HIGHER MORITA–TACHIKAWA CORRESPONDENCE

. Important correspondences in representation theory can be regarded as restrictions of the Morita–Tachikawa correspondence. Moreover, this correspondence motivates the study of many classes of algebras like Morita algebras and gendo-symmetric algebras. Explicitly, the Morita–Tachikawa correspondence describes that endomorphism algebras of generators-cogenerators over ﬁnite-dimensional algebras are exactly the ﬁnite-dimensional algebras with dominant dimension at least two. In this paper, we introduce the concepts of quasi-generators and quasi-cogenerators which generalise generators and cogenerators, respectively. Using these new concepts, we present higher versions of the Morita–Tachikawa correspondence that takes into account relative dominant dimension with respect to a self-orthogonal module with arbitrary projective and injective dimension. These new versions also hold over Noetherian algebras which are ﬁnitely generated and projective over a commutative Noetherian ring.


Introduction
The celebrated Morita-Tachikawa correspondence connects basic finite-dimensional algebras having dominant dimension at least two with pairs consisting of a basic finite-dimensional algebra and a multiplicity-free generator-cogenerator (see for example [Mue68,Theorem 2]).An important specialisation of this correspondence is the famous Auslander correspondence [Aus71] which marks the starting point for the use of Auslander and Reiten ideas in representation theory.Not only the Auslander's correspondence but also many known correspondences like Iyama's higher Auslander correspondence [Iya07] and Iyama-Solberg's correspondence [IS18] are also specialisations of the Morita-Tachikawa correspondence.Moreover, algebras arising from these correspondences have appeared in many different areas like cluster theory, homological algebra, algebraic Lie theory to name a few (see [KSX01] and for instance see also [CIM22] and the references therein).The Morita-Tachikawa correspondence also brought interest to define and study many new classes of algebras for example: Morita algebras [KY13], gendo-Frobenius algebras [Yı22] and gendo-symmetric algebras [FK11] as the counterparts through Morita-Tachikawa correspondence of self-injective, Frobenius and symmetric algebras, respectively.See also [GK15] for the counterparts of Gorenstein algebras under the Morita-Tachikawa correspondence.Recently, these correspondences started to make appearances also in the context of exact categories [HKvR22,ENI21,Gre22].
On the other hand, this kind of correspondences are intimately connected with the theory of dominant dimension.Recently, there has been growing interest in relative dominant dimension as a tool to study the existence and uniqueness of certain tilting modules (see [CBS17,AT21,CP23] and the references therein) and to study the representation theory of endomorphism algebras of summands of a characteristic tilting module over quasi-hereditary algebras (see for example [KSX01,Cru22b,CE22].Moreover, such developments provide evidence that the theory of relative dominant dimension is still in its infancy stage.In particular, the theory needs a relative analogue to the Morita-Tachikawa correspondence. Going from classical dominant dimension to relative dominant dimension, allows for example to recover properties that are intrinsic to minimal Auslander-Gorenstein algebras into more general finite-dimensional Gorenstein algebras equipped with a self-orthogonal module having nice properties.Such pairs were called relative Auslander-Gorenstein pairs in [CP23].But, so far, no relative Auslander correspondence is known for these pairs aside from a special case when the sum of the projective dimension with the injective dimension of the self-orthogonal module is at most one (see [LZ21]).
Therefore, we should expect the existence of a relative Morita-Tachikawa correspondence that takes into account the relative dominant dimension and at the same time can be used in future work for example to discover a relative analogue of the Iyama-Solberg's correspondence to characterise relative Auslander-Gorenstein pairs.
In this paper, we make this expectation precise and we develop higher versions of the Morita-Tachikawa correspondence.To do this, it is essential to introduce higher versions of generators and cogenerators and to observe that the Morita-Tachikawa correspondence is in itself the intersection of two correspondences.Generators of a module category are the modules that contain all the indecomposable projective modules as direct summands, and cogenerators are their dual concept.For each non-negative integer n, we can consider a higher n-version of generator that we propose to call n-quasi-generator to generalise the concept of generator of a module category.In this terminology, the generators are exactly the 0-quasi-generators.Similarly, we can consider the dual concept n-quasi-cogenerator.In the development of what is now called Morita theory, Morita observed that projectives and generators form a sort of dichotomy.Indeed Morita observed that any projective module affording a double centraliser property is a generator over its endomorphism algebra.In the same spirit, n-quasi-generators over finite-dimensional algebras over a field arise from the study of self-orthogonal modules with projective dimension n affording a double centraliser property.More precisely, using n-quasi-generators instead of just generators we obtain the following correspondence: Theorem (see Theorem 3.3).Let k be a field.For every non-negative integer n, there are bijections Φ and Ψ between: B is a finite-dimensional algebra over k M is an n-quasi-generator of mod-B and A is a finite-dimensional algebra over k, M is a finitely generated A-module with projective dimension exactly n, without self-extensions, that is, given as follows: Observe that Γ ∩ Λ is the class of n-tilting modules over a finite-dimensional algebra over a field.By fixing n = 0, this intersection Γ ∩ Λ is the class of projective generators which are the main objects of study in Morita theory.The modules M satisfying Ext i>0 A (M, M ) = 0 are known in the literature as self-orthogonal modules.
The correspondence in Theorem 3.3 restricts to a higher version of the classical Morita-Tachikawa correspondence when combined with its dual version.Indeed, classical Morita-Tachikawa correspondence corresponds to the case n = m = 0 in the following correspondence.
Theorem (see Theorem 3.5).Let k be a field.For every pair of non-negative integers n and m, there is a one-to-one correspondence between: A is a finite-dimensional algebra over k, M is a finitely generated A-module with projective dimension exactly n, and injective dimension exactly m, without self-extensions, that is, Both Theorem 3.5 and Theorem 3.3 are actually formulated for Noetherian algebras which are finitely generated and projective over a commutative Noetherian ring.
In the end, we explore the link between self-orthogonal quasi-generators and the homological conjectures.In particular, we prove that there exists an intermediate conjecture between the Wakamatsu tilting conjecture and the Auslander-Reiten conjecture involving self-orthogonal quasi-generators.

Preliminaries
Throughout the paper, R is a commutative Noetherian ring with identity and A is a projective Noetherian R-algebra, that is, an associative R-algebra so that A is finitely generated projective as R-module.By A-mod we denote the category of finitely generated left A-modules and by mod-A the category of finitely generated right A-modules.We write A-proj to denote the full subcategory of A-mod whose modules are projective.Given M ∈ A-mod (or M ∈ mod-A) we denote by add A M the full subcategory of A-mod (resp.mod-A) whose modules are direct summands of direct sums of M over A. By A op we mean the opposite algebra of A and by D the standard duality functor Hom R (−, R) : A-mod → A op -mod.Given M ∈ A-mod, we denote by End A (M ) the endomorphism algebra of M so that the multiplication f g is the composition f • g of g and f for f, g ∈ End A (M ).By pdim A M we mean the projective dimension of the M over A. By an (A, R)-exact sequence we mean an exact sequence of A-modules which splits as sequence of R-modules.We say that A useful tool to detect the existence of double centraliser properties is relative dominant dimension.We shall now recall the definition of relative dominant dimension and some of its technicalities following [Cru22b] and [Cru22a].
In particular, if there exists an (A, R)-exact sequence 0 Analogously, we can consider the relative codominant dimension of M with respect to Q by defining Q-codomdim (A,R) M := DQ-domdim (A op ,R) DM .We will simply write Q-domdim A M whenever A is a finite-dimensional algebra over a field R and Q-domdim A instead of Q-domdim A A.
Over finite-dimensional algebras over a field, the classical dominant dimension is recovered from relative dominant dimension by fixing Q to be a faithful projective-injective module.Similar to the classical case, relative dominant dimension also possesses some left-right symmetries, for instance A very important property of relative dominant dimension is that it can be characterised by the vanishing properties of certain Tor groups.
In particular, the assumption on Q is satisfied whenever Moreover, in such a case B = End A (Q) op has the base change property (for example see [Cru22a, Proposition 2.3, Proposition 2.1]): as S-algebras, for every commutative Noetherian ring S which is an R-algebra.In the majority of cases (when End A (Q) op is again a projective Noetherian R-algebra having a base change property) and M has a certain base change property with respect to Q, the computations of Q-domdim (A,R) M can be reduced to computations over a finite-dimensional algebra over an algebraically closed field (see [Cru22b, Theorem 3.2.5.]).This is in particular the case when DQ ⊗ A Q and DQ ⊗ A M both are finitely generated projective R-modules.
It follows from Theorem 2.2 that a module Q ∈ A-mod ∩R-proj satisfying Hom A (Q, Q) ∈ R-proj and Q-domdim (A,R) A ≥ 2 affords a double centralizer property.Indeed, such a module Q affords a double centralizer property if and only if Dχ r DA is an isomorphism (see [Cru22b, Lemma 5.1.2.]).In general, the map Dχ r DM admits the following simplification: Lemma 2.3.The map Dχ r DM fits into a commutative diagram Here, ω is the natural transformation between the identity functor on A-mod ∩R-proj and D 2 , ϕ is the canonical map Hom A (M, Q) → Hom A (DQ, DM ) and κ is the isomorphism given by Tensor-Hom adjunction.
Proof.Indeed for any m ∈ M , f ∈ Hom A (M, Q), h ∈ DQ the following holds: and For ϕ being an isomorphism see for example [Cru22a, Proposition 2.2.].
2.2.Generators.It is nowadays a classic result due to Morita that generators are projective modules over its endomorphism algebra.Another approach to see this (over finite-dimensional algebras) is the following.Assume that A is a finite-dimensional algebra over a field and let M be a right A-module which is a generator of A. So, minimal add M -approximations always exist (see [AS80]).Further, for every X ∈ A-mod the minimal right add M -approximation M → X is surjective.Indeed, the induced sequence Hom A (A, M ) → Hom A (A, X) is surjective since A ∈ add M .Therefore, M -codomdim A DA = +∞.By [Cru22b, Corollary 3.1.5.],M -domdim A = M -codomdim A DA = +∞.
Corollary 2.5.Let A be a projective Noetherian R-algebra and let M ∈ A op -mod ∩R-proj.If M is a generator of A as right module and M ⊗ A DM ∈ R-proj, then M -domdim(A, R) ≥ 2.

Proof. By assumption,
and only if DM is a generator of A op .So, all the results for generators can be dualised for (A, R)-cogenerators.Observe, in particular, that Morita's theorem for (A, R)-cogenerators reads as follows: and M affords a double centralizer property.[Hoc56].The (A, R)-injective modules which are projective over the ground ring are exactly the injective objects in A-mod ∩R-proj.Indeed, they can be characterised in the following way: Given M ∈ A-mod ∩R-proj, we say that M has (A, R)-injective dimension n, denoted as idim (A,R) M , if and only if Ext n+1 A (N, M ) = 0 for all N ∈ A-mod ∩R-proj and Ext n+2 A (X, M ) = 0 for some X ∈ A-mod ∩R-proj.

Relative injective dimension. (A, R)-injective modules made their first appearence in
By Proposition 2.7, (A, R)-injective modules which are projective as R-modules are acyclic objects with respect to Hom A (X, −) for every X ∈ A-mod ∩R-proj.So, coresolutions by (A, R)-injective modules can be used to compute (A, R)-injective dimensions.Given M ∈ A-mod ∩R-proj, such a coresolution can be obtained by applying D to a projective resolution of DM .
Lemma 2.8.Let M ∈ A-mod ∩R-proj.The following assertions are equivalent. (1) Proof.Assume that (b) holds.Then M admits an (A, R)-injective coresolution of length at most n+1 (counting X 0 ).Since (A, R)-injective coresolutions are acyclic with respect to Hom A (X, −) for all X ∈ A-mod ∩R-proj it follows that Ext n+1 A (X, M ) = 0 for all X ∈ A-mod ∩R-proj.Conversely, consider an (A, R)-exact sequence 0 Using the fact that all X i are acyclic with respect to Hom A (X, −) we get by applying Hom A (X, −) to previous exact sequence that 0 Remark 2.9.An elementary consequence of Lemma 2.8 is the equality This fact can also be deduced directly from the following result.
Proof.For the first equality see [Cru22a, Lemma 2.15].For the second, observe that for all i ≥ 0, where N • denotes a deleted projective left resolution of N over A.

Quasi-generators
We will now introduce quasi-generators and quasi-cogenerators to prove our main result.
Definition 3.1.Let M ∈ mod-A ∩ R-proj and n be a non-negative integer.We say that M is an n-quasi-generator of mod-A if M ⊗ A DM ∈ R-proj and n is the minimal integer so that there exists an (A, R)-exact sequence which remains exact under Hom A (−, M ) and each M i belongs in add A M .
Dually, we define the concept of relative quasi-cogenerator.
Definition 3.2.Let M ∈ mod-A ∩ R-proj and n be a non-negative integer.We say that M is an n-quasi-(A, R)-cogenerator if M ⊗ A DM ∈ R-proj and n is the minimal integer so that there exists an (A, R)-exact sequence which remains exact under Hom A (M, −) and each M i belongs in add A M .
In this way, a right module M is an n-quasi-generator of mod-A if and only if DM is an n-quasi-(A op , R)-cogenerator.This terminology is chosen to be consistent with the one used in [Cru22a].Similarly, the same definitions can be considered for left modules.
Theorem 3.3.Let R be a commutative Noetherian ring.For each non-negative integer n, there are bijections Φ and Ψ between: B is a projective Noetherian R-algebra, M is an n-quasi-generator of mod-B and A a projective Noetherian R-algebra, M ∈ A-mod ∩R-proj has the following properties: with P i ∈ A-proj.Applying DQ ⊗ A − yields the exact sequence Applying D to (7) we obtain an (B, R)-exact sequence (see also [Cru22a, Proposition 2.1.]) Observe that each Hom A (P i , Q) ∈ add B Hom A (A, Q) = add B Q. To see that (8) remains exact under Hom B (−, Q), consider the following commutative diagram 0 Since Q-domdim(A, R) ≥ 2, the maps χ r DP i are isomorphisms and Q ⊗ B DQ ≃ DA ∈ R-proj by [Cru22b, Theorem 3.1.3.].By Lemma 2.3, the commutativity of the previous diagram and the exactness of the upper row implies that (8) remains exact under Hom B (−, Q).This shows that Q is an i-quasi-generator of mod-B for some i ≤ n.It is indeed an n-quasi-generator of mod-B because applying Hom B (−, Q) to an exact sequence (4) of length i yields a projective resolution of Q over A of length i; which would contradict the projective dimension of Q over A being and A a projective Noetherian R-algebra, M ∈ A-mod ∩R-proj has the following properties: In particular, this generalises [Cru22a, Theorem 4.1] and by consequence also [Mue68, Theorem 2] whose published proof builds on the works of [Tac64,Mor58].

Self-orthogonal quasi-generators
Assume now that A is finite-dimensional algebra over a field k.Recall the definition of tilting module.
Definition 4.1.Given n ∈ N ∪ {0}, a module T ∈ A-mod is called an n-tilting module if it satisfies the following conditions: (a) T has projective dimension at most n over A; Since an n-tilting module is self-orthogonal, the last condition can be replaced by saying that T is an i-quasi-generator for some 0 ≤ i ≤ n.
Conversely, what conditions are necessary and sufficient for a quasi-generator to be a tilting module?We conjecture the following: Conjecture 4.2.Let A be a finite-dimensional algebra over a field k.Every self-orthogonal n-quasi-generator has projective dimension at most n.
Hence, this conjecture implies that every self-orthogonal n-quasi-generator is actually an ntilting module.Observe that if n = 0, then Conjecture 4.2 is exactly the Auslander-Reiten conjecture proposed in [AR75].A similar conjecture is the Wakamatsu tilting conjecture originated in [Wak88].
Conjecture 4.3 (Wakamatsu tilting conjecture).Let A be a finite-dimensional algebra over a field.Let T ∈ A-mod.If T is self-orthogonal with finite projective dimension such that T -domdim A A = +∞, then T is a tilting module.
We can see that the Wakamatsu tilting conjecture is actually stronger than Conjecture 4.2.Proposition 4.4.If the Wakamatsu tilting conjecture is true for all finite-dimensional algebras over a field, then Conjecture 4.2 is also true.
Proof.Let M be a self-orthogonal n-quasi-generator (as left A-module).In particular, 0 = D Ext i>0 A (M, M ) = Tor A i>0 (M, DM ).By Theorem 3.3, M is self-orthogonal and it has projective dimension exactly n as End A (M )-module and M -domdim End A (M ) End A (M ) ≥ 2. Denote by B the endomorphism algebra End A (M ).Since End B (M ) op ≃ A we obtain by [Cru22b, Theorem 3.1.4]that M -domdim B B = +∞.So M ∈ B-mod is in the conditions of the Wakamatsu tilting conjecture, and thus the Wakamatsu tilting conjecture says that M is an n-tilting module over B. Hence, it is an i-quasi-generator of mod-B for some i = 0, . . ., n.By Theorem 3.3, we obtain that M has projective dimension i over A.
Therefore, if we fix n = 0 we obtain an alternative way to see that the Wakamatsu tilting conjecture is stronger than the Auslander-Reiten conjecture.This fact can also be seen by combining [BR07, Proposition 3.1] with [AR75].Therefore by abbreviating the above mentioned conjectures to their initials, their relationship is given as follows: (W T C) =⇒ (C4.2) =⇒ (ARC).
Another interesting consequence from Conjecture 4.2 is a characterisation of Gorenstein algebras in terms of quasi-generators objects.In fact, if Conjecture 4.2 is true for all finite-dimensional algebras over a field, then a finite-dimensional algebra over a field A is Gorenstein if and only if DA is an n-quasi-generator for some non-negative integer n.

Theorem 2. 4 .
Let M be a right A-module.M is a generator if and only if M is projective over End A (M ) and M affords a double centralizer property.Proof.See for example [Lan02, XVII, Theorem 7.1].Let M ∈ mod-A ∩ R-proj be a generator of mod-A.It follows by the Morita theorem and [Cru22b, Lemma 5.1.2,Theorem 3.1.4]that M -domdim A A ≥ 2 whenever DM ⊗ End A (M ) M ∈ R-proj.Fixing B = End A (M ), Morita's theorem infers that M is projective over B and thus Tor B i>0 (DQ, Hom A (A, Q)) ≃ Tor B i>0 (DQ, Q) = 0. (2) Thus M -domdim A = +∞ by [Cru22b, Theorem 3.1.4].