecomposable non-projective left Λ-module V is Gorenstein-projective if and only if V = Λp, where p is a perfect path in Λ. This results unifies those descriptions for indecomposable Gorenstein-projective modules over Nakayama algebras given in [24] and over gentle algebras given in [21]. An overlap in Λ is given by two perfect paths p and q in Λ that satisfy one of the following conditions: (O1) p = q, and p = p x and q = xq for some non-trivial paths x, p and q with the path p xq non-zero.
(O2) p = q, and p = p x and q = xq for some non-trivial path x with the path p xq non-zero.
We refer the reader to look at [  Remark 3.4. We say that a monomial algebra Λ = kQ/I is quadratic if I is generated by paths of length two. In particular, gentle algebras are quadratic monomial. Note that there is no an overlap in a quadratic monomial algebra, for in this situation all the perfect paths are arrows. Thus, Theorem 3.3 applies also to gentle algebras.
3.3. Universal deformation rings of a certain type of Gorensteinprojective modules Definition 3.5. We say that a left Λ-module V is a semi-rigid stable brick if End Λ (V ) = k and Hom Λ (ΩV, V ) = δ ΩV,V · k, where δ ΩV,V is the Kronecker delta.
Remark 3.6. It follows from the proofs of [14,Prop. 5.9] and [12, Lemma 3.6 (iii)] that if Λ is either an algebra in which there is not overlap or skewed-gentle (as introduced by Ch. Geiß and J. A. de la Peña in [19]), then every indecomposable non-projective Gorenstein-projective left Λ-module V is a semi-rigid stable brick as in Definition 3.5.
The following result unifies the classification of universal deformation rings of indecomposable non-projective Gorenstein-projective modules over algebras in which there is no an overlap and over skewed-gentle algebras; its proof can be obtained verbatim from that of Theorem 3.3.
Theorem 3.7. Let Λ be a finite dimensional k-algebra, and let V be a Gorensteinprojective left Λ-module which is also a semi-rigid stable brick as in Definition 3.5. Then the versal deformation ring R(Λ, V ) is universal and isomorphic either to k or to k[[t]]/(t 2 ).