SPECIAL TILTING MODULES FOR ALGEBRAS WITH POSITIVE DOMINANT DIMENSION

Abstract We study certain special tilting and cotilting modules for an algebra with positive dominant dimension, each of which is generated or cogenerated (and usually both) by projective-injectives. These modules have various interesting properties, for example, that their endomorphism algebras always have global dimension less than or equal to that of the original algebra. We characterise minimal d-Auslander–Gorenstein algebras and d-Auslander algebras via the property that these special tilting and cotilting modules coincide. By the Morita–Tachikawa correspondence, any algebra of dominant dimension at least 2 may be expressed (essentially uniquely) as the endomorphism algebra of a generator-cogenerator for another algebra, and we also study our special tilting and cotilting modules from this point of view, via the theory of recollements and intermediate extension functors.


Introduction
In [8], Crawley-Boevey and the second author associated to each Auslander algebra a distinguished tilting-cotilting module T , with the property that it is both generated and cogenerated by a projectiveinjective module. In this paper, we study more general instances of tilting modules generated by projective-injectives, and cotilting modules cogenerated by projective-injectives. In contrast to the case of Auslander algebras, we consider here tilting and cotilting modules of arbitrary finite projective or injective dimension.
More precisely, let Γ be a finite-dimensional algebra with dominant dimension d + 1 (see Definition 2.1). Then for every 1 ≤ k ≤ d, we may specify a specific 'shifted' k-tilting module T k and 'coshifted' k-cotilting module C k , which are both generated and cogenerated by projectiveinjectives. We are also interested in the resulting shifted and coshifted algebras B k = End Γ (T k ) op and B k = End Γ (C k ) op . Some of our results will also apply to the degenerate cases of k = 0 and k = d + 1.
Despite the relatively simple construction of these modules and algebras, they appear not to have been studied in much detail, particularly for k > 1, until very recently. Indeed, the only general result we are aware of prior to our work is by Chen and Xi [6], who call the T k 'canonical tilting modules' and obtain results on the dominant dimension of the shifted algebras B k . Some of our results on the single tilting module T 1 , notably those of Section 3 on minimal 1-Auslander-Gorenstein algebras, were obtained independently by Nguyen, Reiten, Todorov and Zhu [21].
The primary purpose of this paper is to collect information about the modules T k relevant to tilting theory; for example, by computing various subcategories determined by the tilting module and used in Miyashita's generalisation of the Brenner-Butler theorem, and by studying recollements involving the tilted algebras B k . We take our lead from the questions answered in [8] for T 1 in the case that Γ is an Auslander algebra. These considerations are applied in [8] to construct desingularisations for varieties of modules, but the geometric arguments depend crucially on working in a low homological dimension. Nevertheless, we are able to extend most of the homological results of [8] to our much higher level of generality-in some cases, also simplifying the arguments-and we expect these properties to be of independent interest. We note that the geometric statements of [8] can also be extended by generalising in a somewhat different direction, as we explain in [22].
The non-degenerate shifted and coshifted modules are defined when Γ has dominant dimension at least 2, and in this case Γ ∼ = End A (E) op for a generating-cogenerating module E over a finitedimensional algebra A. In fact, assuming for simplicity that all objects are basic, the assignment (A, E) → End A (E) op  with objects considered up to isomorphism on each side 1 [19,26], a result sometimes called [10,25] the Morita-Tachikawa correspondence. In this context, it will often be convenient for us to express results on the special cotilting Γ-modules in terms of the pair (A, E) on the other side of the correspondence, and as such the following definition will be convenient throughout the paper. for Π a maximal projective-injective summand of Γ, and D the usual duality over the base field.
The structure of the paper is as follows. We give the definitions and preliminary results in Section 2, including the observation (Proposition 2.13) that gldim B ≤ gldim Γ whenever B is one of the shifted or coshifted algebras of Γ, and a characterisation (Proposition 2.12) of the algebras B arising in this way. In Section 3, we investigate the modules T k and C k in the context of higher Auslander-Reiten theory, which provides a wealth of examples of algebras with high dominant dimension. The main result of this section is the following.
Theorem 1 (Theorem 3.2). Let Γ be a finite-dimensional non-selfinjective algebra with domdim Γ = d + 1. The following are equivalent: (i) Γ is a minimal d-Auslander-Gorenstein algebra, (ii) T k = C d+1−k for all 0 ≤ k ≤ d + 1, and (iii) there exists M ∈ Γ-mod that is both a shifted and a coshifted module. Under these conditions, Γ is a d-Auslander algebra if and only if gldim Γ < ∞.
If Π is the maximal projective-injective summand of Γ, it is a summand of every tilting or cotilting Γ-module. Thus if B is the endomorphism algebra of such a module, it has an idempotent given by projection onto Π, yielding a recollement involving the categories B-mod and End Γ (Π) op -mod; note that if domdim Γ ≥ 2 then End Γ (Π) op is the algebra A from the Morita-Tachikawa triple involving Γ. In Section 4, we study these recollements for the shifted and coshifted algebras. In particular, we give in Theorems 4.8 and 4.12 an explicit formula for the intermediate extension functor in such a recollement; this functor is, by definition, the image of the universal map from the restriction functor's left adjoint to its right adjoint.
To obtain this formula, we show that each shifted and coshifted algebra of Γ can be described in terms of its Morita-Tachikawa partner (A, E), as in the following theorem.
where B k and B k are the shifted and coshifted algebras of Γ, and E k and E k are certain bounded complexes of A-modules, defined explicitly in Theorem 4.4. 1 We say that (A, E) and (A ′ , E ′ ) are isomorphic if there is an isomorphism ϕ : The reader is warned that (A, E) ∼ = (A, E ′ ) does not imply that E ∼ = E ′ as A-modules, but only that E ∼ = ϕ * E ′ for some ϕ ∈ Aut(A).
This result generalises [8,Prop. 5.5] for the case that Γ is an Auslander algebra and k = 1. The proof we give here is different, and more conceptual.
A k-tilting or k-cotilting Γ-module with endomorphism algebra B defines k + 1 pairs of equivalent subcategories in Γ-mod and B-mod; in the classical case k = 1, the two subcategories on each side form a torsion pair. In Section 5, we describe these subcategories for the shifted and coshifted modules, often in terms of generation or cogeneration by certain projective or injective modules.
In Section 6, we consider again the recollements involving B-mod and A-mod, where B is one of the shifted or coshifted algebras of an algebra Γ in a Morita-Tachikawa triple (A, E, Γ). Recall from general tilting theory that B k , as a tilt of Γ by T k , has a preferred cotilting module DT k . Similarly B k has the preferred tilting module DC k . We prove the following, again generalising the results of [8] for the case that Γ is the Auslander algebra of A.
Theorem 3 (Theorems 6.3, 6.5). Let (A, E, Γ) be a Morita-Tachikawa triple and 0 < k < domdim Γ. Denoting by c k and c k the intermediate extension functors in the recollements relating B k -mod and B k -mod respectively with A-mod, we have Throughout the paper, all algebras are finite-dimensional K-algebras over some field K, and, without additional qualification, 'module' is taken to mean 'finitely-generated left module'. As mentioned above, D = Hom K (−, K) denotes the K-linear dual. Morphisms are composed right-to-left.

Shifted modules and algebras
Throughout this section, we fix a finite-dimensional algebra Γ, assumed for simplicity to be basic, over a field K. The goal of this section is to define certain special tilting and cotilting Γ-modules in the case that Γ has positive dominant dimension, and list some of their basic properties.
Definition 2.1. Let k be a non-negative integer. We say that Γ has dominant dimension at least k, and write domdim Γ ≥ k, if the regular module Γ Γ has an injective resolution with Π 0 , . . . , Π k−1 projective-injective; when k = 0, this condition is taken to be empty. As the notation suggests, we write domdim Γ = k if domdim Γ ≥ k and domdim Γ ≥ k + 1.
Ext i Γ (T, T ) = 0 for i > 0, and (T3) there is an add T -coresolution of Γ of length k, i.e. an exact sequence with t j ∈ add T for 0 ≤ j ≤ k. We say a k-tilting module T is P -special for a projective module P if P ∈ add T and there is a sequence as in (T3) with t j ∈ add P for 0 ≤ j ≤ k − 1.
Dually, we say that C is a k-cotilting module if (C1) idim C ≤ k, (C2) Ext i Γ (C, C) = 0 for i > 0, and (C3) there is an add C-resolution of DΓ of length k, i.e. an exact sequence We say a k-cotilting module C is I-special for an injective module I if I ∈ add C and there is a sequence as in (C3) with c j ∈ add I for 0 ≤ j ≤ k − 1.
Proposition 2.4. Let Γ be a finite-dimensional algebra, and let Π be a maximal projective-injective summand of Γ. Then there exists a basic Π-special k-tilting Γ-module and a basic Π-special k-cotilting Γ-module C k if and only if domdim Γ ≥ k. These modules are unique up to isomorphism.
Proof. We prove the statements involving T k , those for C k being dual. If Γ has a Π-special k-tilting module, then domdim Γ ≥ k by (T3). Conversely, if domdim Γ ≥ k, there is an exact sequence Let T k be a basic module with add T k = add(T ⊕Π). Then T k satisfies (T1) and (T3), and is Π-special, by (2.1). A standard homological argument, involving the application of the functors Hom Γ (T k , −) and Hom Γ (−, T k ) to the short exact sequences coming from (2.1), shows that Ext i Γ (T k , T k ) = Ext i Γ (Γ, Γ) = 0 for i > 0, so T k satisfies (T2). Any two Π-special k-tilting Γ-modules are, by definition, k-th cosyzygies of the regular module Γ. Thus if T ′ is an arbitrary k-th cosyzygy of Γ, it differs from T k only by the possible removal of projective-injective summands and addition of injective summands, so T ∈ add T ′ , where T is as in (2.1). If T ′ is tilting then we must also have Π ∈ add T ′ , so T k ∈ add T ′ . If T ′ is basic, it then follows that T ′ ∼ = T k since all tilting modules have the same number of indecomposable summands up to isomorphism.
Definition 2.5. For Γ a finite-dimensional algebra with domdim Γ ≥ k, write T k and C k for basic Πspecial k-tilting and k-cotilting modules respectively, these modules being unique up to isomorphism by Proposition 2.4. We call T k the k-shifted module of Γ, and C k the the k-coshifted module of Γ. The algebras , are called respectively the k-shifted and k-coshifted algebras of Γ.
The modules T k appeared briefly as an example in a paper of Chen and Xi [6], where they are called 'canonical tilting modules'. It is well-known that if T is a k-tilting Γ-module with B = End Γ (T ) op , then the right derived functor of Hom Γ (T, −) and the left derived functor of D Hom B (−, DT ) are quasi-inverse triangle equivalences between the bounded derived categories D b (Γ) and D b (B), cf. [7, Thm. 2.1]. In particular, Γ is derived equivalent to all of its k-shifted and k-coshifted algebras.
The proof of Proposition 2.4 illustrates that the shifted and coshifted modules are related to Γ and DΓ analogously to the way in which an arbitrary module over a selfinjective algebra is related to its shifts in the stable module category (hence our choice of terminology). Despite this analogy, the case in which Γ is selfinjective does not provide any interesting examples of our constructions, since in this case T k ∼ = Γ ∼ = C k for all k ≥ 0-indeed, there are no other tilting or cotilting Γ modules. More interestingly, any non-selfinjective algebra displays very different behaviour, with no coincidences between any two of the shifted modules. This follows from the following observation.
Proof. Let T • k be the maximal non-projective-injective summand of T k , and let P be the maximal non-injective summand of Γ, which is non-zero by assumption. As in the proof of Proposition 2.4, taking the minimal injective resolution of P and truncating yields an exact sequence with Π j ∈ add Π projective for all j, so this sequence is a minimal projective resolution of T • k . Since T k = T • k ⊕ Π with Π projective, we conclude that pdim T k = k. Remark 2.8. It follows from Proposition 2.7 that any counterexample to the Nakayama conjecture (i.e. a non-selfinjective algebra of infinite dominant dimension) would have a tilting module of each possible projective dimension. This observation shows that the truth of the finitistic dimension conjecture (for a family of algebras) implies the truth of the Nakayama conjecture (for the same family), and is essentially equivalent to Tachikawa's proof of this fact [27, §8].
To give a slightly different characterisation of the modules T k and C k , we introduce the following definitions, which will also be useful in Section 5. Definition 2.9. Let A be an abelian category, and X ∈ A an object. For k ≥ 0, define gen k (X) to be the full subcategory of A on objects M such that there exists an exact sequence remaining exact under the functor Hom A (X, −). Dually, cogen k (X) is the full subcategory of A on objects N such that there exists an exact sequence with X i ∈ add X for all 0 ≤ i ≤ k, remaining exact under the functor Hom A (−, X). Note that the conditions involving the Hom-functor are automatic when X is projective or injective respectively, or when k ≤ 1. When k = 0, we omit it from the notation and refer simply to gen(X) and cogen(X). It is both natural and convenient to define gen −1 (X) = A = cogen −1 (X).
it has projective dimension at most k, and the minimal projective resolution of T is of the form for Π i ∈ add Π and P projective. Without loss of generality, we may assume T , like Γ, is basic. Then the number of indecomposable summands of P is the number of non-projective-injective summands of T , which is the number of non-projective-injective summands of Γ. Thus there is an exact sequence from which it follows simultaneously that domdim Γ ≥ k and that T is Π-special, hence isomorphic to T k by Proposition 2.4.
It is possible to identify those algebras that may be obtained as k-shifted or k-coshifted algebras intrinsically, via the existence of cotilting or tilting modules with special properties. As usual, we write ν = D Hom A (−, A) and ν − = Hom A (DA, −) for the Nakayama functors of an algebra A.
Lemma 2.11. Let T be a k-tilting Γ-module with endomorphism algebra B. By the Brenner-Butler tilting theorem [5], C = DT is a k-cotilting B-module with endomorphism algebra Γ.
(1) If T is P -special for a projective Γ-module P , then C is I P -special for I P = D Hom Γ (P, T ).
Dually, if C is I-special for an injective B-module I, then T is P I -special for P I = Hom B (C, I).
Proof. As usual, we give the proof only for the first item in each pair of dual statements.
(1) This follows by applying D Hom Γ (−, T ) to the exact sequence from (T3), using that T is P -special.

Proposition 2.12. A finite-dimensional basic algebra B is isomorphic to a k-shifted algebra if and only if there is an injective B-module I and an
Dually, a finite-dimensional basic algebra B is isomorphic to a k-coshifted algebra if and only if there exists a projective B-module P and a P -special k-tilting B-module T with νP ∈ add T . Under this isomorphism, T is the dual of the k-coshifted module.
Proof. Let T k be the k-shifted module of an algebra Γ with maximal projective-injective summand Π. Then by Lemma 2.11(1), DT k is an Conversely, assume B, C and I are as in the statement, replacing C and I by basic modules with the same additive closure if necessary. Then Γ = End B (C) op has a basic k-tilting module T = DC, which is P I = Hom B (C, I)-special by Lemma 2.11(1). By Lemma 2.11(3), P I is projective-injective. If Π is the maximal projective-injective summand of Γ, then Π is a summand of T since T is k-tilting, so Π ∈ gen(P I ) since T is P I -special. It follows that add P I = add Π, and so T ∼ = T k is the k-shifted module of Γ by Proposition 2.4.
The second statement is proved dually, reversing the roles of Γ and B in Lemma 2.11.
To close this section, we observe that if B k is the k-shifted algebra of Γ, then gldim B k ≤ gldim Γ, thus obtaining a tighter bound on this global dimension than is possible for endomorphism algebras of arbitrary tilting Γ-modules.
Proof. Write n = gldim Γ, which without loss of generality we may assume to be finite. Since T k is k-tilting, it is well-known (see, for example, [12, Prop. III.3.4]) that Since T k is a k-th cosyzygy of Γ and idim Γ ≤ n, it follows that idim T k ≤ n − k. By a result of Gastaminza The second pair of inequalities is proved dually, using that C k is k-cotilting with pdim C k ≤ n − k.

Shifting and coshifting for minimal d-Auslander-Gorenstein algebras
In [8], Crawley-Boevey and the second author considered the 1-shifted and 1-coshifted modules of an Auslander algebra, and noted that these two modules in fact coincide. In this section, we extend this result by showing that the families of shifted and coshifted modules of a general algebra Γ with domdim Γ ≥ 2 intersect if and only if Γ is a minimal d-Auslander-Gorenstein algebra, as defined by Iyama and Solberg [16], and in this case they even coincide completely.
and that it is a d-Auslander algebra if The definition of a d-Auslander algebra is due to Iyama [15] (see also [13,Defn. 4.1] for more general versions), generalising Auslander for d = 1 [2].
Note that any d-Auslander algebra is minimal d-Auslander-Gorenstein, and a minimal d-Auslander-Gorenstein algebra is a d-Auslander algebra if and only if it has finite global dimension [16,Prop. 4.8]. A selfinjective algebra is minimal d-Auslander-Gorenstein for all d, and so is a d-Auslander algebra for all d if and only if it is semisimple. On the other hand, by [16,Prop. 4.1], any minimal d-Auslander-Gorenstein algebra Γ that is not selfinjective satisfies id Γ = d + 1 = domdim Γ, so d is uniquely determined. Similarly, any d-Auslander algebra Γ that is not semisimple has gldim Γ = d + 1 = domdim Γ.
These classes of algebras are also interesting from the point of view of the Morita- Tachikawa  As promised, we may characterise d-Auslander and minimal d-Auslander-Gorenstein algebras via their shifted and coshifted modules, as follows.

is both a shifted and a coshifted module. Under these conditions, Γ is a d-Auslander algebra if and only if gldim Γ < ∞.
Proof. We start by showing that (i) implies (ii), so assume that Γ is minimal d-Auslander-Gorenstein. The assumptions on the homological dimensions of Γ imply that the regular module has a minimal injective resolution 0 with each Π j projective-injective. The number of indecomposable summands of I is equal to the number of non-injective indecomposable summands of Γ (cf. [4, Thm. 5.2]) and so, assuming without loss of generality that Γ is basic, the indecomposable direct summands of I are the indecomposable non-projective injective Γ-modules, each appearing with multiplicity one. It follows that we have DΓ = I ⊕ Π for Π the maximal projective-injective summand of Γ. Thus, by adding the identity map Π → Π to the right-hand end of the above injective resolution, we obtain a sequence in which each Π j is projective-injective. This is simultaneously an injective resolution of Γ and a projective resolution of DΓ, and has the appropriate number of projective-injective terms for computing shifted and coshifted modules, so these modules must coincide as claimed.
where Π j ∈ add Π for each j, and an injective resolution of the form with Π j ∈ add Π for each j. Taking the Yoneda product of these two sequences produces a sequence which shows that id Γ ≤ m + n ≤ domdim Γ, i.e. that Γ is minimal (m + n − 1)-Auslander-Gorenstein.
We also see from this sequence that m + n = domdim Γ = d + 1, else DΓ would be projective, contradicting our assumption that Γ is not self-injective.
The final statement is the previously noted fact that d-Auslander algebras are precisely minimal d-Auslander-Gorenstein algebras of finite global dimension [16,Prop. 4.8].
We remark that this result remains morally true when Γ is selfinjective; in this case properties (i)-(iii) hold for any positive integer d. Combining it with Proposition 2.10, we obtain the following corollary, the statement of which is more directly comparable with [

Recollements and homotopy categories
Given a Morita-Tachikawa triple (A, E, Γ), the module category of each shifted and coshifted algebra of Γ is naturally part of a recollement, also involving A-mod. Before describing and discussing these specific recollements, we will recall some facts about idempotent recollements in general.

Idempotent recollements.
Let B be a finite-dimensional algebra, let e ∈ B be an idempotent element and write A = eBe for the corresponding idempotent subalgebra (sometimes called the corner or boundary algebra). We obtain from e a diagram Various properties of the above six functors, including that both (ℓ, e) and (e, r) are adjoint pairs, mean that this diagram forms a recollement of abelian categories; we do not give the general definition here since we will only consider recollements of module categories determined by idempotents as above (cf. [23]). For a Γ-module M , one obtains the same A-module eM either by applying the functor e in this diagram, or by multiplying on the left by the idempotent e, hence the abuse of notation. Write η ℓ : 1 → eℓ and ε ℓ : ℓe → 1 for the unit and counit of the adjunction (ℓ, e), and similarly η r and ε r for the unit and counit of the adjunction (e, r). A special property of idempotent recollements is that the unit η ℓ and the counit ε r are natural isomorphisms. This means that there is a natural isomorphism  [17], which, like ℓ and r, is fully faithful. In the sequel, we will implicitly use the natural epimorphism ℓ → c and monomorphism c → r composing to the natural map ζ. We now give some alternative descriptions of the kernels and images of the functors in our recollement (4.1), including the categories ker q and ker p appearing in this TTF-triple, in terms of the categories gen k (X) and cogen k (X) associated to X ∈ B-mod as in Definition 2.9. , if X ∈ ker q then the counit map ℓeX → X is an epimorphism. Take a projective cover Q → eX; since ℓ preserves epimorphisms we obtain an epimorphism ℓQ → ℓeX → X. Since ℓA = P , we have ℓQ ∈ add P and thus X ∈ gen(P ). Conversely, gen(P ) ⊆ ker q since qP = qℓA = 0 and q preserves epimorphisms. Using instead [8, Lem/Def. 2.3], one similarly proves that ker p = cogen(I). Finally, the equality im c = ker p ∩ ker q is the first statement of [9,Prop. 4.11].
Now let (A, E, Γ) be a Morita-Tachikawa triple, with Π the maximal projective-injective summand of Γ. Recall from the Morita-Tachikawa correspondence that A ∼ = End Γ (Π) op . If T is any tilting (or cotilting) Γ-module, we must have Π ∈ add T . It follows that there is an idempotent e ∈ B = End Γ (T ) op , given by projection onto the summand Π of T , such that Thus we get a recollement as in (4.1). In particular, this holds for the shifted and coshifted algebras B k and B k of Γ. In this section, we explain how these different recollements are related, for different values of k, and give an explicit formula for the intermediate extension functor in each case.

4.2.
Recollements for shifted and coshifted algebras. We first introduce some notation for our preferred idempotents. Let Γ be a finite-dimensional algebra and k ≤ domdim Γ. We denote by e k the idempotent of the k-th shifted algebra B k of Γ given by projection onto Π ∈ add T k , and by e k the idempotent of the k-th coshifted algebra B k given by projection onto Π ∈ add C k . The algebras e k B k e k and e k B k e k are all isomorphic to A := End Γ (Π) op , so A-mod appears on the right-hand side of all of our recollements. In the case of the quotient algebras B k /B k e k B k and B k /B k e k B k appearing on the other side of the recollements, we have the following. where Ω(X) is the kernel of a minimal projective cover of X, and Ω − (Y ) is the cokernel of a minimal injective hull of Y ; for X ∈ gen(Π), such a projective cover is a minimal left add Π-approximation as referred to in [4,Thm. 5.2], and the corresponding statement holds for Y ∈ cogen(Π). Now, noting that for k = 0 there is nothing to prove, the result for 1 ≤ k ≤ domdim Γ follows inductively using the fact that, by construction, T k ∈ gen(Π) and Ω(T k ) agrees with T k−1 up to a summand in add Π, i.e. Ω(T k ) ∼ = T k−1 ∈ Γ-mod/ add Π. The result for C k is proved dually.
It follows from Lemma 4.3 that the families of shifted and coshifted modules each provide a family of recollements, such that the left-hand side of the recollement is constant in each family, and the right-hand side is constant across both families. More precisely, for each 0 ≤ k ≤ domdim Γ, we get a pair of recollements as follows.
We denote the intermediate extension functors in these recollements by c k and c k respectively.

Homotopy categories.
We now turn to the problem of computing the intermediate extension functor in each recollement from (4.2). To do this, it will be useful to give a new description of the shifted and coshifted algebras as endomorphism algebras in the bounded homotopy category of A-modules, rather than in the category of Γ-modules, generalising a result of Crawley-Boevey and the second author [8,Prop. 5.5] in the case that Γ is an Auslander algebra. Our proof is also somewhat simpler and more conceptual, using only standard homological algebra.
We begin with the following general considerations. Let A be a finite-dimensional algebra, E ∈ A-mod, and Γ = End A (E) op . The bounded homotopy categories K b (Γ-proj) and K b (Γ-inj) of complexes of projective and injective Γ modules respectively admit tautological functors to the unbounded derived category D(Γ), equivalences onto their images, which we treat as identifications. These subcategories may be characterised intrinsically as the full subcategories of D(Γ) on the compact and cocompact objects (in the context of additive categories) respectively. Extending the Yoneda equivalences Using the intrinsic description of K b (Γ-proj) and K b (Γ-inj) above, we see that F induces respective equivalences from the subcategories of compact and cocompact objects of T to K b (Γ-proj) and K b (Γ-inj) respectively, and thus realises thick E as a full subcategory of T (in two ways). This holds in particular when T = D(B) for some algebra B derived equivalent to Γ, such as the endomorphism algebra of a tilting or cotilting Γ-module.
Given a derived equivalence D(B) ∼ − → D(Γ), it follows from Rickard's Morita theory for derived categories [24] that the image of the stalk complex B in K b (Γ-proj) is a tilting complex for Γ with endomorphism algebra B. The preimage of this complex under the Yoneda equivalence is an object of thick E ⊆ K b (A), again with endomorphism algebra B. Similarly, the image of DB ∈ K b (B-inj) in K b (Γ-inj) is a cotilting complex, related by the dual Yoneda equivalence to another object of thick E with endomorphism algebra B.
Our conclusion is that when Γ is the endomorphism algebra of an A-module E (or more generally an object E ∈ K b (A)), any algebra B derived equivalent to Γ must also appear as an endomorphism algebra in thick E ⊆ K b (A). In general, B need not be an endomorphism algebra in A-mod. When E is a generator-cogenerator and B is one of the shifted or coshifted algebras of Γ, we may compute the relevant objects of thick E explicitly, and obtain a particularly straightforward answer.

Theorem 4.4. Let (A, E, Γ) be a Morita-Tachikawa triple with all objects basic, and let
where the first summand denotes the complex whose non-zero part is given by the first k terms of a minimal projective resolution of E, with E in degree 0, and the second denotes the stalk complex with A in degree −k. Then with the idempotent e k ∈ B k corresponding to projection onto the summand where the first summand denotes the complex whose non-zero part is given by the first k terms of a minimal injective resolution of E, with E in degree 0, and the second denotes the stalk complex with DA in degree k. Then with the idempotent e k ∈ B k corresponding to projection onto the summand DA[−k].
Proof. As usual, we only prove (a), since (b) is dual. By definition, B k is the endomorphism algebra of the k-cotilting Γ-module C k , so that the image of DB k in K b (Γ-inj) is given by an injective resolution of C k . By the preceding discussion, we need only show that the dual Yoneda equivalence maps E k to such a resolution (up to a degree shift). But this equivalence sends E k to the complex begins a minimal projective resolution of DΓ, and Π denotes as usual the maximal projective-injective summand of Γ. This complex is exact except in degree −k, and by comparing to the definition of the coshifted modules, we see that its cohomology in this degree is precisely C k , as required.
Remark 4.5. When Γ is an Auslander algebra, so A is representation-finite and add E = A-mod, the category add E 1 is equivalent to the category H from [8, §3]. Moreover, C 1 = T 1 , and so Theorem 4.4(a) recovers [8,Prop. 5.5] in this case. In contrast to the proof given in [8], we did not have to identify the module over End K b (A) (E 1 ) op corresponding to the B 1 -module DC 1 .

4.4.
Intermediate extensions. As in [8], the advantage of describing B k as the endomorphism algebra of a complex, as in Theorem 4.4, is that it allows for convenient descriptions of some of the functors in the recollements (4.2), as we will demonstrate in this section. We will also state the dual results for B k . Throughout, we treat the isomorphisms of Theorem 4.4, as well as the natural isomorphisms with which they are compatible, as identifications. We also write f k : A⊕P k−1 → P k−2 for the leftmost non-zero map in the complex E k (see Theorem 4.4(a)); this notation includes, in the case k = 1, the convention that P −1 = E (and so this object is typically not projective).
Lemma 4.6. For M ∈ A-mod, we have Proof. Under the isomorphisms of Theorem 4.4, we have recalling that these isomorphisms identify the idempotent e k with projection onto the summand A[k] of E k . Thus the adjoints to e k arise from usual tensor-hom adjunction, which immediately gives the required formula for the left adjoint, and the formula for the right. By a standard computation in the homotopy category, we have an A-module isomorphism providing the first claimed formula for r k . The second then follows by observing, directly from the definition, that E k is isomorphic to the stalk complex ker Proof. Let P M 1 → P M 0 → M → 0 be a projective presentation of M , whence we obtain the exact sequence Lemma 4.6, and there are natural isomorphisms , identifying ℓ k (M ) with the cokernel of the map . For any N ∈ A-mod, we may compute Hom K b (A) (E k , N [k]) via the exact sequence From this observation and our projective presentation of M , we may construct the commutative diagram  Proof. We first deal separately with the case k = 1. By [8,Lem. 4.2], and im f 1 = E in this case since P 0 → E is a projective cover, giving the desired result. Assume now that k ≥ 2, and denote by (f k ) * : Hom A (P k−2 , M ) → Hom A (A ⊕ P k−1 , M ) the map induced by f k , so that ℓ k (M ) = coker (f k ) * by Lemma 4.7. Since (f k ) * factors through the inclusion which we claim is the natural transformation ζ k : of ℓ k and r k when k ≥ 2, one can see that the canonical map ζ k : ℓ k → r k agrees with that coming from the Verdier localisation functor with Hom A (X k , M )/ im (f k ) * identifies the set of maps factoring through an acyclic complex, which is the kernel of the Verdier localisation functor, with Hom A (im f k , M )/ im (f k ) * .
We now state the corresponding dual results for ℓ k , r k and c k , using the notation g k : Q k−2 → Q k−1 ⊕ DA for the rightmost non-zero map in E k .

Tilting subcategories for shifted modules
When two algebras are related via tilting, a result of Miyashita provides equivalences between various subcategories of their module categories. In this section, we will first recall this result, and then provide convenient descriptions of the relevant subcategories in the case of shifted and coshifted modules. In fact, our results will hold for arbitrary special tilting or cotilting modules, in the sense of Definition 2.3.
To begin with, let Γ be any finite-dimensional k-algebra, and let T ∈ Γ-mod be a tilting module of any finite projective dimension. We set B := End Γ (T ) op and note that DT is a k-cotilting left B-module. We define, for i ≥ 0, subcategories which we refer to collectively as the tilting subcategories associated to T . If pdim T = k, both T i (T ) and C i (DT ) are zero for i > k. These are the subcategories involved in Miyashita's equivalences, which are as follows. In the case k = 1, in which T is a classical tilting module, the pair (T 0 (T ), T 1 (T )) is a torsion pair in Γ-mod, and (C 1 (DT ), C 0 (DT )) is a torsion pair in B-mod. In this case Miyashita's result recovers Brenner-Butler's famous theorem [5] (see also [1,§VI.3]), stating that the torsion class in each of these pairs is equivalent to the torsion-free class in the other.
We will now, over the course of a lemma and three propositions, calculate the tilting subcategories for a special tilting or cotilting module. In each case the proof is provided for tilting modules, and can be dualised to provide an argument for cotilting modules. Dually, if C is an I-special k-cotilting Γ-module for some projective I and k ≥ 1, then ker Hom Γ (−, C) = ker Hom Γ (−, I).
Proposition 5.3. If k ≥ 1 and T is a P -special k-tilting Γ-module for some projective P , then T 0 (T ) = gen k−1 (P ).
Dually, if C is an I-special k-cotilting module for some injective I, then C 0 (C) = cogen k−1 (I).
Proof. Assume X ∈ gen k−1 (P ), so we have an exact sequence 0 Y P k−1 · · · P 0 X 0 with P i ∈ add P . Since P ∈ add T and T is k-tilting, a standard homological argument with long exact sequences shows that for j ≥ 1 we have We prove the converse by induction on k. In the case k = 1, note that P is a direct summand of T ∈ gen P , and hence gen(P ) = gen(T ), and the latter coincides with T 0 (T ) (e.g. by [1,Thm. VI.2.5]). Now let T be P -special k-tilting for k > 1, so that there is an exact sequence with P i ∈ add P and T • ∈ add T . It follows directly from this sequence that T ′ = P ⊕ker ψ is P -special (k − 1)-tilting, and that T ′′ = P ⊕ coker ϕ is P -special 1-tilting. By induction, we may assume that gen k−2 (P ) = T 0 (T ′ ). Now let X ∈ T 0 (T ). It follows from (5.1) that T 0 (T k ) ⊆ T 0 (T k−1 ) = gen k−2 (Π), and so we have an exact sequence with P i ∈ add P . Thus we only need to see that Z ∈ gen(P ), or equivalently, by the base case of the induction, that Z ∈ T 0 (T ′′ ). We claim that Ext 1 Γ (T ′′ , Z) ∼ = Ext k Γ (T, Z) ∼ = Ext 1 Γ (T, X) = 0. The first isomorphism follows from (5.1), the second follows from (5.2), and Ext 1 Γ (T, X) = 0 by assumption since X ∈ T 0 (T ). Thus Z ∈ ker Ext 1 Γ (T ′′ , −) = T 0 (T ′′ ) = gen(P ), as required.
Proposition 5.4. If k ≥ 1 and T is a P -special k-tilting Γ-module for some projective P , then for any 0 < j < k. Dually, if C is an I-special k-cotilting module for some injective I, then C j (C) = {0} for any 0 < j < k.
Proof. Let T ′′ be the P -special 1-tilting module defined as in the proof of Proposition 5.3. As in this previous proof, it follows from the exact sequence (5.1) that ker Ext k Γ (T, −) = ker Ext 1 Γ (T ′′ , −) = T 0 (T ′′ ), and so for j = k we have T j (T ) ⊆ T 0 (T ′′ ).
On the other hand, if j = 0 then we can use Lemma 5.2 to see that Thus T j (T ) ⊆ T 0 (T ′′ ) ∩ T 1 (T ′′ ), but this intersection is {0} since T ′′ is 1-tilting, meaning that its two tilting subcategories form a torsion pair.
Proposition 5.5. If k ≥ 1 and T is a P -special k-tilting Γ-module for some projective P , then T k (T ) = ker Hom Γ (P, −).
Dually, if C is an I-special k-cotilting module for some injective I, then C k (C) = ker Hom Γ (−, I).
Proof. Since k ≥ 1, the inclusion follows from Lemma 5.2, and so it remains to show the reverse inclusion. Assume that Hom Γ (P, X) = 0, and consider again the exact sequence (5.1). It follows from this sequence that P ⊕T • is a k-tilting module, meaning that its additive closure, which is a priori contained in add(T ), is even equal to add(T ). Thus, to show that Ext i Γ (T, X) = 0 for 0 < i < k, it suffices to show that Ext i Γ (T • , X) = 0 for such i. But writing M i for the kernel of the map in (5.1) starting at P i (and M −1 = T • ), a standard homological argument shows that, for 0 < i ≤ k, we have We now apply the preceding results to the shifted and coshifted modules, to obtain the main results of this section.
Theorem 5.6. Let Γ be a finite-dimensional algebra with domdim Γ = d + 1 > 0 and maximal projective-injective module Π, and let 1 ≤ k ≤ d + 1. Write I k = D Hom Γ (Π, T k ). Then I k is an injective summand of the B k -module DT k , and the tilting subcategories associated to the shifted module T k are otherwise.
Proof. That I k is an injective summand of DT k follows from the fact that Π is a summand of both Γ and T k . The rest of the statement is a direct application of Propositions 5.3, 5.4 and 5.5, using that T k is Π-special k-tilting, and that DT k is I k -special k-cotilting (see Lemma 2.11(1)).
By combining Theorem 5.6 with Miyashita's equivalences from Theorem 5.1, we obtain the following. In particular, the categories C k (DT k ) for 1 ≤ k ≤ d + 1 are all equivalent to each other, and if k ≥ 2 there is a fully faithful functor C 0 (DT k ) → C 0 (DT k−1 ) sending DT k to DT k−1 .
Proof. The equivalences are immediate from Theorems 5.1 and 5.6. Using the second equivalence, we see that C k (DT k ) ≃ ker Hom Γ (Π, −) for any 1 ≤ k ≤ d + 1. For the final statement, the fully faithful functor is provided by the composition where the equivalences are those in Theorem 5.1, and the middle map is the natural inclusion. The first equivalence takes DT k to DΓ, which is then taken to DT k−1 by the second equivalence.
Using the versions of Propositions 5.3-5.5 for cotilting modules, we obtain the following dual statements.
Theorem 5.8. Let Γ be a finite-dimensional algebra with domdim Γ = d + 1 > 0 and maximal projective-injective module Π, and let 1 ≤ k ≤ d + 1. Write P k = Hom Γ (C k , Π). Then P k is a projective summand of the B k -module DC k , and the tilting subcategories associated to the coshifted module C k are otherwise, Corollary 5.9. In the setting of Theorem 5.6, there are equivalences of categories In particular, the categories T k (DC k ) for 1 ≤ k ≤ d + 1 are all equivalent to each other, and if k ≥ 2 there is a fully faithful functor T 0 (DC k ) → T 0 (DC k−1 ) sending DC k to DC k−1 .

Tilting modules as intermediate extensions
As usual, let Γ be a finite-dimensional algebra with domdim Γ = d+1 > 0. In this section, we assume d ≥ 1, so that Γ forms part of a Morita-Tachikawa triple (A, E, Γ), and consider the intermediate extension functors in our preferred recollements involving the shifted and coshifted algebras B k and B k of Γ, which we denote by c k and c k respectively. Our main result is that, for any 1 ≤ k ≤ d, the distinguished cotilting module DT k for the k-th shifted algebra B k of Γ is the intermediate extension c k E. Similarly, c k E = DC k is the distinguished tilting module for the coshifted algebra B k . We first give some general results, for arbitrary tilting or cotilting modules. (ii) hold for m = n = 1. For the converse, note that these conditions become stronger as m and n increase, so it suffices to show that if they hold for m = n = 1 then DT = cDΠ. In this case we have DT ∈ gen(P ) ∩ cogen(I) = im c, so DT = ceDT . But by Proposition 6.1, we have eDT = DΠ, and the result follows.
As an application of this result, we obtain the promised result for the shifted modules and algebras of the algebra Γ appearing in a Morita-Tachikawa triple. Theorem 6.3. Let (A, E, Γ) be a Morita-Tachikawa triple, so domdim Γ = d + 1 for d ≥ 1, and write Π for a maximal projective-injective summand of Γ. For each 0 ≤ k ≤ d + 1, consider the shifted module T k , its endomorphism algebra B k , and let c k : Γ-mod → B k -mod be the intermediate extension functor from the recollement in (4.2). Writing P k = Hom Γ (T k , Π) and I k = D Hom Γ (Π, T k ), we have DT k ∈ gen d−k (P k ) ∩ cogen k−1 (I k ).
We close the section by stating the dual results for coshifted modules and algebras. Again in the setting of Proposition 6.1, write P ′ = Hom Γ (C, Π), I ′ = D Hom Γ (Π, C), noting that P ′ is projective, I ′ = νP ′ is injective, and P ′ ⊕ I ′ ∈ add DC. The dual of Proposition 6.2, obtained by swapping the roles of the two algebras, is as follows. This result can then be used to prove the following dual to Theorem 6.3. Theorem 6.5. Let (A, E, Γ) be a Morita-Tachikawa triple, so domdim Γ = d + 1 for d ≥ 1, and write Π for a maximal projective-injective summand of Γ. For each 0 ≤ k ≤ d + 1, consider the coshifted module C k , its endomorphism algebra B k , and let c k : Γ-mod → B k -mod be the intermediate extension functor from the recollement in (4.2). Writing P k = Hom Γ (C k , Π) and I k = D Hom Γ (Π, C k ), we have DC k ∈ gen d−k (P k ) ∩ cogen k−1 (I k ).
modulo all paths of length 2 except that with middle vertex k + 1. It follows that gldim B k = max{n − k, k}.
Example 7.3. It can happen that the dominant dimension of a shifted algebra is again positive, allowing us to iterate sequences of shifts and coshifts. We illustrate this on the Auslander algebra of the path algebra of a linearly oriented A 3 quiver. We may compute that the first shifted algebra is