On faithfully balanced modules, F-cotilting and F-Auslander algebras
aa r X i v : . [ m a t h . R A ] J un ON FAITHFULLY BALANCED MODULES,F-COTILTING AND F-AUSLANDER ALGEBRAS
BIAO MA AND JULIA SAUTER
Abstract.
We revisit faithfully balanced modules. These are faithful modules having the doublecentralizer property. For finite-dimensional algebras our main tool is the category cogen ( M ) of mod-ules with a copresentation by summands of finite sums of M on which Hom( − , M ) is exact. For afaithfully balanced module M the functor Hom( − , M ) is a duality on these categories - for cotiltingmodules this is the Brenner-Butler theorem. We also study new classes of faithfully balanced modulescombining cogenerators and cotilting modules. Then we turn to relative homological algebra in thesense of Auslander-Solberg and define a relative version of faithfully balancedness which we call 1- F -faithful. We find relative versions of the best known classes of faithfully balanced modules (including(co)generators ,(co)tilting and cluster tilting modules). Here we characterize the corresponding modulesover the endomorphism ring of the faithfully balanced module - this is what we call a correspondence .Two highlights are the relative (higher) Auslander correspondence and the relative cotilting correspon-dence - the second is a generalization of a relative cotilting correspondence of Auslander-Solberg to aninvolution (as the usual cotilting correspondence is). Introduction
Let Λ be a ring and M a left Λ-module. We write endomorphisms of Λ M on the left, thus fortwo endomorphisms f, g ∈ End Λ ( M ) and an element m ∈ M the image of m under gf is g ( f ( m )).Then M can be considered naturally as a left End Λ ( M )-module, and moreover as a left End Λ ( M )-left Λ-bimodule. We say M is faithful / balanced / faithfully balanced if the natural map of ringsΛ → End
End Λ ( M ) ( M ) is injective/surjective/bijective. Balanced modules are also known as moduleswith the double centralizer property , see for example [DR72]. In [Wis00], M is faithfully balancedmeans Λ is M -static. In [BS98], a faithfully balanced module is a module of faithful dimension atleast 2.In this paper, we restrict to study finite-dimensional algebras and finite-dimensional modules overthem. For a module Λ M we define add( M ) to be the category consisting of direct summands of finitedirect sums of M andcogen ( M ) = { X | ∃ → X → M → M exact , M i ∈ add( M ) and Hom Λ ( − , M ) exact on it } . Dually, one can define gen ( M ). If Λ M is a faithfully balanced module, then we have a dualityHom Λ ( − , Λ M ) : cogen ( Λ M ) ←→ cogen ( Γ M ) : Hom Γ ( − , Γ M )where Γ = End Λ ( M ). Buan and Solberg [BS98] first observed the symmetry: Λ ∈ cogen ( Λ M )is equivalent to D Λ ∈ gen ( Λ M ) and both are equivalent to M being faithfully balanced (see alsoLemma 2.8). We will consider tuples (Λ , M , . . . , M t ) consisting of an algebra and several modulesup to an equivalence relation which identifies two such tuples (Λ , M , . . . , M t ) and (Λ ′ , M ′ , . . . , M ′ t )if there is a Morita equivalence from Λ to Λ ′ which sends each add( M i ) to add( M ′ i ). We denote by[Λ , M , . . . , M t ] the equivalence class of (Λ , M , . . . , M t ). It is easy to see that faithfully balancednessof a module is preserved under this equivalence (cf. [CR72]).( ∗ ) The assignment [Λ , Λ M ] [End Λ ( M ) , End Λ ( M ) M ] is an involution on the set of pairs [Λ , Λ M ]with M a faithfully balanced module. Date : June 12, 2019.2010
Mathematics Subject Classification.
Key words and phrases. tilting, cluster tilting, relative homological algebra, Auslander algebra. In [AF92, section 4], faithfully balancedness is only defined for bimodules. A faithfully balanced module Λ M in thispaper is called faithful and balanced in loc. cit., which is equivalent to say that Λ M End Λ ( M ) op is a faithfully balancedbimodule. For example in [SW09] faithfully balancedness is used for left modules as in this article. t is a generally intriguing problem to establish an Endo -dictionary explaining which properties of Λand Λ M are translated into which properties of End Λ ( M ) and End Λ ( M ) M . A restriction of ( ∗ ) to abijection between two sets of such pairs (or related tuples) will be called a correspondence .Two classes of well-studied faithfully balanced modules are (co)tilting modules [BB80, Miy86] and(co)generators [Tac69, Theorem 3] - and their special cases: generator-cogenerators [Mor71, Tac70]and Auslander generators (i.e., the additive generator of the module category [Aus99]). Starting withM¨uller’s results [M¨ul68] there are also higher versions of any of these. One of our motivations wasto understand the interplay between correspondences and relative homological algebra in the sense ofAuslander-Solberg [AS93b]. In this paper we explain a relative version of faithfully balancedness andthen systematically look at the relative analogues of the well-known correspondences. Let us give abrief overview of previously studied correspondences (see the following table) in representation theoryof finite-dimensional algebras (or more generally, artin algebras). The relative versions can be foundin the corresponding theorems in the second column. Classical case Relative case (co)generator correspondence(=Wedderburn correspondence and Hom( − , ring)) Corollary 5.17 (1) (2)Morita-Tachikawa correspondence(=generator-cogenerator correspondence) Corollary 5.17 (3)M¨uller correspondence Lemma 5.8(higher) Auslander correspondence Theorem 6.7Auslander-Solberg correspondence Theorem 6.4(co)tilting correspondence(=Brenner-Butler theorem) Theorem 8.9correspondence of Gorenstein algebras Corollary 8.15We give a summary of the content (but in the introduction we restrict to the easy versions).In section 2, we study some basic properties of faithfully balanced modules and dualities (equiva-lences) of subcategories.We start the relative theory in section 5. We consider an additive subbifunctor F ⊆ Ext ( − , − ) ofthe form F = F G = F H for a generator G and a cogenerator H - this is equivalent to consider theexact structure on finite-dimensional Λ-modules induced by the functor F (cf. [DRSS99]), meaningan exact sequence is F -exact if and only if it remains exact after applying the functor Hom Λ ( G, − )(or equivalently after applying the functor Hom Λ ( − , H )). We define cogen F ( M ) ⊆ cogen ( M ) to bethe full subcategory of modules X such that there exists an exact sequence 0 → X → M → M with M , M ∈ add( M ) and Hom Λ ( − , H ⊕ M ) is exact on it (analogously we define gen F ( M )). We alsointroduce the notion of 1- F -faithfulness (meaning G ∈ cogen F ( M )) as the relative analogue of thenotion of faithfully balancedness. Let Λ M be 1- F -faithful, then we have a dualityHom Λ ( − , Λ M ) : cogen F H ( M ) ←→ cogen F R ( M ) : Hom Γ ( − , Γ M )where Γ = End Λ ( M ) and R = D Hom Λ ( M, H ). There is also a dual version of the above duality whichinvolves the modules G and L := Hom Λ ( G, M ). Then we observe the following relationship between
G, H and
L, R Λ G D D ( − ,M ) (cid:26) (cid:26) τ ( ( ❤ ❞ ❴ ❬ ❱ Λ H τ − h h ❤❞❴❬❱ Γ L Ω − M ( ( ❧ ❤ ❞ ❴ ❬ ❱ ❘ Γ R (cid:4) (cid:4) D( M, − ) Z Z Ω M h h ❧❤❞❴❬❱❘ Here the upper dashed arrows mean H = τ G ⊕ D Λ and G = τ − H ⊕ Λ whereas the lower dashedarrows mean R = Γ M ⊕ Ω − M L and L = Γ M ⊕ Ω M R . As in the classical case, we have G ∈ cogen F ( M )is equivalent to H ∈ gen F ( M ) (Theorem 5.6). e consider the assignment (AS) (referring to Auslander and Solberg):(AS) The assignment [Λ , Λ M, G ] [Γ = End Λ ( M ) , Γ M, L = Hom Λ ( G, M )] with M faithfully bal-anced and G a generator. Generator correspondence.
A generator Λ G is, by definition, a module such that Λ ∈ add( G )which is automatically a faithfully balanced module. Theorem 1.1. (Generator correspondence, [Azu66, Tac69]) The assignment ( ∗ ) restricts to a bijection { [Λ , G ] : G is a generator } ←−−→ { [Γ , P ] : P is a f.b. projective module } where f.b. is the abbreviation of faithfully balanced.The generator correspondence can also be expressed as Auslander’s Wedderburn correspondencecomposed with the duality Hom Γ ( − , Γ), see [AS93d].Our relative generalization is the following
Theorem 1.2. (= Corollary 5.17 (1)) The assignment (AS) restricts to an involution on the set oftriples [Λ , M, G ] with Λ ⊕ M ∈ add( G ) and M is - F G - faithfulIn Corollary 5.17 we also give relative versions of the cogenerator correspondence and the generator-cogenerator correspondence (also known as the Morita-Tachikawa correspondence [Mor71, Tac70], seealso [Rin07]). The most famous special case is the Auslander correspondence, see below. Auslander-Solberg and Auslander correspondence . The Auslander-Solberg correspondence,which is defined by Iyama and Solberg [IS18], characterizes algebras Λ with domdim Λ ≥ k + 1 ≥ id Λ.In the case k = 1, this result is due to Auslander-Solberg [AS93a].Our relative generalization is the following Theorem 1.3. (= Theorem 6.4, k = 1) The assignment (AS) restricts to an involution on the set oftriples [Λ , M, G ] with Λ ∈ add( G ) , F = F G , M is both 1 - F - faithful and F - projective - injective , anddomdim F Λ ≥ ≥ id F G . As a special case of the Auslander-Solberg correspondence, Iyama’s higher Auslander correspon-dence ([Iya07]) characterizes algebras Λ with domdim Λ ≥ k + 1 ≥ gldim Λ. The case k = 1 is thewell-known Auslander correspondence [Aus99].Our relative generalization is the following Theorem 1.4. (= Theorem 6.7, k = 1) The assignment (AS) restricts to an involution on the set oftriples [Λ , M, G ] with Λ ∈ add( G ) , F = F G , M is both 1 - F - faithful and F - projective - injective , anddomdim F Λ ≥ ≥ gldim F Λ . Cotilting correspondence . The main result on relative cotilting modules (cf. Definition 8.1) ofAuslander-Solberg is the following
Theorem 1.5. ([AS93c, Theorem 3.13] , [AS93d, Theorem 2.8]) The assignment (AS) restricts to abijection between the following two sets of triples(1) [Λ , M, G ] with Λ ∈ add( G ) , F = F G , M is F -cotilting, and(2) [Γ , N, L ] with N ∈ add( L ) , L ∈ cogen ( N ) and L is a cotilting module.To improve this result, we need the 4-tuple assignment[Λ , M, L, G ] [Γ , N, e L, e G ]with Γ = End Λ ( M ), N = Γ M , e L = Hom Λ ( G, M ), e G = Hom Λ ( L, M ). Then we have
Theorem 1.6. (= Theorem 8.9)
The -tuple assignment restricts to an involution on the set of -tuples [Λ , M, L, G ] satisfying Λ ∈ add( G ) , F = F G , L is F -cotilting and L ∈ cogen F ( M ) . It is well known that a cotilting module will induce a triangle duality, see [Hap88, CPS86]. We provea relative analogue of this result (Proposition 8.12): In the situation of the previous theorem we have atriangle duality between D b F G (Λ-mod) and D b F e G (Γ-mod) where Γ = End Λ ( M ) and e G = Hom Λ ( L, M ).We illustrate the above results by the following easy examples which are special cases of F -Auslanderalgebras from Example 6.9(4). xample 1.7. (1) Let Λ be the path algebra of the quiver 1 → → M = P ⊕ ( P ⊕ τ − P ), G = P ⊕ M and H = I ⊕ M . Then we have F G = F H =: F . It iseasy to see that domdim F Λ = 2 = gldim F Λ, and hence Λ is a 1- F -Auslander algebra. Now,we see End Λ ( M ) ∼ = Λ, End Λ ( M ) M ∼ = Λ M and Hom Λ ( G, M ) ∼ = Λ G . It follows that the triple[Λ , M, G ] is a fixed point of the assignment (AS).(2) The same idea leads to a 2- F -Auslander algebra structure (i.e. domdim F Λ ≥ ≥ gldim F Λ)on Λ = K (1 → → → M = P ⊕ ( P ⊕ τ − P ) ⊕ ( P ⊕ τ − P ⊕ τ − P ), G = P ⊕ M and H = I ⊕ M . Then we have F G = F H =: F and one easily sees Λ isa 2- F -Auslander algebra. We also define L = τ − P ⊕ M , the F -exact sequence sequence0 → L → τ − P ⊕ M → τ − P ⊕ M → H → L is a 2- F -cotiltingmodule and L ∈ cogen F ( M ). Then we have • (cid:27) (cid:27) ✼✼✼✼ Γ = End Λ ( M ) : • C C ✞✞✞✞ (cid:27) (cid:27) ✼✼✼✼ • (cid:27) (cid:27) ✼✼✼✼ Γ M =
10 10 0 1 ⊕
11 10 1 1 ⊕
11 11 1 0 ⊕
11 01 0 0 • C C ✞✞✞✞ • C C ✞✞✞✞ • e G := Hom Λ ( L, M ) =
00 10 1 0 ⊕ Γ, e H :=
10 10 0 1 ⊕ D Γ and F e G = F e H =: e F . We also define e L := Hom Λ ( G, M ) =
00 00 0 1 ⊕
00 10 0 1 ⊕
00 10 1 1 ⊕ Γ M and an e F -exact sequence0 → e L → Γ M ⊕ (
10 10 0 1 ) ⊕
11 10 1 1 → Γ M ⊕ (
11 01 0 0 ) ⊕
11 11 1 0 → e H → . can be used to see that e L is a 2- e F -cotilting module and e L ∈ cogen e F ( M ). This exampleis an instance of a more general class of examples which we call special (co)tilting modulessystematically studied in section 9, for this particular example see subsubsection 9.1.1. Acknowledgements:
The first author is supported by the China Scholarship Council. The secondauthor is supported by the Alexander von Humboldt Foundation in the framework of the Alexandervon Humboldt Professorship endowed by the German Federal Ministry of Education and Research.The second author also wishes to thank William Crawley-Boevey who contributed Lemma 2.2.2.
On categories generated or cogenerated by a module
We fix a finite-dimensional algebra Λ (over a field K ) and denote by Λ-mod the category of finitelygenerated (or equivalently, finite-dimensional) left Λ-modules. Let M ∈ Λ-mod and Γ = End Λ ( M ) beits endomorphism ring. Then M can be naturally viewed as a left Γ-module. We write Γ M when weconsider M as a left Γ-module. We will study the following four contravariant functorsHom Λ ( − , M ) : Λ-mod ←→ Γ-mod : Hom Γ ( − , M )D Hom Λ ( M, − ) : Λ-mod ←→ Γ-mod : D Hom Γ ( M, − )where D = Hom K ( − , K ) is the standard K -dual functor.In order to keep the formulas and diagrams in reasonable length we will often use the conventions( − , Λ M ) := Hom Λ ( − , M ) and D( Λ M, − ) := D Hom Λ ( M, − ). If there is no ambiguity we may omit thesubscript and write ( − , Λ M ) (or ( − , Γ M )) as ( − , M ).We begin with the Yoneda embedding which is known as projectivization ([ARS95]). Lemma 2.1. ([AS93c, Lemma 3.3][ARS95, Proposition 2.1])
Let M ∈ Λ - mod and Γ = End Λ ( M ) . (1) (( − , Λ M ) , ( − , Γ M )) is an adjoint pair of contravariant functors and it restricts to a duality ( − , Λ M ) : add( Λ M ) ←→ add(Γ) = P (Γ) : ( − , Γ M ) . (2) (D( Λ M, − ) , D( Γ M, − )) is an adjoint pair of contravariant functors and it restricts to a duality D( Λ M, − ) : add( Λ M ) ←→ add(D Γ) = I (Γ) : D( Γ M, − ) . or every non-negative integer k we associate to a module M ∈ Λ-mod two full subcategories ofΛ-modcogen k ( M ) := (cid:26) N (cid:12)(cid:12)(cid:12)(cid:12) ∃ exact seq. 0 → N → M → · · · → M k with M i ∈ add( M ) , and s.t.( M k , M ) → · · · → ( M , M ) → ( N, M ) → (cid:27) gen k ( M ) := (cid:26) N (cid:12)(cid:12)(cid:12)(cid:12) ∃ exact seq. M k → · · · → M → N → M i ∈ add( M ) , and s.t.( M, M k ) → · · · → ( M, M ) → ( M, N ) → (cid:27) . Recall that a map f : N → M with M ∈ add( Λ M ) is called a left add( M )-approximation if the mapHom Λ ( f, M ) : Hom Λ ( M , M ) → Hom Λ ( N, M ) is an epimorphism, and this approximation is calledminimal if any endomorphism θ : M → M satisfying θf = f is an automorphism. It is well-knownthat every left add( M )-approximation has a minimal version which is unique up to isomorphism,see [ARS95, Theorem 2.4]. Dually, we can define right (minimal) add( M )-approximation. We definecogen ∞ ( M ) to be the full subcategory consisting of modules N such that there exists an exact sequence0 → N f −→ M f −→ M · · · f n −→ M n → · · · such that f i factors through coker f i − → M i which is aminimal left add( M )-approximation for every i ≥
0. The definition of gen ∞ ( M ) is dual.The following lemma will be used frequently, the case k = 0 is well known and can be found in[ASS06, Lemma VI 1.8]. Lemma 2.2.
Let ≤ k ≤ ∞ . (1) The following are equivalent for N ∈ Λ - mod . (1a) N ∈ cogen k ( M ) . (1b) The natural map N → Hom Γ (Hom Λ ( N, M ) , M ) = (( N, M ) , M ) , n ( f f ( n )) is anisomorphism and Ext i Γ (Hom Λ ( N, M ) , M ) = 0 for ≤ i ≤ k − . (2) The following are equivalent for N ∈ Λ - mod . (2a) N ∈ gen k ( M ) . (2b) The natural map D( M, D( M, N )) ∼ = Hom Λ ( M, N ) ⊗ Γ M → N , f ⊗ m f ( m ) is anisomorphism and Ext i Γ ( M, D( M, N )) = 0 for ≤ i ≤ k − .Proof. (1) Let N ∈ cogen k ( M ), that means we have an exact sequence0 → N → M → · · · → M k with M i ∈ add( M ) and such that the functor Hom Λ ( − , M ) is exact on it, i.e., we get an exactsequence ( M k , M ) → · · · → ( M , M ) → ( N, M ) → . This sequence is a projective resolution of Hom Λ ( N, M ) as a left Γ-module. Applying thefunctor Hom Γ ( − , M ) to it yields a complex0 → (( N, M ) , M ) → (( M , M ) , M ) → · · · → (( M k , M ) , M ) . Now, consider the natural map N → Hom Γ (Hom Λ ( N, M ) , M ), this gives a commutative dia-gram, 0 / / (( N, M ) , M ) / / (( M , M ) , M ) / / · · · / / (( M k , M ) , M )0 / / N O O / / M O O / / · · · / / M k O O The map M ′ → Hom Γ (Hom Λ ( M ′ , M ) , M ) is an isomorphism for M ′ ∈ add( M ) because it is inthe case of M ′ = M . This implies that all vertical maps are isomorphisms, in particular N → Hom Γ (Hom Λ ( M, N ) , M ) is an isomorphism and since the second row is exact, the complex inthe first row is also exact. This implies Ext i Γ (Hom Λ ( N, M ) , M ) = 0 for 1 ≤ i ≤ k − Λ ( N, M )as a left Γ-module as follows( M k , M ) → · · · → ( M , M ) → ( N, M ) → nd apply Hom Γ ( − , M ) to compute Ext i Γ (Hom Λ ( N, M ) , M ), 1 ≤ i ≤ k −
1. Since by assump-tion Ext i Γ (Hom Λ ( N, M ) , M ) = 0, 1 ≤ i ≤ k − N → Hom Γ (Hom Λ ( N, M ) , M ) is anisomorphism. The complex gives an exact sequence0 → N → M → · · · → M k . If we apply Hom Λ ( − , M ) to this sequence we get the projective resolution from before, so itis exact which shows that N is in cogen k ( M ).(2) By using the facts that N ∈ gen k ( M ) if and only if D N ∈ cogen k (D M ) and End Λ op (D M ) ∼ =End Λ ( M ) op , we see that the statement (2) can be deduced from the right module version of(1). (cid:3) Corollary 2.3.
For ≤ k ≤ ∞ , the categories cogen k ( M ) and gen k ( M ) are closed under direct sumsand summands. Furthermore, we have cogen ∞ ( M ) = \ ≤ k< ∞ cogen k ( M ) , gen ∞ ( M ) = \ ≤ k< ∞ gen k ( M ) . We will need the following useful lemma which already appeared for the specific situation of a relativecotilting module in [AS93c, Lemma 3.3 (b)] and [AS93c, Proposition 3.7]. For a finite-dimensionalalgebra Λ we write ν Λ = D( − , Λ) , ν − Λ = (D Λ , − ) for the Nakayama functors (cf. [ASS06]). Lemma 2.4.
Let M ∈ Λ - mod and Γ = End Λ ( M ) . (1) A module X ∈ cogen ( M ) if and only if the natural map Hom Λ ( Y, X ) → Hom Γ (( X, M ) , ( Y, M )) is an isomorphism for all Y ∈ Λ - mod . Furthermore, in this case we have ν Γ ( X, M ) = D((
X, M ) , ( M, M )) ∼ = D( M, X ) . Dually, a module Y ∈ gen ( M ) if and only if the natural map Hom Λ ( Y, X ) → Hom Γ (D( M, X ) , D( M, Y )) is an isomorphism for all X ∈ Λ - mod . Furthermore, in this case ν − Γ D( M, Y ) = (D(
M, M ) , D( M, Y )) ∼ = ( Y, M ) . (2) For k ≥ , X ∈ cogen k +1 ( M ) if and only if the natural maps Ext i Λ ( Y, X ) → Ext i Γ (( X, M ) , ( Y, M )) , ≤ i ≤ k are isomorphisms for all Y ∈ T ki =1 ker Ext i Λ ( − , M ) . Dually, Y ∈ gen k +1 ( M ) if and only if thenatural maps Ext i Λ ( Y, X ) → Ext i Γ (D( M, X ) , D( M, Y )) , ≤ i ≤ k are isomorphisms for all X ∈ T ki =1 ker Ext i Λ ( M, − ) .Proof. (1) Assume X ∈ cogen ( M ), then there exists an exact sequence 0 → X → M → M suchthat M i ∈ add( M ) and Hom Λ ( − , M ) is exact on it. We apply Hom Λ ( Y, − ) to get an exactsequence 0 → Hom Λ ( Y, X ) → Hom Λ ( Y, M ) → Hom Λ ( Y, M ) . Now, we consider the commutative diagram0 / / ( Y, X ) / / ( − ,M ) (cid:15) (cid:15) ( Y, M ) ( − ,M ) ∼ = (cid:15) (cid:15) / / ( Y, M ) ( − ,M ) ∼ = (cid:15) (cid:15) / / (( X, M ) , ( Y, M )) / / (( M , M ) , ( Y, M )) / / (( M , M ) , ( Y, M )) . The second row also can be obtained by applying first Hom Λ ( − , M ) then Hom Γ ( − , ( Y, M )) tothe exact sequence 0 → X → M → M , so it remains exact. The induced isomorphism of thekernels is the map in the claim. Conversely, by taking Y = Λ we obtain a natural isomorphism X ∼ = −→ (( X, M ) , M ) which implies X ∈ cogen ( M ).
2) Assume X ∈ cogen k +1 ( M ), then we have an exact sequence 0 → X → M → · · · → M k +1 suchthat M i ∈ add( M ) and Hom Λ ( − , M ) is exact on it. Applying Hom Λ ( − , M ) yields an exact se-quence ( M k +1 , M ) → · · · → ( M , M ) → ( X, M ) → X, M )as a left Γ-module. Now assume Y ∈ T ki =1 ker Ext i Λ ( − , M ). To compute Ext i Γ (( X, M ) , ( Y, M ))for 1 ≤ i ≤ k we apply Hom Γ ( − , ( Y, M )) to this projective resolution and delete the term((
X, M ) , ( Y, M )) to get a complex · · · → → (( M , M ) , ( Y, M )) → (( M , M ) , ( Y, M )) →· · · → (( M k +1 , M ) , ( Y, M )) → → · · · which fits into the following commutative diagram · · · / / / / ( Y, M ) / / ( − ,M ) ∼ = (cid:15) (cid:15) ( Y, M ) / / ( − ,M ) ∼ = (cid:15) (cid:15) · · · / / ( Y, M k +1 ) ( − ,M ) ∼ = (cid:15) (cid:15) / / / / · · ·· · · / / / / (( M , M ) , ( Y, M )) / / (( M , M ) , ( Y, M )) / / · · · / / (( M k +1 , M ) , ( Y, M )) / / / / · · · where the complex in the first row is obtained by applying Hom Λ ( Y, − ) to 0 → X → M →· · · → M k +1 and deleting the term ( Y, X ). Our assumption Y ∈ T ki =1 ker Ext i Λ ( − , M ) impliesthat the i-th cohomology of the first row is Ext i Λ ( Y, X ). Now the isomorphism of the twocomplexes induces the claimed natural isomorphisms. To prove the other implication, justtake Y = Λ. (cid:3) We also prove the following simple criterion.
Lemma 2.5.
Let
M, N ∈ Λ - mod and Γ = End( M ) . If N and Hom Γ (Hom Λ ( N, M ) , M ) are isomorphicas Λ -modules, then we have N ∈ cogen ( Λ M ) .Proof. The essential image of the functor ( − , M ) is contained in cogen( M ) since if Y = ( Z, M ), thenwe may choose a projective cover P → Z and apply ( − , M ) to see that Y ∈ cogen( M ).This means N ∼ = (( N, M ) , M ) ∈ cogen( M ). This implies that the natural map N → (( N, M ) , M )mapping n ( f f ( n )) is a monomorphism. Since both vector spaces have the same dimension itis an isomorphism. This implies by Lemma 2.2 that N ∈ cogen ( M ). (cid:3) Faithfully balanced modules.
Faithfully balanced modules can be defined for any ring. Forfinite-dimensional algebras, Lemma 2.2 allows us to give the following internal definition.
Definition 2.6.
We call a finitely generated (left or right) Λ-module M faithfully balanced if Λ ∈ cogen ( M ).The following surprising and also well-known result says every module becomes faithfully balancedwhen considering as a module over its endomorphism ring. Lemma 2.7. ([AF92, Proposition 4.12] [AS93a, Lemma 2.2])
Let M ∈ Λ - mod and Γ = End Λ ( M ) and consider M as a left Γ -module. Then Γ M is faithfully balanced. In [BS98], a faithfully balanced module is also known as a module of faithful dimension at least 2.The following lemma (the same as [BS98, Proposition 2.2]), which characterizes modules of faithfuldimension at least k + 1, can be obtained as an immediate consequence of Lemma 2.2. Lemma 2.8.
The following are equivalent for every ≤ k ≤ ∞ . (1) Λ ∈ cogen k ( M ) . (2) The natural map Λ → End Γ ( M ) is an isomorphism and Ext i Γ ( M, M ) = 0 , ≤ i ≤ k − . (3) D Λ ∈ gen k ( M ) .Proof. The equivalence between (1) and (2) is a special case of Lemma 2.2. The equivalence to (3)follows again by seeing that the equivalence between (1) and (2) also works for right modules. Thenpass with the duality from the right module statement for (1) to (3). (cid:3)
The following lemma plays a fundamental role in this paper.
Lemma 2.9. ( cf. [Xi00, Proposition 5.1]) Let M be a faithfully balanced Λ -module and Γ = End Λ ( M ) .Then the functors ( − , Λ M ) : Λ - mod ←→ Γ - mod : ( − , Γ M ) restrict to a duality of categories cogen ( Λ M ) ←→ cogen ( Γ M ) . hey restrict further to a duality cogen k ( Λ M ) ←→ cogen ( Γ M ) ∩ k − \ i =1 ker Ext i Γ ( − , Γ M ) . Dually, the functors D( Λ M, − ) : Λ - mod ←→ Γ - mod : D( Γ M, − ) restrict to a duality of categories gen ( Λ M ) ←→ gen ( Γ M ) . They restrict further to a duality gen k ( Λ M ) ←→ gen ( Γ M ) ∩ k − \ i =1 ker Ext i Γ ( Γ M, − ) . Proof.
By Lemma 2.4 the functor ( − , Λ M ) is fully faithful on cogen ( Λ M ). Let Λ-Γ M be a Λ-Γ-bimodule and Λ N a left Λ-module, and Γ N ′ a left Γ-module. We denote by α N : N → (( N, M ) , M )and α N ′ : N ′ → (( N ′ , M ) , M ) the two natural maps. Then the compositions( N, M ) α ( N,M ) / / ((( N, M ) , M ) , M ) ( α N ,M ) / / ( N, M )( N, M ) α ( N ′ ,M ) / / ((( N ′ , M ) , M ) , M ) ( α N ′ ,M ) / / ( N ′ , M )are both identities, since by Lemma 2.1 the functors ( − , Λ M ) and ( − , Γ M ) form an adjoint pair.Therefore, if α N (resp. α N ′ ) is an isomorphism, then so is α ( N,M ) (resp. α ( N ′ ,M ) ). Since M isfaithfully balanced the dualities follow from Lemma 2.2. (cid:3) Remark 2.10.
We have already seen in Lemma 2.2 that cogen ( M ) consists of the modules N suchthat α N is an isomorphism. It is also straightforward to see that cogen( M ) consists of the modules N with α N a monomorphism.If we now consider a faithfully balanced Λ-module M , Γ = End Λ ( M ) and Im( − , M ) the essentialimage of the functor ( − , M ), then we havecogen ( Γ M ) ⊆ Im( − , M ) ⊆ cogen( Γ M ) . Let Im( − , M ) ⊕ be the full subcategory of Γ-mod whose objects are summands of modules in Im( − , M ).Then it is easy to see from the previous proof that Im( − , M ) ⊕ consists of those modules N such that α N is a split monomorphism.If Λ M is a cogenerator, then Im( − , M ) = cogen ( Γ M ) and in particular Im( − , M ) is closed undersummands in this case. Corollary 2.11.
Let k ≥ . Let M ∈ Λ - mod be faithfully balanced and assume id Γ M ≤ k − , thenwe have cogen k ( M ) = cogen k +1 ( M ) = · · · = cogen ∞ ( M ) . Corollary 2.12.
Let k ≥ and M be a faithfully balanced Λ -module and Ext i Λ ( M, M ) = 0 for ≤ i ≤ k − . Then we have (1) The functors ( − , Λ M ) , ( − , Γ M ) restrict to a duality { M ′ ∈ add( Λ M ) | pd M ′ ≤ k } ←→ { P ∈ add(Γ) | Ω − ( k +1) M P = 0 } . (2) The functors D( Λ M, − ) , D( Γ M, − ) restrict to a duality { M ′ ∈ add( Λ M ) | id M ′ ≤ k } ←→ { J ∈ add(D Γ) | Ω ( k +1) M J = 0 } . Proof.
It is straightforward to check that the duality from Lemma 2.9 restricts to these equivalences. (cid:3)
We also recall the following result of Wakamatsu.
Theorem 2.13.
Let M be a faithfully balanced Λ -module and Γ = End Λ ( M ) . Assume M is self-orthogonal both as left Λ -module and right Γ -module ( i.e., Ext > ( Λ M, Λ M ) = 0 = Ext > ( Γ M, Γ M ) ) .Then we have the following (1) If id Λ M < ∞ and id Γ M < ∞ ( or resp. pd Λ M, pd Γ M < ∞ ) , then they are equal. If id Λ M, id Γ M < ∞ ( or resp. pd Λ M, pd Γ M < ∞ ) , then we have | Γ | = | Λ M | = | Γ M | = | Λ | and M is cotilting ( or resp. tilting ) .Proof. (1) is the main result in [Wak88]. If id Λ M, id Γ M < ∞ , then it follows from the previouscorollary that the is a k such that Ω kM D Λ = 0, this implies that M is cotilting and in particular | Λ M | = | Λ | . (cid:3) Example 2.14.
Assume Λ is a self-injective algebra. Then a finite-dimensional Λ-module is a faithfullybalanced if and only if it is a cogenerator. In particular, any faithfully balanced module has at least | Λ | summands.3. Dualizing summands and the Auslander-Solberg assignment
Auslander and Solberg introduced (in [AS93d, section 2]) the following notion.
Definition 3.1.
Let
M, L ∈ Λ-mod and assume M is a summand of L . We say M is a dualizingsummand of L if L ∈ cogen ( M ). For k ≥
0, we say M is a k -dualizing summand if L ∈ cogen k ( M ).Thus a dualizing summand of L is the same as a 1-dualizing summand of L .By using the duality from Lemma 2.9 it is easy to find modules having a given faithfully balancedmodule as a dualizing summand. Corollary 3.2.
Let M be a faithfully balanced Λ -module and Γ = End( M ) . Then the assignments G ( G, M ) , L ( L, M ) give inverse bijections between (1) isomorphism classes of Λ G ∈ cogen ( M ) with Λ ∈ add( G ) , and (2) isomorphism classes of modules L ∈ Γ - mod having Γ M as a dualizing summand. Lemma 3.3.
Let
M, L ∈ Λ - mod , Γ = End Λ ( M ) and assume M is a summand of L . Then M is adualizing summand of L if and only if cogen ( L ) = cogen ( M ) .Proof. The “if” part is obvious. For the “only if” part, assume M is a dualizing summand of L and X ∈ cogen ( L ). Then there exists an exact sequence 0 → X → L X → L X with L Xi ∈ add( L ) and ( − , L )exact on it. We apply ( − , Λ M ) to it and the resulting complex remains exact, since M ∈ add( L ). Nowapply ( − , Γ M ) to see X ∼ = (( X, M ) , M ). This proves cogen ( L ) ⊆ cogen ( M ). To prove cogen ( M ) ⊆ cogen ( L ), take any Y ∈ cogen ( M ) and take the minimal left add( L )-approximations f : Y → L Y and coker f → L Y . Then we get a complex 0 → Y f −→ L Y → L Y . We need to show it is exact. Byconstruction, we will obtain an exact sequence ( L Y , M ) → ( L Y , M ) → ( Y, M ) → − , Λ M ). Now apply ( − , Γ M ) to yield an exact sequence 0 → (( Y, M ) , M ) → (( L Y , M ) , M ) → (( L Y , M ) , M ) which is naturally isomorphic to the complex 0 → Y f −→ L Y → L Y , as desired. (cid:3) There is another subcategory of Λ-mod that is closely related to cogen k ( M ):copres k ( M ) := { N | ∃ exact seq. 0 → N → M → · · · → M k with M i ∈ add( M ) } . This subcategory is useful in characterizing tilting modules (see [Wei10]). It follows from the definitionsthat cogen ( M ) = cogen( M ) = copres ( M ) and cogen k ( M ) ⊆ copres k ( M ) for any M and k ≥
1. Inparticular, if M is injective then cogen k ( M ) = copres k ( M ) for any k ≥
0. We observe the following
Lemma 3.4.
Let
M, N ∈ Λ - mod and L = M ⊕ N . For k ≥ , if N ∈ cogen k ( M ) ( i.e., M is a k -dualizing summand of L ) , then M is faithfully balanced if and only if L is faithfully balanced. Inthis case we have cogen k ( M ) = cogen k ( L ) . Furthermore, if additionally copres k ( L ) = cogen k ( L ) ,then we also have copres k ( M ) = cogen k ( M ) .Proof. According to Lemma 3.3, we may assume k >
1. Since L ∈ cogen k ( M ) ⊆ cogen ( M ), itfollows from Lemma 3.3 that cogen ( L ) = cogen ( M ) and hence M is faithfully balanced if andonly if L is faithfully balanced. Let us from now on assume that M, L are faithfully balanced. Wewant to see that cogen k ( L ) = cogen k ( M ). Let Γ = End Λ ( M ). Since L ∈ cogen ( M ) we can find agenerator G ∈ Γ-mod such that L = ( G, M ) by Corollary 3.2. We observe that L ∈ cogen k ( M ) impliesExt i Γ (( L, M ) , M ) = Ext i Γ ( G, M ) = 0 for 1 ≤ i ≤ k −
1. In other words Γ M ∈ T k − i =1 ker Ext i Γ ( G, − ). Butsince G is a generator we have that gen k +1 ( G ) = Γ-mod. We set B = End Λ ( L ) ∼ = End Γ ( G ) op and take ∈ cogen k ( Λ M ). Now, observe ( X, L ) ∼ = ((( X, M ) , M ) , ( G, M )) ∼ = ( G, ( X, M )) is an isomorphism ofleft B -modules. The dual statement in Lemma 2.4 (2) gives that we have natural isomorphismsExt i Γ (( X, M ) , M ) → Ext iB (( G, ( X, M )) , ( G, M )) ∼ = Ext iB (( X, L ) , L )for 1 ≤ i ≤ k −
1. This implies by Lemma 2.2 that cogen k ( M ) = cogen k ( L ). Furthermore, sincecogen k ( M ) ⊆ copres k ( M ) ⊆ copres k ( L ) are always fulfilled, an equality cogen k ( M ) = copres k ( L )implies they are all equal. (cid:3) Remark 3.5.
Let M be a faithfully balanced module. Morita [Mor58, Theorem 1.1] has shown thatfor every indecomposable module N the following are equivalent:(1) M ⊕ N is faithfully balanced,(2) N ∈ gen( M ) or N ∈ cogen( M ).In particular, M ⊕ P ⊕ I is faithfully balanced for every projective module P and injective module I . Example 3.6.
Let H be a cogenerator, then every summand of H of the form D Λ ⊕ X is a k -dualizingsummand for every k ≥ , M, G ) where Λ is a finite-dimensional algebra and M and G arefinite-dimensional left Λ-modules. We define the following equivalence relation between these triples:(Λ , M, G ) is equivalent to (Λ ′ , M ′ , G ′ ) if there is a Morita equivalence Λ-mod → Λ ′ -mod restrictingto equivalences add( M ) → add( M ′ ) and add( G ) → add( G ′ ). We denote by [Λ , M, G ] the equivalenceclass of a triple. Definition 3.7.
We consider the following assignment[Λ , M, G ] [Γ , N, L ]with Γ = End( M ), N = Γ M , L = ( G, M ) and call this the Auslander-Solberg assignment.There is a dual assignment [Λ , M, H ] [Γ , N, R ]with Γ , N as before and R = D( M, H ) which we call the dual Auslander-Solberg assignment.From Corollary 3.2 we see that the Auslander-Solberg assignment gives a one-to-one correspondencebetween the following(1) [Λ , M, G ] with Λ ∈ add( G ), G ∈ cogen ( M ),(2) [Γ , N, L ] with N ∈ add( L ), Γ ⊕ L ∈ cogen ( N ).The previous bijection has an obvious dual version using the dual Auslander-Solberg assignment andgen, H and R instead of cogen, G and L , respectively.We are going to refine this assignment, our first refinement needs the following definition. Herewe denote for Γ-modules N and X by Ω N X the kernel of the minimal right add( N )-approximation N X → X . For k ≥ kN X := Ω N X if k = 1 and Ω kN X := Ω N (Ω k − N X ) for k ≥
2. Dually, we define Ω − N X as the cokernel of a minimal left add( N )-approximation X → N X andΩ − kN X inductively as before. Definition 3.8.
Let k be a non-negative integer and L, N, R ∈ Λ-mod. An exact sequence0 → L → N → N → · · · → N k → R → k -add( N )-dualizing sequence from L to R if(i) N i ∈ add( N ) for i ∈ { , . . . , k } ,(ii) the functors ( − , N ) and D( N, − ) are exact on it,(iii) add( R ) = add( N ⊕ Ω − ( k +1) N L ) and add( L ) = add( N ⊕ Ω k +1 N R ).In this case we say L is the left end and R is the right end of this k -add( N )-dualizing sequence.This has the following consequences for the ideal quotient categories add( L ) / add( N ) , add( R ) / add( N )(for the definition of an ideal quotient category see [ASS06, A.3]): Lemma 3.9.
Let → L → N → N → · · · → N k → R → be a k - add( N ) -dualizing sequence from L to R for some k ≥ in Λ - mod . Then we have an equivalence Ω − ( k +1) N : add( L ) / add( N ) ←→ add( R ) / add( N ) : Ω k +1 N . roof. We claim that given a short exact sequence η : 0 → U f −→ N g −→ V → N ∈ add( N )and such that the functors ( − , N ) and ( N, − ) are exact on it, then we have an equivalence Ω − N :add( U ) / add( N ) ↔ add( V ) / add( N ) : Ω N . Take a map α : X → Y in add( U ) and consider thefollowing commutative diagram η X : 0 / / X f X / / α (cid:15) (cid:15) N X g X / / β (cid:15) (cid:15) ✤✤✤ Ω − N X / / γ (cid:15) (cid:15) ✤✤✤ η Y : 0 / / Y f Y / / N Y g Y / / Ω − N Y / / η X and η Y are both direct summands of η by our assumption. In particular, we have Ω N Ω − N X ∼ = X and Ω N Ω − N Y ∼ = Y in add( U ) / add( N ). Assume there is another map β ′ : N X → N Y such that β ′ f X = f Y α and denote by γ ′ the induced map on cokernels. Then we have ( β − β ′ ) f X = 0 andthus there exists a unique θ : Ω − N X → N Y such that β − β ′ = θg X . It follows that γ − γ ′ = g Y θ .Now assume α factors as α = α α though an object N ′ ∈ add( N ), then since f X is a left add( N )-approximation there is a map φ : N X → N ′ such that α = φf X . Thus we have α = α α = ( α φ ) f X ,and a diagram chasing gives a map ψ : Ω − N X → N Y such that γ = g Y ψ . These proves that the mapHom add( U ) / add( N ) ( X, Y ) → Hom add( V ) / add( N ) (Ω − N X, Ω − N Y ) , α γ is well defined. Similarly, we havea map Hom add( V ) / add( N ) (Ω − N X, Ω − N Y ) → Hom add( U ) / add( N ) ( X, Y ) , γ α . Clearly, these two mapsare mutually inverse and this proves the claim. Now the lemma follows by induction on k . (cid:3) Let X ∈ Λ-mod and k ≥ τ k X = τ (Ω k − X ) and τ − k X := τ − (Ω − ( k − X ).We occasionally use the conventions X ⊥ ∼ k := T ki =1 ker Ext i ( X, − ) and ∼ k ⊥ X := T ki =1 ker Ext i ( − , X ). Lemma 3.10.
Let M be a faithfully balanced Λ -module and Γ = End Λ ( M ) . Then, for k ≥ , theassignment X, Y ( X, M ) , D( M, Y ) gives a self-inverse bijection ( up to seeing X, Y as Λ or as Γ -modules ) between the following sets of pairs of Λ -modules and Γ -modules { Λ G, Λ H | G = τ − k H ⊕ Λ ∈ cogen ( M ) ∩ ∼ ( k − ⊥ MH = τ k G ⊕ D Λ ∈ gen ( M ) ∩ M ⊥ ∼ ( k − } and { Γ L, Γ R | ∃ a k - Γ M -dualizing sequence from L to R } . If all modules are basic, we have D( M, τ k G ) = Ω − ( k +1) M ( G, M ) .Proof. The bijection follows from Lemma 2.9 and the observation ν Γ ( M ′ , M ) = D( M, M ′ ) for every M ′ ∈ add( M ) ⊆ cogen ( M ) from Lemma 2.4. The rest statements are obvious. (cid:3) Corollary 3.11.
Let
G, H be as in the bijection of Lemma 3.10, then we have an equivalence τ k : add( G ) ←→ add( H ) : τ − k , where add( G ) (resp. add( H ) ) denotes the projective (resp. injective) stable category, see [AB69] .Proof. This follows from the equivalence of Lemma 3.9 by pre- and postcomposing with ( − , M ) andD( M, − ) and then use the previous Lemma 3.10. (cid:3) Example 3.12.
Obviously triples [Λ , M, M ] correspond to triples [Γ , N,
Γ] and since M is a generatorwe conclude that N is a projective Γ-module. The module M is a generator-cogenerator (i.e., Λ ⊕ D Λ ∈ add( M )) if and only if N is projective-injective.Furthermore, add( M ) = add( τ k M ⊕ Λ) and M being a generator-cogenerator with Ext i ( M, M ) = 0,1 ≤ i ≤ k − → Γ → N → · · · → N k → D Γ → N i projective-injective and P ( G ) a projective generator, I (Γ) an injective cogenerator. But this isequivalent to Γ being k -minimal Auslander-Gorenstein which means by definition id Γ Γ ≤ k + 1 ≤ domdim Γ Γ. These algebras have been studied by Iyama and Solberg in [IS18].Our previous results also enable us to understand all faithfully balanced modules in an easy example. xample 3.13. Let Λ n = K (1 → → · · · → n ). Then, the faithfully balanced modules for Λ are themodule which are generator or cogenerators. In general every tilting (and automatically cotilting) Λ n -module T that is coming from a slice in the Auslander-Reiten quiver fulfills that every indecomposablemodule is either cogenerated or generated by T . By Remark 3.5 we conclude that every module having T as a summand is faithfully balanced. Clearly, faithfully balanced modules must have P = I n as asummand. But even if a module has a tilting module as a summand it is not necessarily faithfullybalanced, for example T = P ⊕ S ⊕ S is a tilting Λ -module but P ⊕ S ⊕ S ⊕ S is not faithfullybalanced. The 21 faithfully balanced modules for Λ are: T ⊕ I ⊕ P for a projective P and an injective I , modules with one of the other four tilting modules as a summand (since these four come from slices)and the module P ⊕ I ⊕ P .We call two modules N, M equivalent if cogen ( N ) = cogen ( M ) and gen ( N ) = gen ( M ). Then weconsider the partial order on equivalence classes N ≤ M ⇔ cogen ( N ) ⊆ cogen ( M ) , gen ( N ) ⊇ gen ( M )The Hasse diagram for the 20 equivalence classes (the 2 generator-cogenerators are equivalent) offaithfully balanced modules for Λ is the following.D ΛD Λ | S D Λ | P | S D Λ | P D Λ | P D Λ | P | S P | I | S P | | S | I P | | S | I D Λ | Λ P | | S | I P | P | I P | P | I Λ | S | I Λ | S | I P | S | P Λ | S Λ | I Λ | I Λwhere for example P | | S | I = P ⊕ P ⊕ S ⊕ I .4. Combining the cogenerator and the cotilting correspondence.
As an application of faithfully balanced modules we give a simultaneous generalization of the co-generator and the cotilting correspondence. We look at modules M which are of the form M = C ⊕ X with C a cotilting module and X ∈ < ⊥ C . If C = D Λ, then M is an arbitrary cogenerator. If X = 0,then M is a cotilting module. By Lemma 3.4 we know that M is faithfully balanced and if id C ≤ k ,then cogen t ( M ) = cogen t ( C ) for all t ≥ k −
1. So, what is the corresponding pair to a pair [Λ , M ] asjust described?We will need the following definition.
Definition 4.1.
Let Γ be a finite-dimensional algebra and
N, J a left Γ-modules with J injective. Wesay that N is a J -restricted k -cotilting module if the following holds(i) there is an exact sequence 0 → N → J → · · · → J k → J i ∈ add J , 0 ≤ i ≤ k ,(ii) N is self-orthogonal,(iii) there is an exact sequence 0 → N k → · · · → N → J → N i ∈ add N , 0 ≤ i ≤ k . he following is straightforward to see. Lemma 4.2. If N is in cogen ( J ) for an injective Γ -module J , the following are equivalent (1) N is J -restricted k -cotilting, (2) D( N, J ) is a left k -cotilting B := End( J ) op -module and Ext iB (D J, D( N, J )) = 0 for ≤ i ≤ k .Proof. Let B = End Γ ( J ) op . The injective module Γ J induces a restriction functor D( − , J ) : Γ − mod → B − mod which has a fully faithful right adjoint r = ( B D J, − ). By Lemma 2.9 we get an equivalenceof categories D( − , J ) : cogen k +1 ( J ) ←→ T ki =1 ker Ext iB (D J, − ) : r for every k ≥ B D J isa generator.Assume (1), then it is easy to check that D( N, J ) is a k -cotilting B -module since D( − , J ) is exactand apply Lemma 2.4 (2) to prove the self-orthogonality. Since N ∈ cogen k +1 ( J ), we can use theequivalence just mentioned to see that (2) is fulfilled.Assume (2), since Ext iB (D J, D( N, J )) = 0 for 1 ≤ i ≤ k we have that r is exact on an injectivecoresolution of S := D( N, J ) and on the exact sequence 0 → S k → · · · → S → D B → S i ∈ add S . Since N = (D J, S ), we can use again Lemma 2.4 (2) to see that N is self-orthogonal. (cid:3) Lemma 4.3.
The assignment [Λ , M ] [Γ = End( M ) , N = Γ M ] restricts to a bijection between (1) [Λ , M ] such that cogen k − ( M ) is the perpendicular category of a k -cotilting module C which isa summand of M . (2) [Γ , N ] such that N is faithfully balanced J -restricted k -cotilting module for some injective mod-ule J .Furthermore, if [Λ , M = C ⊕ X ] is mapped to [Γ , N ] as explained before, then we have | M | = | Λ | + | X | and | N | = | Γ | − | X | and End( C ) ∼ = End Γ ( J ) op , under this isomorphism End( C ) C ∼ = D( N, J ) and End
End( J ) ( J ) ∼ = End End( C ) (( M, C )) ∼ = End Λ ( M ) = Γ , therefore Γ J is also faithfully balanced. We remark that in the previous lemma in (1) the tilting module C does not have to be mentionedsince it can be reobtained as the Ext-injectives in cogen k − ( M ). Recall that a module I ∈ cogen k − ( M )is Ext-injective if Ext ( N, I ) = 0 for all N ∈ cogen k − ( M ). Similar, in (2) a restricted cotilting module N is restricted to a unique injective module which is obtained as the direct sum of the injectivesappearing in an injective coresolution of N . Proof. (1) (2) : Let [Λ , M ] be as in (1). Since D Λ ∈ cogen t ( M ) for all t ≥ Γ M = N is self-orthogonal by Lemma 2.4 (2). Since C isa k -cotilting module, we have two exact sequences of Λ-modules(1) 0 → C k → C k − → · · · → C → D Λ → → C → I → · · · → I k − → I k → C i ∈ add( C ) , I i ∈ add D Λ, 0 ≤ i ≤ k . Since M ∈ < ⊥ C we have Ext i Λ ( M, C ) for all i ≥
1, thisimplies that D( M, − ) is exact on both of the sequences, so if we denote J = D( M, C ) ∈ add D Γ, thenwe obtain two exact sequences with N = Γ M = D( M, D Λ)(1 ′ ) 0 → N → J → · · · → J k − → J k → ′ ) 0 → N k → N k − → · · · → N → J → J i ∈ add J, N i ∈ add N , 0 ≤ i ≤ k . Sequence (1 ′ ) implies id N ≤ k . Since N is self-orthogonal,we see that the functor D( N, − ) is exact on sequence (1 ′ ) and (2 ′ ). This implies using sequence (2 ′ )that J ∈ gen k ( N ) and Ω k +1 N J = 0.(2) (1) Let Λ = End Γ ( N ) and M = Λ N , C = Λ D( N, J ) ∈ add( M ). If N is a J -restricted k -cotilting, we get an isomorphism 0 = Ext i Γ ( J, D Γ) → Ext i Λ ( M, C ) for every i ≥ M ∈ T i ≥ ker Ext i Λ ( − , C ) and C self-orthogonal. It is straight-forward to see that thefunctor D( N, − ) is exact on the two exact sequences in the definition of the J -restricted k -cotiltingmodule and that these yield the two exact sequences to see that C is a k -cotilting module. (cid:3) We can refine the previous two lemmas, for that we will use the following four assignments fortriples of finite-dimensional algebras together with two modules: the Auslander-Solberg assignment (AS), • the dual Auslander-Solberg assignment (dual AS), • swap s ([Λ , M, G ]) := [Λ , G, M ], and • passing to the opposite algebra ( − ) op ([Λ , M, G ]) := [Λ op , D M, D G ].We remark that if all involved modules are faithfully balanced then each of the assignments is self-inverse. Theorem 4.4.
We consider the following triples (1) [Λ , M, C ] such that C ∈ add( M ) , cogen k − ( M ) = cogen k − ( C ) = < ⊥ C , (2) [Γ , N, J ] such that J ∈ add D Γ , N is faithfully balanced J -restricted k -cotilting module, (3) [ B, G, Q ] such that Q is a k -cotilting module, B ∈ add G and G ∈ cogen k − ( Q ) .Then the following diagram of bijective assignments is well-defined and commutes (1) (dual AS) / / s ◦ ( AS ) ◦ s ❆❆❆❆❆❆❆❆ (2) ( − ) op ◦ ( AS ) ◦ s ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ (3) Proof.
The correspondence between (1) and (2) is Lemma 4.3, using also its proof to see J = D( M, C )in this case. The correspondence between (2) and (3) is Lemma 4.2.Let [Λ , M, C ] be as in (1) corresponding to [Γ , N, J ] under the dual AS assignment. Now, we observein Lemma 4.3 also that we have End Γ ( J ) op ∼ = End Λ ( C ) =: B and using this isomorphism we have B D J ∼ = B ( C, M ), B D( N, J ) ∼ = B C . This implies that the whole diagram is commutative and thecorrespondence between (1) and (3) is a consequence of this. (cid:3) Example 4.5.
Let Λ be the path algebra of the quiver 1 → → M be theΛ-module P ⊕ P ⊕ S ⊕ I . The we have M = C ⊕ X for C = P ⊕ I ⊕ S a 1-cotilting module and X = P ∈ cogen( C ). We identify Γ with the commuting square b β (cid:30) (cid:30) ❃❃❃❃ a α ? ? (cid:0)(cid:0)(cid:0)(cid:0) γ (cid:31) (cid:31) ❃❃❃❃ cd δ @ @ (cid:0)(cid:0)(cid:0)(cid:0) , βα − δγ = 0via a = [ P ] , b = [ P ] , c = [ I ] , d = [ S ]. Then Γ M = I b ⊕ I c ⊕ P b is faithfully balanced, self-orthogonaland it has injective dimension 1, the injective coresolution of P b is given by 0 → P b → I c → I d → M, − ) is exact on it. We consider the injective module J = I b ⊕ I c ⊕ I d , theprevious exact sequence shows J ∈ gen ( M ) and Ω M J = 0.5. On categories relatively cogenerated by a module
Let M ∈ Λ-mod. We recall from [AS93b] that one can associate two additive subbifunctors F M , F M ⊆ Ext ( − , − ) to the subcategory add( M ) defined for ( C, A ) ∈ (Λ-mod) op × Λ-mod as follows F M ( C, A ) = { → A → B → C → | Hom Λ ( − , M ) is exact on it } F M ( C, A ) = { → A → B → C → | Hom Λ ( M, − ) is exact on it } . An exact sequence in Λ-modules is called F M exact if and only if Hom Λ ( − , M ) is exact on it and thecategory I ( F M ) = add( M ⊕ D Λ) is called the category of F M -injectives.An exact sequence in Λ-modules is called F M exact if and only if Hom( M, − ) is exact on it and thecategory P ( F M ) = add( M ⊕ Λ) is called the category of F M -projectives.If F ⊆ Ext ( − , − ) is a subbifunctor, we will say a monomorphism f : X → Y is an F -monomorphismif the short exact sequence 0 → X f −→ Y → coker f → F -exact, dually we define F -epimorphism.We say a left exact sequence of morphisms is F -exact if all inclusions of images are F -monomorphisms,dually we define a right exact map to be F -exact if all epimorphisms on cokernels are F -epimorphisms.Compositions of F -monomorphisms (resp. F -epimorphisms) are again F -monomorphisms (resp. F -epimorphisms). n the two new exact structures, we have(1) cogen k ( M ) is the category of modules N such that there exists an F M -exact sequence0 → N → M → · · · → M k with M i ∈ add( M ). Since M is F M -injective, this sequence can be seen as the beginning ofan F M -injective coresolution.(2) gen k ( M ) is the category of modules N such that there exists an F M -exact sequence M k → · · · → M → N → M i ∈ add( M ). Since M is F M -projective, this sequence can be seen as the beginning ofan F M -projective resolution.By [AS93b, Proposition 1.7], we have that F M , F M are both additive subbifunctors of Ext ( − , − )with enough projectives and enough injectives. Therefore, one can define for F ∈ { F M , F M } thederived right functors Ext i F ( − , − ), i ≥
1, these are defined by using F -injective coresolutions or F -projective resolutions.There exist additive subbifunctors of Ext which are not of the form F M or F M , see [Bua01] or[DRSS99]. However, according to [AS93c], the existence of F -cotilting modules is equivalent to F is ofthe form F = F G = F H for a generator G and a cogenerator H , and in this case H = τ G ⊕ D Λ and G = τ − H ⊕ Λ. Such a functor is called an additive subbifunctor (of Ext ) of finite type . As one of ourmain results, we will prove (in section 8) the relative (co)tilting correspondence. So, in this paper, wewill only consider the additive subbifunctors of finite type. Note that, by definition, for any module Λ M we have F M = F M ⊕ Λ and F M = F M ⊕ D Λ .We define two new full subcategories of Λ-modcogen k F ( M ) := (cid:26) N (cid:12)(cid:12)(cid:12)(cid:12) ∃ F -exact seq. 0 → N → M → · · · → M k with M i ∈ add( M ) , and s.t.Hom( M k , M ) → · · · → Hom( M , M ) → Hom(
N, M ) → (cid:27) gen F k ( M ) := (cid:26) N (cid:12)(cid:12)(cid:12)(cid:12) ∃ F -exact seq M k → · · · → M → N → M i ∈ add( M ) , and s.t.Hom( M, M k ) → · · · → Hom(
M, M ) → Hom(
M, N ) → (cid:27) . Similarly, we can define copres k F ( M ) and pres F k ( M ). Then, we have cogen k ( M ) = copres k F M ( M ) =cogen k F M ( M ) and gen k ( M ) = pres F M k ( M ) = gen F M k ( M ). Example 5.1.
Let F = F G = F H for a generator G and a cogenerator H and M be a module withExt i Λ ( G, M ) = 0 (resp. Ext i Λ ( M, H ) = 0), 1 ≤ i ≤ k + 1 for some k ≥
0. Then one hascogen k F ( M ) = cogen k ( M ) ∩ k +1 \ i =1 ker Ext i Λ ( G, − ) (resp. gen F k ( M ) = gen k ( M ) ∩ k +1 \ i =1 ker Ext i Λ ( − , H ) ) . Lemma 5.2.
Let F = F G = F H for a generator G and a cogenerator H . A module Z ∈ cogen k ( M ) is in cogen k F ( M ) if and only if the short exact sequences → Ω − iM Z f i −→ M i → Ω − ( i +1) M Z → with f i minimal left add( M ) -approximation are F -exact for ≤ i ≤ k .Proof. It is enough to observe the following: If f : X → Y is an F -monomorphism and F = F H , thenthis is equivalent to ( f, H ) being surjective. So if an F -monomorphism f factors as f = αβ , then β also has to be an F -monomorphism. (cid:3) Example 5.3.
Let M be any module and k ≥
0, then cogen k ( M ) = cogen k F M ( M ) = copres k F M ( M ) isclosed under summands and is F M -extension closed since M is F M -injective.For k ≥
1, it is closed under kernels of F M -epimorphisms X → Y with X, Y ∈ cogen k ( M ). For k = ∞ it is also closed under cokernels of F M -monomorphisms X → Y with X, Y ∈ cogen ∞ ( M ). So,one can define the derived category D b F M (cogen k ( M )), see [Nee90, Kel96]. It is completely unknownwhich informations these encode. .1. The relative version of faithfully balancedness.
Recall that for a finite-dimensional algebraΛ a module Λ M is faithful if and only if Λ ∈ cogen( M ) = cogen ( M ), and it is faithfully balanced ifand only if Λ ∈ cogen ( M ). So it makes sense to call a faithful module 0-faithful and call a faithfullybalanced module 1-faithful. Of course one can define the notion of k -faithful module for any non-negative integer k . Since in the relative setting balancedness doesn’t make sense, we introduce thefollowing definition. Definition 5.4.
Let F ⊆ Ext ( − , − ) be an additive subbifunctor of finite type and k a non-negativeinteger. We say a module M is k - F -faithful if P ( F ) ⊆ cogen k F ( M ). In particular, a 1- F Λ -faithfulmodule is just a faithfully balanced module.Easy examples of 1- F -faithful modules are F -(co)tilting modules (see section 8) and modules whichhave G or H as a summand. Here is an other easy example. Example 5.5. (1) Let Λ be a finite-dimensional algebra, P , . . . , P n its indecomposable projec-tives and assume that there is a subset I ⊆ { , . . . , n } such that M := L i ∈ I L j ≥ τ − j P i is finite-dimensional and faithfully balanced. Then G = M ⊕ Λ and H = M ⊕ D Λ fulfill F G = F H =: F . Clearly, we have G ∈ cogen F ( M ), so M is 1- F -faithful.(2) Let Λ be a basic Nakayama algebra and assume M = L X : indec, not simple X is faithfullybalanced . Let G = M ⊕ L P i : simple proj P i and H = M ⊕ L I i : simple inj I i . Then we claim: { F -faithful modules } = { M ′ ⊕ S | S semi-simple , add( M ′ ) = add( M ) } Since M is F -projective-injective, M has to be summand of every 1- F -faithful module. Onthe other hand, let S be a semi-simple module, we want to see that M ⊕ S is 1- F -faithful.Assume that there is a simple projective P / ∈ add( S ), since ( P, S ) = 0 = (
S, P ) we have thatthe minimal left add( M ⊕ S ) equals the minimal left add( M ) and the minimal left add( H )-approximation, in particular G ∈ cogen F ( M ⊕ S ). Now, we look at the cokernel of theapproximation X = Ω − M ⊕ S P , since M is faithfully balanced we have X ∈ cogen( M ), inparticular X has no simple injective summand. So, every simple summand S ′ / ∈ add( S ) of X has a minimal left add( H )-approximation which coincides with a minimal left add( M )- andadd( M ⊕ S )-approximation which is an F -monomorphism and therefore, we conclude that G ∈ cogen F ( M ⊕ S ).The main result of this subsection is the following Theorem 5.6.
Let F ⊆ Ext ( − , − ) be an addtive subbifunctor of the form F = F G = F H for agenerator G and a cogenerator H . The following are equivalent for every module M and every k ≥ . (1) G ∈ cogen k F ( M ) . (2) H ∈ gen F k ( M ) . Let M ∈ Λ-mod and Γ = End Λ ( M ). We defineΣ = End Λ ( H ) and ∆ = End Λ ( G ) . We first remark that generators and cogenerators are faithfully balanced, in particular this applies to H and G and we haveΛ-mod = cogen( H ) = cogen ( H ) = cogen ( H ) = · · · = cogen ∞ ( H )Λ-mod = gen( G ) = gen ( G ) = gen ( G ) = · · · = gen ∞ ( G ) . By Lemma 2.9 we have dualities of categories( − , Λ H ) : Λ-mod ←→ cogen ( Σ H ) : ( − , Σ H )D( Λ G, − ) : Λ-mod ←→ gen ( ∆ G ) : D( ∆ G, − ) . The key step in the proof is given by the following lemma.
Lemma 5.7.
Keep the above notations. For ≤ k ≤ ∞ we have (1) The following are equivalent this is the case if Λ has no simple projective-injective and τ − S is not simple injective for every S simple projective- for example A n fulfills this for n ≥ N ∈ cogen k F ( M ) . (1b) Σ ( N, H ) ∈ gen k ( Σ ( M, H )) . (1c) Consider the natural map ( M, H ) ⊗ Γ ( N, M ) → ( N, H ) , f ⊗ g f ◦ g . (i) For k = 0 : It is an epimorphism . (ii) For k ≥ : It is an isomorphism and Ext i Γ (( N, M ) , D( M, H )) = 0 for ≤ i ≤ k − . (2) The following are equivalent (2a) N ∈ gen F k ( M ) . (2b) ( G, N ) ∆ ∈ gen k (( G, M ) ∆ ) . (2c) Consider the natural map ( M, N ) ⊗ Γ ( G, M ) → ( G, N ) , f ⊗ g f ◦ g . (i) For k = 0 : It is an epimorphism. (ii) For k ≥ : It is an isomorphism and Ext i Γ (( G, M ) , D( M, N )) = 0 for ≤ i ≤ k − .Proof. It is easy to see the equivalence of (1a) and (1b) using that the duality ( − , H ) restricts to aduality of categories( − , Λ H ) : cogen k F ( M ) ←→ cogen ( Σ H ) ∩ gen k ( Σ ( M, H )) : ( − , Σ H ) . To see that the map from the right to the left is well-defined it is important to observe that Σ H isan injective module (since H is a cogenerator), therefore the functor ( − , Σ H ) is exact. Similarly, it iseasy to see the equivalence of (2a) and (2b) using the second equivalence mentioned above. For theequivalence of (1b) and (1c) we translate the statement of (1c) into the characterization from Lemma2.2. The most important observation is the following E := End Σ (( M, H )) = Γ op . The natural mapfrom Lemma 2.2 (for the category gen k Σ ( M, H )) is:Hom Σ (( M, H ) , ( N, H )) ⊗ E ( M, H ) → ( N, H ) f ⊗ g f ( g ) . First observe E = Γ op means left (resp. right) E -modules are naturally right (resp. left) Γ-modulesand X E ⊗ E E L ∼ = L Γ ⊗ Γ Γ X . Secondly, since H is a cogenerator we haveHom Σ (( M, H ) , ( N, H )) = (
N, M )With this identifications the map from before becomes the natural map mentioned in (1c).The equivalence of (2b) and (2c) is analogue. We set C = End ∆ (( G, M )) = Γ. By lemma 2.2 we haveto look at the natural map Hom ∆ (( G, M ) , ( G, N )) ⊗ C ( G, M ) → ( G, N ) f ⊗ g f ( g )We have an isomorphism of right Γ-modules since G is a generatorHom ∆ (( G, M ) , ( G, N )) = (
M, N )With this identifications the map from before becomes the natural map in (2c). (cid:3)
We observe that the proof of Theorem 5.6 is a direct consequence of the previous lemma: By setting N = G in part (1) and N = H in part (2), we obtain the same maps in (1c), (2c) and therefore theclaim follows. Lemma 5.8.
Let F = F G = F H for a generator G and a cogenerator H . Let M ∈ Λ - mod , Γ =End Λ ( M ) , L = ( G, M ) and R = D( M, H ) .If we assume that Λ ∈ cogen F ( M ) and H ∈ gen ( M ) then the duality ( − , Λ M ) : cogen ( M ) ↔ cogen ( M ) : ( − , Γ M ) restricts to a duality cogen F H ( Λ M ) ↔ cogen F R ( Γ M ) . Furthermore, it restrictsto a duality ( − , Λ M ) : cogen k F H ( M ) ←→ cogen F R ( M ) ∩ k − \ i =1 ker Ext i Γ ( − , R ) : ( − , Γ M ) In particular, G ∈ cogen k F H ( M ) is equivalent to L ∈ cogen F R ( M ) and Ext i Γ ( L, R ) = 0 for ≤ i ≤ k − . roof. Since H ∈ gen ( M ) we have that D( M, R ) = D( M, D( M, H )) → H is an isomorphism. So, itis enough to proof that ( − , Λ M ) maps cogen F H ( M ) to cogen F R ( M ) and use R ∈ gen ( Γ M ) to get thequasi-inverse by symmetry.Let X ∈ cogen F H ( M ). We choose a projective presentation P → P → X →
0. By applying( − , Λ M ) we get an exact sequence of Γ-modules 0 → ( X, M ) → ( P , M ) → ( P , M ) is with ( P i , M ) ∈ add( M ). We apply ( − , R ) to get a complex (( P , M ) , R ) → (( P , M ) , R ) → (( X, M ) , R ) →
0. Wewould like to see that it is exact. By Hom-Tensor adjunction it identifies with the first row in thefollowing commutative diagramD[(
M, H ) ⊗ Γ ( P , M )] / / D[(
M, H ) ⊗ Γ ( P , M )] / / D[(
M, H ) ⊗ Γ ( X, M )] / / P , H ) / / O O D( P , H ) O O / / D( X, H ) / / O O M, H ) ⊗ Γ ( Y, M ) → ( Y, H ) given by f ⊗ g f ◦ g . By Lemma 5.7 we know that this natural map is an isomorphism if and only if Y ∈ cogen F H ( M ). By assumption we have P , P , X ∈ cogen F H ( M ) and the first row identifies withthe complex in the second row. But the exactness of the second row follows since D( − , H ) is rightexact. This proves ( X, M ) ∈ cogen F R ( M ). For the symmetry, we need to see Γ ∈ cogen F R ( M ). ButΓ = ( M, M ) and M ∈ cogen F H ( M ) implies the claim by the argument just given.The further restriction follows directly from Lemma 5.7. (cid:3) Of course there is a dual version of the previous lemma which we will leave out.If Λ M is 1- F H -faithful, then Γ M does not have to be 1- F R -faithful (with Γ = End Λ ( M ) and R = D( M, H )). We give an example for this:
Example 5.9.
Let Λ be the path algebra of 1 α −→ β −→ βα = 0. Let G = Λ ⊕ S , H = D Λ ⊕ S and F = F H = F G . Then M := G is clearly 1- F H -faithful and pd F M = 0. Let uslook at Γ = End Λ ( M ), since we have irreducible morphisms S → → → S , we can identifyit with the following bound path algebra d → c → b → a modulo all path of length 2. We have Γ M = ( P , M ) ⊕ ( P , M ) ⊕ ( P , M ) = P b ⊕ P c ⊕ P d and Γ R := D( M, H ) = Γ M ⊕ D( M, S ). Weapply D( M, − ) to an injective coresolution 0 → S → I → I to obtain a projective presentation P b = ( P , M ) → P c = ( P , M ) → D( M, S ) →
0. This implies D(
M, S ) ∼ = S c and therefore τ − R = τ − S c = S d . It is easy to see that S d / ∈ cogen( Γ M ) implying τ − R / ∈ cogen F R ( M ). This shows Γ M isnot 1- F R -faithful.Thus the property of being 1- F -faithful is not as nicely symmetric as being faithfully balanced.Nevertheless, we can get the symmetry again if we restrict to the following special case. Proposition 5.10.
Let F = F G = F H for a generator G and a cogenerator H . Let M be a faithfullybalanced Λ -module, Γ = End Λ ( M ) , L = ( G, M ) and R = D( M, H ) .If M ∈ add( H ) ( or equivalently, D Γ ∈ add( R ) ) , then the following are equivalent: (1) Λ M is - F H -faithful. (2) Γ M is - F R -faithful.Dually, if M ∈ add( G ) , then Λ M is - F G -faithful if and only if Γ M is - F L -faithful.Proof. We assume M ∈ add( H ). Assume G ∈ cogen F H ( M ), we have to see τ − R ∈ cogen F R ( M ).Since H ∈ gen F H ( M ) implies that we have an F -exact sequence0 → Ω M H → M → M → H → M i ∈ add( M ). Since M ∈ add( H ), this implies Ω M H ∈ cogen F ( M ). We apply D( M, − ) tothe last three terms of the four term sequence and obtain an injective copresentation of R . We apply( − , M ) to the first three terms and observe and get an exact sequence( M , M ) → ( M , M ) → (Ω M H, M ) = τ − R → (cid:3) .2. Strong dualizing sequences.Definition 5.11.
Let 0 → L → M → M → · · · → M k → R → k -add( M )-dualizing sequencein Γ-mod for some non-negative integer k . We say it is strong if D( L, − ) is exact on it.We can characterize it as follows. Lemma 5.12. A k - add( M ) -dualizing sequence as in the above definition is strong if and only if one(equivalently all) of the following equivalent statement is fulfilled: (1) D( L, − ) is exact on it, i.e., it is an F L -exact sequence ( or equivalently, R ∈ gen F L k ( M ) ) . (2) ( − , R ) is exact on it, i.e., it is an F R -exact sequence ( or equivalently, L ∈ cogen k F R ( M ) ) . (3) Consider the natural map ( M, R ) ⊗ Λ ( L, M ) → ( L, R ) , where Λ = End Γ ( M ) . (i) For k = 0 : It is an epimorphism . (ii) For k ≥ : It is an isomorphism and Ext i Λ (( L, M ) , D( M, R )) = 0 for ≤ i ≤ k − .Proof. We will prove (1) and (3) are equivalent and the equivalence of (2) and (3) can be proveddually.We consider the following commutative diagram / / D( L, R ) i ′ / / i (cid:15) (cid:15) D( L, M k ) f / / ∼ = (cid:15) (cid:15) D( L, M k − ) / / ∼ = (cid:15) (cid:15) · · · / / D( L, M ) ∼ = (cid:15) (cid:15) / / D((
M, R ) ⊗ ( L, M )) j / / ∼ = (cid:15) (cid:15) D((
M, M k ) ⊗ ( L, M )) g / / ∼ = (cid:15) (cid:15) D((
M, M k − ) ⊗ ( L, M )) / / ∼ = (cid:15) (cid:15) · · · / / D((
M, M ) ⊗ ( L, M )) ∼ = (cid:15) (cid:15) / / (( L, M ) , D( M, R )) / / (( L, M ) , D( M, M k )) / / (( L, M ) , D( M, M k − )) / / · · · / / (( L, M ) , D( M, M )) . Assume (1), then the first row is exact. Since the functor ((
L, M ) , − ) is left exact, the sequence0 → D((
M, R ) ⊗ ( L, M )) → D((
M, M k ) ⊗ ( L, M )) → D((
M, M k − ) ⊗ ( L, M )) is exact. For k = 0, wehave ji is a monomorphism and so is i . This shows the natural map ( M, R ) ⊗ Λ ( L, M ) → ( L, R ) is anepimorphism. For k ≥
1, we have an induced isomorphism on kernelsD(
L, R ) = ker f ∼ = −→ ker g = D(( M, R ) ⊗ ( L, M )) . This proves the natural map (
M, R ) ⊗ Λ ( L, M ) → ( L, R ) is an isomorphism. Now the exactness of thefirst row implies the exactness of the last row which is equivalent to Ext i Λ (( L, M ) , D( M, R )) = 0 for1 ≤ i ≤ k −
1. Conversely, assume (3). If k = 0, then the map i is a monomorphism and so is i ′ . If k ≥
1, then the last row is exact and the natural map (
M, R ) ⊗ Λ ( L, M ) → ( L, R ) is an isomorphismwill imply the first row is isomorphisc to the last row. So we have, in both cases, that the first row isexact. Since the functor D( L, − ) is right exact, (1) follows from the exactness of the first row. (cid:3) Remark 5.13.
From the proof of the above lemma we see that for any X if N ∈ cogen F X ( M )then the natural map ( M, X ) ⊗ ( N, M ) → ( N, X ) is an isomorphism. The converse holds true if X is a cogenerator (cf. Lemma 5.7). Similarly, we have if N ∈ gen F X ( M ) then the natural map( M, N ) ⊗ ( X, M ) → ( X, N ) is an isomorphism.
Lemma 5.14.
Let Γ be a finite-dimensional algebra and → L → M → · · · → M k → R → be a k - add( M ) -dualizing sequence of Γ -modules with M faithfully balanced. Define Λ = End Γ ( M ) , G = ( L, M ) and H = D( M, R ) . If Γ ∈ cogen F R ( M ) and R ∈ gen ( M ) then for every k ≥ thefunctor ( − , M ) restricts to a duality cogen k F R ( M ) ←→ cogen F H ( M ) ∩ k − \ i =1 ker Ext i Λ ( − , H ⊕ M ) . In particular, L ∈ cogen k F R ( M ) is equivalent to Ext i Λ ( G, H ⊕ M ) = 0 , ≤ i ≤ k − .Proof. The case k = 1 follows directly from Lemma 5.8. For k > F R = F R ⊕ D Γ andthen apply Lemma 5.8 using the cogenerator R ⊕ D Γ (in place of H ). (cid:3) emma 5.15. Let M be a faithfully balanced Λ -module and Γ = End Λ ( M ) . Let k ≥ . Then, theassignment X, Y ( X, M ) , D( M, Y ) gives a self-inverse bijection ( up to seeing X, Y as Λ or as Γ -modules ) between the following sets of pairs of Λ -modules and Γ -modules { Λ G, Λ H | G = τ − k H ⊕ Λ ∈ cogen F H ( M ) ∩ ∼ ( k − ⊥ ( M ⊕ H ) H = τ k G ⊕ D Λ ∈ gen F G ( M ) ∩ ( M ⊕ G ) ⊥ ∼ ( k − } and { Γ L, Γ R | ∃ a strong k - Γ M -dualizing sequence from L to R } . Proof.
This follows from Lemma 3.10, Lemma 5.12 and Lemma 5.14. (cid:3)
Example 5.16.
Let M be a faithfully balanced Λ-module and assume that it has a summand X ⊕ τ − X with X not injective. We define G = Λ ⊕ τ − X , H = D Λ ⊕ X and F = F G = F H . Then, by defi-nition we have G ∈ cogen ( M ) = cogen F ( M ) and H ∈ gen ( M ) = gen F ( M ). Therefore, we obtainfor Γ = End Λ ( M ) a strong add( Γ M )-dualizing sequence with a projective-plus- M left end and aninjective-plus- M right end.Now, we can formulate a relative version of the generator/ cogenerator and Morita-Tachikawacorrespondence. Corollary 5.17. (1) ( relative generator correspondence ) The Auslander-Solberg assignment [Λ , M, G ] [End( M ) , M, ( G, M )] is an involution on theset of triples [Λ , M, G ] with Λ ⊕ M ∈ add( G ) and M is - F G -faithful. (2) ( relative cogenerator correspondence ) The dual Auslander-Solberg assignment [Λ , M, H ] [End( M ) , M, D( M, H )] is an involutionon the set of triples [Λ , M, H ] with D Λ ⊕ M ∈ add( H ) and M is - F H -faithful. (3) ( relative Morita-Tachikawa correspondence ) The assignment [Λ , M, G, H ] [End( M ) , M, L = ( G, M ) , R = D( M, H )] is a bijection between * [Λ , M, G, H ] with Λ ∈ add( G ) , D Λ ∈ add( H ) , G = Λ ⊕ τ − H and M ∈ add( G ) ∩ add( H ) is - F G -faithful, and * [Γ , N, L, R ] with L, R are the ends of a strong add( N ) -dualizing sequence with Γ ∈ add( L ) and D Γ ∈ add( R ) . F-dualizing summands.
Of course, we can also consider relative dualizing summands.
Definition 5.18.
Let F = F G = F H , M, L ∈ Λ-mod and assume M is a summand of L . We say M is an F -dualizing summand of L if L ∈ cogen F ( M ). For k ≥
0, we say it is a k - F -dualizing summandif L ∈ cogen k F ( M ).Relative dualizing summands have the properties which we expect from them: Lemma 5.19.
Let F = F G = F H and M, N be Λ -modules and L = M ⊕ N , k ≥ . If N ∈ cogen k F ( M )( i.e., M is k - F -dualizing summand of L ) , then M is - F -faithful if and only if L is - F -faithful.If H ∈ gen ( M ) , then cogen k F ( M ) = cogen k F ( L ) . Furthermore, in this case if also copres k F ( L ) =cogen k F ( L ) then we have copres k F ( M ) = cogen k F ( M ) .In particular, if M is - F -faithful, then M ⊕ P ⊕ I is - F -faithful for every F -projective module P and F -injective module I .Proof. Let Σ = End Λ ( H ). We consider the duality for M from Lemma 5.7:( − , H ) : cogen k F ( M ) ←→ cogen ( Σ H ) ∩ gen k ( Σ ( M, H )) : ( − , H )and also for L we have( − , H ) : cogen k F ( L ) ←→ cogen ( Σ H ) ∩ gen k ( Σ ( L, H )) : ( − , H ) . Since (
M, H ) is a summand of (
L, H ) and (
L, H ) ∈ gen k ( M, H ) follows that gen ( L, H ) ⊆ gen ( M, H )(dual argument to 1-dualizing summand situation).Furthermore, we claim: if H ∈ gen ( M ), then Σ ( M, H ) is faithfully balanced (and therefore, the claimfollows from the dual of Lemma 3.4 and using the duality from above again). So, assume there is n exact sequence M → M → H → M i ∈ add( M ) and ( M, − ) exact on it. Apply ( − , H )to it, to obtain an exact sequence 0 → Σ → ( M , H ) → ( M , H ). Apply ( − , ( M, H )) to it andusing ((
X, H ) , ( Y, H )) = (
Y, X ) for all Λ-modules
X, Y you can identify the result with the complex(
M, M ) → ( M, M ) → ( M, H ) → H ∈ gen ( M ). This provesΣ ∈ cogen (( M, H )) and therefore the claim. The remaining claims are proven as in Lemma 3.4. (cid:3)
Example 5.20.
Let G be a generator and F = F G . Then a 1- F -faithful summand of G is the sameas an F -dualizing summand of G . These are easily determined as follows, let H = D Λ ⊕ τ G and P → P → H → F -presentation with P i ∈ add( G ). Then, the 1- F -faithful summands of G are the summands P of G with P ⊕ P ∈ add( P ). Of course, with a dual statement one can findthe 1- F -faithful (i.e., the F -codualizing) summands of H .6. Relative Auslander-Solberg and Auslander correspondence
We generalize the notion of dominant dimension to the relative setting.
Definition 6.1.
Let Γ be a finite-dimensional algebra and F = F G = F H for a generator Γ G and acogenerator Γ H . Consider the minimal F -coresolution of G by F -injectives0 → G → H → H → H → · · · . We define domdim F Γ = k if there exists an integer k such that H i ∈ add( G ) for 0 ≤ i ≤ k − H k / ∈ add( G ). If H i ∈ add( G ) for all i ≥ F Γ = ∞ . Remark 6.2.
As is in the classical case, our definition of F -dominant dimension is left-right symmetricin the following sense: A functor F = F G = F H determines a functor F D H = F D G =: F ∗ in thecategory Γ op -mod and vice versa, and domdim F Γ = k if and only if domdim F ∗ Γ op = k .6.1. Relative Auslander-Solberg correspondence.Lemma 6.3.
Let F = F G = F H with G and H basic and assume Γ M is a module such that add( M ) =add( G ) ∩ add( H ) . Then the following are equivalent for every k ≥ . (1) There is an F -exact sequence → G → M → M → · · · → M k → H → with M i ∈ add( M ) . (2) domdim F Γ ≥ k + 1 ≥ id F G. (3) domdim F Γ ≥ k + 1 ≥ pd F H. Proof. (1) ⇒ (2) and (1) ⇒ (3) are obvious. We prove (2) ⇒ (1) and (3) ⇒ (1) is dual.Assume (2) then we have an F -exact sequence0 → G → M → M → · · · → M k − → M ′ k → H ′ → M i ∈ add( M ) for 0 ≤ i ≤ k − M ′ k ∈ add( M ) and H ′ ∈ add( H ). We may assume this F -exactsequence is a successive composition of minimal left add( M )-approximations of the cokernels. By thedual version of Lemma 8.3 we have M ⊕ H ′ is an F -tilting module with id F ( M ⊕ H ′ ) = 0. By Lemma8.8 (1) we know that M ⊕ H ′ is basic and hence M ⊕ H ′ = H . Now the desired F -exact sequence in(1) can be obtained by adding M −→ M to M ′ k → H ′ . (cid:3) Theorem 6.4.
Let Λ M be a faithfully balanced module and Γ = End Λ ( M ) . The assignment X, Y ( X, M ) , D( M, Y ) gives a self-inverse bijection between the following sets of pairs of modules (1) { Λ L, Λ R | Λ M ⊕ Λ ∈ add( L ) , Λ M ⊕ D Λ ∈ add( R ) , L = τ − k R ⊕ Λ , R = τ k L ⊕ D Λ , Ext i Λ ( L, R ) = 0 , ≤ i ≤ k − such that there exists a strong add( Λ M ) -dualizing sequence with left end L andright end R } . (2) { Γ G, Γ H | M ⊕ Γ ∈ add( G ) , M ⊕ D Γ ∈ add( H ) , G = τ − H ⊕ Γ , H = τ G ⊕ D Γ such that thereexists a strong k - add( Γ M ) -dualizing sequence with left end G and right end H } .Proof. Combine Lemma 5.15 and Lemma 6.3. (cid:3) .2. Relative Auslander correspondence.Lemma 6.5.
Let k ≥ and assume domdim F Γ ≥ k + 1 . Let Γ M be a module with add( M ) =add( G ) ∩ add( H ) . Then we have cogen k F ( M ) = Ω k +1 F (Γ - mod) and gen F k ( M ) = Ω − ( k +1) F (Γ - mod) .Furthermore, the following are equivalent: (1) cogen k F ( M ) = add( G ) . (2) gen F k ( M ) = add( H ) . (3) gldim F Γ ≤ k + 1 .Proof. Since domdim F Γ ≥ k + 1 and add( M ) = add( H ) ∩ add( G ), we have clearly add( G ) ⊆ cogen k F ( M ) ⊆ Ω k +1 F (Γ-mod). On the other hand, we prove in Lemma 8.3 that in this case:cogen k F ( M ) = \ i ≥ ker Ext i F ( − , C )for C = M ⊕ Ω k +1 M H and id F C ≤ k + 1. So given X ∈ Ω k +1 F (Γ-mod), there is an Y ∈ Γ-mod suchthat X = Ω k +1 F Y and then by dimension shift for i ≥ i F ( X, C ) = Ext i + k +1 F ( Y, C ) = 0since id F C ≤ k + 1. In particular, X ∈ cogen k F ( M ). One can prove gen F k ( M ) = Ω − ( k +1) F (Γ-mod) withthe dual argument.Now clearly, gldim F Γ ≤ k + 1 is equivalent to Ω k +1 F (Γ-mod) ⊆ add( G ) and by the just proved result,we conclude it is equivalent to (1). The equivalence of (3) and (2) can be proven with the analogousargument. (cid:3) Definition 6.6.
Let M ∈ Λ-mod and assume that there is a strong add( M )-dualizing sequence withleft end L and right end R .We say that M is a k -( L, R )-cluster tilting module if(i) Λ ∈ cogen F R ( M ) and D Λ ∈ gen F L ( M ),(ii) cogen F R ( M ) ∩ T k − i =1 ker Ext i Λ ( − , R ) = add( L ) and gen F L ( M ) ∩ T k − i =1 ker Ext i Λ ( L, − ) = add( R ).Let Γ be a finite-dimensional algebra and F = F G for a generator G . Then we say Γ is a k - F -Auslanderalgebra if domdim F Γ ≥ k + 1 ≥ gldim F Γ. Theorem 6.7. ( relative Auslander correspondence ) Let k ≥ . There is a one-to-one correspondence between isomorphism classes of basic k - ( L, R ) -clustertilting modules Λ M ( for some L, R ) and finite-dimensional algebras Γ with an exact structure givenby F = F G = F H such that domdim F Γ ≥ k + 1 ≥ gldim F Γ . The correspondence is induced by theassignment [Λ , M, L, R ] [Γ = End Λ ( M ) , Γ M, G = (
L, M ) , H = D( M, R )] . Proof.
Let M be an k -( L, R )-cluster tilting moduleand Γ = End Λ ( M ), G = ( L, M ) , H = D( M, R ) and F = F G = F H . Since L ∈ cogen F R ( M ) ∩ T k − i =1 ker Ext i Λ ( − , R ), we have G ∈ cogen k F ( M ) by Lemma5.8. Similarly, from Λ ∈ add( L ) , D Λ ∈ add( R ) we conclude that Γ M ∈ add( G ) ∩ add( H ) and thereforedomdim F Γ ≥ k + 1. By the same lemma, we also have cogen k F ( M ) = add( G ) and therefore by Lemma6.5 gldim F Γ ≤ k + 1.Conversely, by Lemma 6.5 and Lemma 5.8 we can also conclude the other implication. (cid:3) The easiest example can be found for k = 1. Here, for a 1-cluster tilting pair ( L, R ) with respectto M we have G = L is a generator, H = R is a cogenerator with F = F G = F H and the definitionshortens to a module M such that cogen F ( M ) = add( G ) and gen F ( M ) = add( H ) is fulfilled.Here are some easy examples of 1- F -Auslander algebras. Example 6.8. (1) Let F = F Λ and M be a projective-injective module such that cogen ( M ) =add(Λ) and gen ( M ) = add(D Λ). Then, by Lemma 6.5 it is easy to see that this is equivalentto domdim Λ ≥ ≥ gldim Λ and it is well-known that this characterizes Λ to be an Auslanderalgebra.
2) Assume F = F G = F H and G = H is a generator-cogenerator, in this case we say Λ is F -selfinjective. A classification of F -selfinjective algebras can be found in [AS93a, section 5]. Forexample, if G is an Auslander generator (= 1-cluster tilting module), this is fulfilled. Then,if we choose M = G = H , then we have cogen F ( M ) = cogen ( M ) = add( M ) = gen ( M ) =gen F ( M ) and this gives us another example.(3) Let Γ be the path algebra of 1 → → M = P ⊕ P ⊕ I . We define G := Γ ⊕ M and H := D Γ ⊕ M , then it is easy to see F G = F H =: F and cogen F ( M ) = add( G ) , gen F ( M ) =add( H ).(4) Let Γ be the path algebra of the following quiver: 1 (cid:15) (cid:15) / / / / . Let M := P ⊕ τ − P ⊕ τ − P , G = M ⊕ P ⊕ P ⊕ P , H = M ⊕ I ⊕ I ⊕ I and F := F G = F H .Then we have F -exact sequences0 → P → P → τ − P → I → → P → τ − P → τ − P → I → → P → τ − P → τ − P → I → F Γ = 2. It also easy to see that 2 = max X { pd F X } (= gldim F Γ) , sincethe three missing indecomposables which are not in add G or add H are 2 , , which appearas cosyzygies of the three injectives in the F -exact sequences and so all have pd F = 1. Wehave Λ = End Γ ( M ) is given by the following quiver with relations • α / / β / / • γ / / δ / / • γα = δβ = δα + γβ = 0 . (5) Let Λ be the path algebra modulo the relations: b β ( ( ❘❘❘❘ a α ♠♠♠♠ γ ) ) ❙❙❙❙ d, βα − δγ = 0 c δ ❦❦❦❦ Its Auslander-Reiten quiver is drawn in the following graphic, in the square boxes you find M = P d ⊕ b dc ⊕ P a ⊕ ba c ⊕ I a and together with the remaining circled modules G = M ⊕ P b ⊕ P c ?>=<89:; dc " " ❉❉❉❉❉❉ b (cid:27) (cid:27) a c ❆❆❆❆❆❆ (cid:23)(cid:22) (cid:21)(cid:20)(cid:16)(cid:17) (cid:18)(cid:19) P a ' ' PPPP (cid:23)(cid:22) (cid:21)(cid:20)(cid:16)(cid:17) (cid:18)(cid:19) d > > ⑥⑥⑥⑥⑥⑥ & & ▼▼▼▼▼ (cid:23)(cid:22) (cid:21)(cid:20)(cid:16)(cid:17) (cid:18)(cid:19) b dc ' ' ◆◆◆◆◆ ♥♥♥♥ (cid:23)(cid:22) (cid:21)(cid:20)(cid:16)(cid:17) (cid:18)(cid:19) ba c < < ③③③③③ ' ' ❖❖❖❖❖ (cid:23)(cid:22) (cid:21)(cid:20)(cid:16)(cid:17) (cid:18)(cid:19) a GFED@ABC b d ♦♦♦♦♦ c ♣♣♣♣♣ ba qqqqq It is very easy to see that M is faithfully balanced and F = F G = F M ⊕ D Λ fulfills domdim F Λ =2 = gldim F Λ. Now we look at Γ = End Λ ( M ), this is given by the path algebra of the followingquiver with the overlapping zero-relations3 δ (cid:29) (cid:29) ❁❁❁❁ α / / β / / γ A A ✂✂✂✂ ε / / , βα = 0 = δγα, εβ = 0 = εδγ the vertices 1 , , , , d, b dc , P a , ba c , a in the given order. Tocalculate Γ M = (Λ , M ) = D( M, D Λ) we look at its four indecomposable summands P = ( P a , M ) = D( M, I a ) = I I = D( M, I d ) = ( P d , M ) = P ( P b , M ) = D( M, I b ) ( P c , M ) = D( M, I c )then we apply ( − , M ) to the F -exact sequence 0 → P b → b dc → ba c → I b → → P → P → P → ( P b , M ) → P c , M )). From his we conclude that ( P b , M ), ( P c , M ) are two regular modules in different homogeneous tubesfor the full subquiver e A , more precisely:( P b , M ) =: R : K ! ! ❈❈❈ / / K / / = = ④④④ K / / P c , M ) =: R : K ! ! ❈❈❈ / / K / / = = ④④④ K / / τ ( ± ) R j = R j , j = 0 ,
1. We set now e G = P ⊕ P ⊕ P ⊕ M, e H = I ⊕ I ⊕ I ⊕ M anddefine e F := F Γ e G = F Γ e H , observe that add( Γ M ) = add( e G ) ∩ add( e H ). The following sequencesare e F -exact (setting R = R ⊕ R )0 / / P / / R / / I / / I / / / / P / / P / / I / / I / / / / P / / P / / R / / I / / e F Γ = 2 = id e F D Γ. We have End Γ ( e G ) op ∼ = End Λ ( G ) has gldim ≤ F Λ ≤ e F D Γ we conclude gldim e F Γ ≤ F -Auslander algebras. Example 6.9. (1) A k -( L, R )-cluster tilting module M with L = M = R is just the same as a k -cluster tilting module in the sense of [Iya08]. In this case, Γ = End Λ ( M ), G = ( M, M ) = Γ, H = D( M, M ) = D Γ, so F = F Γ and so domdim F Γ = domdim Γ, gldim F Γ = gldim Γ and wereobtain a higher Auslander algebra (this is the Krull-dimension zero case of Iyama’s Auslandercorrespondence, see [Iya07]).(2) Let Γ be the path algebra of 1 → → · · · → n . Let M t := L i = t L j ≥ τ − j P i , G t = M t ⊕ P t , F t = F G t , 1 < t ≤ n , then Γ has the structure of a ( t − F t -Auslander algebra for t ≥ t = 2 we have domdim F Λ = 1 = gldim F Λ. For large n we have that Λ = End Λ ( M ) isa representation-infinite algebra with an F -Auslander structure.(3) We consider the following quiver (of Dynkin type E ) Qfa / / b / / c / / O O d / / e For x ∈ { a, b, d, e, f } we define M x = L y = x L j ≥ τ − j P y , G x = M x ⊕ P x , F x = F G x . Then aninspection if the AR-quiver gives the following for the path algebra Γ = KQ : Γ is a 2- F a - and2- F b -Auslander algebra, a 4- F d - and 4- F f -Auslander algebra and 6- F e -Auslander algebra(4) Let Γ = K (1 → → · · · → n ) for some integer n > M := L n − i =1 L j ≥ τ − j P i , G = M ⊕ P n , H = M ⊕ I and F = F G = F H . We find the minimal F -projective resolutionof I (which is also the minimal F -injective resolution of P n ) as follows0 → P n → n − n → n − n − → · · · → → I → ∗ )from this we conclude pd F D Γ = n − F G = n −
1. One can easily see that thehighest pd F is obtained at an injective module and therefore gldim F Γ = n −
1, so we have an( n − F -Auslander algebra.Let Λ = End Γ ( M ), we denote by P [ M i ] , I [ M i ] , S [ M i ] the projective, injective and semi-simpleΛ-module associated to M i ∈ add( M ). Let L = Λ ( G, M ) = Λ ⊕ ( P n , M ), R = D( M, H ) = D Λ ⊕ D( M, I ) and Λ M ∈ add( L ) ∩ add( R ). Then we have Π := ( L 2. Now, apply ( − , P [ P ] ) to ( ∗∗ ) and obtain K = ( S [ 12 ] , P [ P ] ) ∼ =Ext (( S , M ) , P [ P ] ) = Ext n − ( I [ P ] , P [ P ] ).We would like to see that Λ M is a ( n − L, R )-cluster tilting modulewith respect to L and R as before. Since we easily verify cogen F H ( Γ M ) = add( G ⊕ L ≤ i We fix Γ = K (1 → → · · · → n ) for some integer n ≥ I . Our aim is to describe a family of F -Auslanderalgebras which interpolate between Iyama’s example [Iya08, Example 2.4] and the usual exact structureon Γ-mod. We study the following class of generators G ℓ := Γ ⊕ L ≤ i ≤ ℓ L j> τ − j P i , 1 < ℓ < n − .1. If ( n − ℓ + 1) | n (or equivalently, ( n − ℓ + 1) | ( ℓ − ℓ − n − ℓ +1 )-minimal F ℓ -Auslander-Gorenstein algebra (i.e., domdim F ℓ Γ ≥ ℓ − n − ℓ +1 + 1 ≥ id F ℓ G ℓ ), where F ℓ = F G ℓ . If ℓ < n − n − ℓ + 1 does not divide n , then Γ is not a minimal F ℓ -Auslander-Gorenstein algebra. proof: For ℓ ≤ k ≤ n we look at the F ℓ -injective resolution of P k and here we keep track thesequence of tops (they are all simple) of the F ℓ -injectives appearing, it fulfills a = ℓ, a = k − ( n − ℓ − , a t = a t − − ( n − ℓ + 1) for all t ≥ 3. Now, the condition to be a minimal F ℓ -Auslander-Gorenstein algebra is equivalent to that there is one t (for all k ) such that a t = 1.Since t has to work for all k (and ℓ < n − t has to be uneven, say t = 2 s + 1 (then it is an 2 s -minimal F ℓ -Auslander-Gorenstein algebra). Now, the recursiontells us 1 = a t = a t − − ( n − ℓ + 1) = a t − s − s ( n − ℓ + 1) = ℓ − s ( n − ℓ + 1), so it follows s = ℓ − n − ℓ +1 .2. But from the shape of the Auslander-Reiten quiver of Γ we can conclude that the maximalpd F ℓ is obtained at an injective module, therefore gldim F ℓ Γ = pd F ℓ D Γ and we have:Γ is a k - F ℓ -Auslander algebra (for some k ) if and only if ( n − ℓ + 1) | n and in this case k = 2( ℓ − n − ℓ +1 ).3. Assume I is a 2-sided admissible ideal with { X | IX = 0 } ⊆ { X | dim K X ≥ n − ℓ + 2 } . Wedefine G ℓ := Γ /I ⊗ Γ G ℓ is a generator for Γ /I and we set F ℓ := F G ℓ . Since we can use the same F -projective and F -injective resolutions (because of the choice of the ideal) we have: Γis a k - F ℓ -Auslander algebra) if and only if Γ /I is an k - F ℓ -Auslander algebra)In particular, if we set I = rad n − ℓ +1 (Γ), then we have G ℓ = Γ /I and if ( n − ℓ + 1) | n then weget a (non-relative) 2( ℓ − n − ℓ +1 )-Auslander algebra.If we allow ℓ = n − n − Example 6.11. Let C n be the oriented cycle quiver with arrows i → i + 1(mod n) and J ⊆ KC n bethe ideal generated by the arrows, N ∈ N , we define Γ := KC n /J N (this is a self-injective Nakayamaalgebra). Let n − ℓ + 1 < N and M ℓℓ ≥ n − ℓ +1 be the direct sum of all modules of vector space dimension ≥ n − ℓ + 1 and let X n be the direct sum of all modules having S n as a composition factor and For ℓ = 1 we have F = Ext and observe domdim Γ = 1 = gldim Γ; ℓ = n, n − ector space dimension < n − ℓ + 1, we define G ℓ = M ℓℓ ≥ n − ℓ +1 ⊕ X n and F ℓ = F G ℓ . Then G n is theAuslander generator and for ℓ = n − n − F n − - Auslander algebra. Moreover,for 1 < ℓ < n − k - F ℓ -Auslander algebra (for some k ) if and only if ( n − ℓ + 1) | n , andin this case k = 2 ℓ − n − ℓ +1 . The proof is exactly the same as in the previous example.7. The 4-tuple assignment Now we consider 4-tuples (Λ , M, L, G ) with Λ a finite-dimensional algebra and M, L, G finite-dimensional Λ-modules. We define the following equivalence relation between these 4-tuples: (Λ , M, L, G )is equivalent to (Λ ′ , M ′ , L ′ , G ′ ) if there is an equivalence of categories Λ-mod ∼ −→ Λ ′ -mod restrictingto equivalences add( M ) ∼ −→ add( M ′ ), add( L ) ∼ −→ add( L ′ ) and add( G ) ∼ −→ add( G ′ ). We denote by[Λ , M, L, G ] the equivalence class of a 4-tuple and we may assume the algebra and all the modulesappearing in the equivalence class to be basic.To establish a relative version of cotilting correspondence which is an involution, we will need thefollowing definition. Definition 7.1. We define the following assignment[Λ , M, L, G ] [Γ , N, e L, e G ]with Γ = End Λ ( M ), N = Γ M , e L = ( G, M ), e G = ( L, M ) and call this the balanced Auslander-Solbergassignment or just the 4-tuple assignment.The dual 4-tuple assignment is the following[Λ , M, R, H ] [Γ , N, e R, e H ]with Γ = End( M ), N = Γ M , e R = D( M, H ), e H = D( M, R ). Since, we will always consider pairs( G, H ) and ( L, R ) which determine each other, we will in later proofs combine the two assignmentsinto a 6-tuple assignment [Λ , M, L, R, G, H ] [Γ , N, e L, e R, e G, e H ]with Γ = End( M ), N = Γ M , e L = ( G, M ), e R = D( M, H ), e G = ( L, M ), e H = D( M, R ). Lemma 7.2. Keep the above notations. Then we have (1) The -tuple assignment restricts to an involution on the set of -tuples [Λ , M, L, G ] with Λ ∈ add( G ) , F = F G , M is - F -faithful, M is an F -dualizing summand of L and L is the left endof an F -exact strong add( M ) -dualizing sequence. (2) The dual -tuple assignment restricts to an involution on the set of -tuples [Λ , M, R, H ] with D Λ ∈ add( H ) , F = F H , M is - F -faithful, M is an F -codualizing summand of R and R isthe right end of an F -exact strong M -dualizing sequence.Proof. We take a 6-tuple [Λ , M, L, R, G, H ] with Λ ∈ add( G ) , D Λ ∈ add( H ), F = F G = F H , M F -faithful and there is an F -exact strong M -dualizing sequence 0 → L → M → M → R → 0. We wantto see that applying the 6-tuple assignment gives an involution. So consider [Γ , N, e L, e R, e G, e H ] withΓ = End( M ), N = Γ M , e L = ( G, M ), e R = D( M, H ), e G = ( L, M ), e H = D( M, R ). Clearly, Γ ∈ add( e G ),D Γ ∈ add( e H ) since M ∈ add( L ) ∩ add( R ) and since L and R are ends of an add( Λ M )-dualizingsequence we have F e G = F e H =: e F . Since L is left end of a strong add( M )-dualizing sequence, we haveby Lemma 5.15 that e G = ( L, M ) ∈ cogen e F ( N ), this means N is 1- e F -faithful. Since M is 1- F -faithfulwe get a strong add( N )-dualizing sequence 0 → e L → f N → f N → e R → → N −→ N −→ N −→ N → → e L ′ → N → N → e R ′ → ∗ )with N i ∈ add( N ). The only missing property is that ( ∗ ) is e F -exact. We first observe that N i =D( M, I i ) with 0 → H ′ → I → I is an injective copresentation, H = D Λ ⊕ H ′ . Since ( e G, − ) =(( L, M ) , − ) is left exact, it is enough to check that it is also right exact on ( ∗ ). Now, since L ∈ cogen F H ( M ) we have a natural isomorphism D( L, H ) → (( L, M ) , D( M, H )) by Lemma 5.7 (1). Inparticular, we have a natural isomorphism ( e G, N i ) = (( L, M ) , D( M, I i )) → D( L, I i ) since I i ∈ add( H ). his means when we apply ( e G, − ) to the last three nonzero terms of ( ∗ ) we get an exact sequencewhich identifies under the just mentioned natural isomorphism withD( L, I ) → D( L, I ) → D( L, H ′ ) → (cid:3) Relative cotilting theory Relative cotilting modules are introduced in [AS93c]. Definition 8.1. Let F = F H ⊆ Ext be an additive subbifunctor with H a cogenerator. We call aΛ-module C a k - F -cotilting module if(i) it is F -self-orthogonal (i.e., Ext > F ( C, C ) = 0),(ii) id F C ≤ k , and(iii) there is an F -exact sequence 0 → C k → · · · → C → C → H → C i ∈ add( C ).We recall a result of Wei. Partially, it is already proven in [AR91a]. Theorem 8.2. ([Wei10, Theorem 3.10]) Let F ⊆ Ext ( − , − ) be an additive subbifunctor with enoughprojectives and injectives, C be a Λ -module and let k ≥ . Then the following are equivalent (1) C is a k - F -cotilting module. (2) cogen k − F ( C ) = T i ≥ ker Ext i F ( − , C ) .In this case, we also have copres k − F ( C ) = cogen k − F ( C ) and cogen k − F ( C ) = cogen k F ( C ) = cogen k +1 F ( C ) = · · · = cogen ∞ F ( C ) . Lemma 8.3. Let F = F G = F H ⊆ Ext ( − , − ) be an additive subbifunctor with enough projectivesand injectives. Let k ≥ and M an F -self-orthogonal module. If id F M ≤ and H ∈ gen F k − ( M ) ,then C = M ⊕ Ω kM H is a k - F -cotilting module. Furthermore, we have cogen k − F ( M ) = \ i ≥ ker Ext i F ( − , C ) . Then M is an ( k − - F -dualizing summand of C .Proof. It is straightforward to check id F C ≤ k by induction on k . Now we check C is F -self-orthogonal:(i) using the definition of Ω kM H by approximations one easily checks Ext i F ( M, Ω kM H ) = 0 for all i ≥ M F -selforthogonal one shows Ext i F (Ω kM H, M ) ∼ = Ext i +1 F (Ω k − M H, M ) = 0 for all i ≥ 1, here the last space is zero since id F M ≤ i F (Ω kM H, Ω kM H ) = 0 for all i ≥ F C ≤ k . More precisely, one applies( − , Ω kM H ) to the F -exact sequences 0 → Ω tM H → M t − → Ω t − M H → M t − ∈ add( M ). Wethen can conclude Ext i F (Ω kM H, Ω kM H ) ∼ = Ext i +1 F (Ω k − M H, Ω kM H ) ∼ = · · · ∼ = Ext i + k F ( H, Ω kM H ) = 0 sinceid F C ≤ k .Together with H ∈ gen F k − ( M ), we conclude that C is an k - F -cotilting module. Furthermore, it iseasy to check cogen k − F ( M ) ⊆ T i ≥ ker Ext i F ( − , C ). We prove the other inclusion by induction over k .Let k = 1. By definition we have C ∈ cogen F ( M ) and this implies using Wei’s result ker Ext F ( − , C ) =cogen F ( C ) ⊆ cogen F ( M ).Let k ≥ 2. Since C does depend on k we denote it in this part of the proof with C k . We first observe(i) T i ≥ ker Ext i F ( − , C k ) ⊆ T i ≥ ker Ext i F ( − , C k − ). This is easy to see using that there is an F -exactsequence 0 → C k → M ′ → C k − → M ′ ∈ add( M ).(ii) By induction hypothesis we may assume T i ≥ ker Ext i F ( − , C k ) ⊆ cogen k − F ( M ).Let X ∈ T i ≥ ker Ext i F ( − , C k ), so there exists an F -exact sequence 0 → X → M → · · · → M k − → Z → M i ∈ add( M ), ( − , M ) exact on it. We claim Z ∈ cogen F ( M ) = ker Ext F ( − , C ). We splitthe sequence up in short F -exact sequences X := X , Z := X k − and 0 → X t → M t → X t +1 → ≤ t ≤ k − 2. Since ( − , M ) is exact on the sequence for t = k − 2, we conclude Ext F ( Z, M ) = 0. So,it is enough to see Ext F ( Z, Ω M H ) = 0. We first show:(iii) Ext F ( Z, Ω M H ) ∼ = Ext k F ( Z, Ω kM H ) by applying ( Z, − ) to the sequences 0 → Ω tM H → M t − → Ω t − M H → i F ( Z, Ω t − M H ) ∼ = Ext i +1 F ( Z, Ω tM H ) for all i ≥ 1. Applying this iteratively ives (iii). Now, we prove:(iv) Ext k F ( Z, Ω kM H ) ∼ = Ext ( X, Ω kM H ) by applying ( − , Ω kM H ) to the short exact sequences 0 → X t → M t → X t +1 → i F ( X t , Ω kM H ) ∼ = Ext i +1 F ( X t +1 , Ω kM H ) for all i ≥ 1. Applying thisiteratively gives (iv).But since X ∈ T i ≥ ker Ext i F ( − , C k ) we have Ext ( X, Ω kM H ) = 0 and therefore, using (iii) and (iv)this implies Ext F ( Z, Ω M H ) = 0. (cid:3) Remark 8.4. If C is a 1- F -cotilting module and M an F -dualizing summand, then we have M = C .Therefore, non-trivial F -dualizing summands only appear in the theory of F -cotilting modules withid F > Example 8.5. Let M ∈ Λ-mod be rigid (i.e., Ext ( M, M ) = 0) and also X := Ω − M be rigid, thenfor H = X ⊕ D Λ, F = F H we have id F M ≤ M is F -self-orthogonal and X ∈ gen ( M ).If we now assume additionally that M is faithfully balanced and Ext , ( M ⊕ X, X ) = 0, then we have H ∈ gen F ( M ) = gen ( M ) ∩ \ i =1 ker Ext i ( − , X )(cf. Example 5.1) implying that M is 1- F -faithful. In particular, we have then C := M ⊕ Ω M H is a1- F -cotilting module with cogen F ( M ) = \ i ≥ ker Ext i F ( − , C ) . Example 8.6. Let X be an arbitrary faithfully balanced module and k ≥ 1. If τ X ∈ cogen k − ( X ),then cogen k − ( X ) is the F X -perpendicular category T i ≥ ker Ext i F X ( − , C ) for the F X - k -coltiltingmodule C = X ⊕ Ω kX D Λ. If add( X ) is, for example, τ -stable then τ X ∈ cogen k − ( X ).More generally we will study the F -cotilting modules obtained from a 1- F -faithful F -injective mod-ule as special cotilting modules (in section 9).Let us fix an F -exact resolution by F -projectives of H (with add( H ) = I ( F )) · · · → P → P → P → H → . Then we obtain the relative version of [IZ18, Theorem 1.1] as follows, let cotilt F n (Λ) be the set ofbasic isomorphism classes of n - F -cotilting Λ-modules. It is naturally a poset with respect C ≤ C ′ ifand only if C ∈ T i ≥ ker Ext i F ( − , C ′ ). Lemma 8.7. Let F = F G = F H and n ≥ , we define P := L n − j =0 P j . If id F P ≤ n and id F Ω nP H ≤ n ,then C = P ⊕ Ω nP H is an n - F -cotilting module and it is the minimum element in cotilt F n (Λ) .Furthermore, if id F P j ≤ j + 1 , ≤ j ≤ n − , then id F P ⊕ Ω nP H ≤ n .Proof. We check that id F C ≤ n implies that C is F -selforthogonal: Observe that Ω nP H = Ω n F H andlet i ≥ 1, then we have Ext i F ( C, C ) = Ext i F (Ω n F H, C ) = Ext i + n F ( H, C ) = 0 since id F C ≤ n .Since the last condition is fulfilled by definition of C , we can conclude that C is an n - F -cotiltingmodule.If L ∈ cotilt F n (Λ), then we have by definition of C that Ext i F ( C, L ) = Ext i + n F ( H, L ) = 0 since id F L ≤ n .Therefore C is the minimum.The last claim is a straight forward induction over n . For n = 1 the claim follows from the previouslemma. For the induction step apply ( − , M ) to the F -exact sequence 0 → Ω nP H → P n − → Ω n − P H → 0, by hypothesis id F P n − ≤ n , id F Ω n − P H ≤ n − F Ω nP H ≤ n . (cid:3) In particular, if P , . . . , P n − are F -injective, this will be referred to as F -domdim Λ ≥ n , then theprevious lemma applies.8.1. The relative cotilting correspondence. We give a generalization of the cotilting correspon-dence to a relative set-up together with a relative dualizing summand - this is a generalization ofAuslander-Solberg’s main results in [AS93c, AS93d] which we reobtain as a corollary. We will use the4-tuple assignments for our theorem (see Definition 7.1, Lemma 7.2).As before, we fix an additive subbifunctor F = F G = F H of Ext ( − , − ) for some generator G andcogenerator H . efine K + ,b F (add( H )) = { Y ∈ K + (add( H )) | ∃ n ∈ Z such that H i (Hom Λ ( Y, H )) = 0 for i ≥ n } then we have D b F (Λ-mod) ≃ K + ,b F (add( H )) as triangulated categories, where D b F (Λ-mod) is the boundedderived category of the exact category Λ-mod with the exact structure induced by F . For more on thederived category of an exact category we refer to [Nee90, Kel96, Pan16]. As in the standard case, onecan prove that an F -self-orthogonal Λ-module L is an F -cotilting module if and only if Thick ( L ) = K b (add( H )) where by Thick ( L ) we mean the smallest triangulated subcategory of K b (add( H )) whichcontains L and closed under direct summands. We also have the following lemma which can be provedby the same argument in the standard case (cf. [CHU94, AI12]). Lemma 8.8. Let L = M ⊕ U be a basic F -cotilting module. (1) If there exists an F -exact sequence → U f −→ M → V → with f the left minimal add( M ) -approximation of U , then M ⊕ V is a basic F -cotilting module with id F ( M ⊕ V ) ≤ id F L .Furthermore, this F -exact sequence ( after adding M to f and its cokernel ) gives rise to astrong - add( M ) -dualizing sequence with Ext i ( U ⊕ M, V ⊕ M ) = 0 for i ≥ . (2) If there exists an F -exact sequence → V → M g −→ U → with g the right minimal add( M ) -approximation of U , then M ⊕ V is a basic F -cotilting module with id F ( M ⊕ V ) ≤ id F L + 1 .Again this gives rise to a strong - add( M ) -dualizing sequence with Ext i ( V ⊕ M, U ⊕ M ) = 0 for i ≥ . Now we are ready to present our improvement of Auslander and Solberg’s results. Recall, the4-tuple assignment [Λ , M, L, G ] [Γ , Γ M , e L, e G ]where Γ = End Λ ( M ), e L = ( G, M ) and e G = ( L, M ). We also consider the dual 4-tuple assignment[Λ , M, R, H ] [Γ , Γ M , e R, e H ]where Γ = End Λ ( M ), e R = D( M, H ) and e H = D( M, R ). Theorem 8.9. Keep the above notations. Then we have (1) The -tuple assignment restricts to an involution on the set of -tuples [Λ , M, L, G ] satisfying (1a) Λ ∈ add( G ) , F = F G , (1b) L is F -cotilting and M is an F -dualizing summand of L . (2) The dual -tuple assignment restricts to an involution on the set of -tuples [Λ , M, R, H ] sat-isfying (2a) D Λ ∈ add( H ) , F = F H , (2b) R is F -cotilting and M is an F -codualizing summand of R ( that is, M ∈ add( R ) and R ∈ gen F ( M ) ) .Furthermore, for an assignment [Λ , M, R, H ] [Γ , Γ M , e R, e H ] we have id F H R = id F e H e R. Proof. We prove (1) and (2) together.We want to use Lemma 7.2, so we first prove that (1b) (or (2b)) implies that M is 1- F -faithful.To prove M is 1- F -faithful we need to show the natural map ( M, H ) ⊗ Γ ( G, M ) → ( G, H ) is anisomorphism, where Γ = End Λ ( M ). Since L is F -cotilting it is 1- F -faithful and thus the natural map( L, H ) ⊗ B ( G, L ) → ( G, H ) is an isomorphism, where B = End Λ ( L ). By Lemma 5.7 (1), M being an F -dualizing summand of L is equivalent to that the natural map ( M, H ) ⊗ Γ ( L, M ) → ( L, H ) is anisomorphism. Hence we have( M, H ) ⊗ Γ ( G, M ) ∼ = −→ (( M, H ) ⊗ Γ ( L, M )) ⊗ B ( G, L ) ∼ = −→ ( L, H ) ⊗ B ( G, L ) ∼ = −→ ( G, H )as desired. Since L is F -cotilting and M is an F -dualizing summand of L , we have an F -exact strongadd( M )-dualizing sequence 0 → L → M → M → R → M i ∈ add( M ). By Lemma 8.8 wesee that R is also an F -cotilting module. Now, by Lemma 7.2 the 6-tuple assignment restricts to aninvolution on the set of 6-tuples [Λ , M, L, R, G, H ] satisfying the conditions (1a), (1b), (2a) and (2b)if we prove that e R := D( M, H ) and e L := ( G, M ) are e F -cotilting modules, where e F := F e G = F e H , e G = ( L, M ) and e H = D( M, R ). ssume id F R = n , then we have F -exact sequences0 → R → H → H → · · · → H n − → H n → ∗ )and 0 → R n → R n − → · · · → R → R → H → . ( ∗∗ )The functor ( M, − ) is exact on both ( ∗ ) and ( ∗∗ ). Applying D( M, − ) to ( ∗∗ ) we get an exactsequence0 → D( M, H ) = e R → D( M, R ) → D( M, R ) → · · · → D( M, R n − ) → D( M, R n ) → ⋆⋆ )of Γ-modules, where each D( M, R i ) ∈ add( e H ) is an e F -injective module. We claim that this sequenceis e F -exact which will imply that ( ⋆⋆ ) is an e F -injective resolution of e R and so id e F e R ≤ n . Consider thefollowing commutative diagram0 / / (( L, M ) , D( M, H )) / / ∼ = (cid:15) (cid:15) (( L, M ) , D( M, R )) / / ∼ = (cid:15) (cid:15) · · · / / (( L, M ) , D( M, R n )) / / ∼ = (cid:15) (cid:15) / / D(( M, H ) ⊗ ( L, M )) / / D(( M, R ) ⊗ ( L, M )) / / · · · / / D(( M, R n ) ⊗ ( L, M )) / / / / D( L, H ) / / ∼ = O O D( L, R ) / / ∼ = O O · · · / / D( L, R n ) / / ∼ = O O H, R ∈ gen F L ( M ). The last row is obtainedby applying the functor D( L, − ) to ( ∗∗ ) and it is exact . Hence the first row is exact and the claimfollows.Similarly, apply the functor D( M, − ) to ( ∗ ) we will get an e F -exact sequence0 → D( M, H n ) → D( M, H n − ) → · · · → D( M, H ) → D( M, H ) → D( M, R ) = e H → ⋆ )with D( M, H i ) ∈ add( e R ). Now applying the functor D( e R, − ) = D(( M, H ) , − ) to ( ⋆⋆ ) we will get thefirst row of the following commutative diagram0 / / (D( M, H ) , D( M, H )) / / (D( M, H ) , D( M, R )) / / · · · / / (D( M, H ) , D( M, R n )) / / / / ( H, H ) / / ∼ = D( M, − ) O O ( R , H ) / / ∼ = D( M, − ) O O · · · / / ( R n , H ) / / ∼ = D( M, − ) O O ∗∗ ) is F -exact and the vertical arrows are isomorphisms because H, R ∈ gen ( M ). Therefore the upper row is exact and this means Ext i e F ( e R, e R ) = 0 for i > ⋆ ) and ( ⋆⋆ ), we see that e R is an e F -cotilting module. According to the proof of Lemma7.2, there is a strong add( Γ M )-dualizing sequnce 0 → e L → f M → f M → e R → f M i ∈ add( Γ M ).Again by Lemma 8.8, we conclude that e L is an e F -cotilting module.Finally, since the dual 4-tuple assignment restricts to an involution we have id F R = id e F e R . (cid:3) Corollary 8.10. (1) The functors ( − , Λ M ) : Λ - mod ←→ Γ - mod : ( − , Γ M ) restrict to dualities < ⊥ F L ←→ < ⊥ e F e L and < ⊥ F R ←→ < ⊥ e F e R . (2) We have id F R ≤ id F L ≤ id F R + 2 and id e F e R ≤ id e F e L ≤ id e F e R + 2 .Proof. (1) Given X ∈ < ⊥ F L we need to show that ( X, Λ M ) ∈ < ⊥ e F e L and it is enought to show( X, Λ M ) ∈ copres ∞ e F ( e L ) by Theorem 8.2. Taking an F -projective resolution · · · → P → P → X → X and applying ( − , Λ M ) to get a complex 0 → ( X, M ) → ( P , M ) → ( P , M ) → · · · . A standardargument shows that it is e F -exact and therefore ( X, Λ M ) ∈ copres ∞ e F ( e L ). Now given Y ∈ < ⊥ F R wewill prove that ( Y, Λ M ) ∈ < ⊥ e F e R . Applying (( Y, M ) , − ) to the e F -injective resolution ( ⋆⋆ ) of e R givesa complex 0 → (( Y, M ) , e R ) → (( Y, M ) , D( M, R )) → · · · → (( Y, M ) , D( M, R n )) → 0. One can easilycheck that it is in fact exact and thus ( Y, Λ M ) ∈ < ⊥ e F e R .(2) follows from Lemma 8.8. (cid:3) emark 8.11. In particular, if we take M = L to be the trivial F -dualizing summand then we have[Λ , L, L, G ] [Γ , Γ L, e L = ( G, L ) , Γ] and thus e L is a cotilting Γ-module, Γ L is a dualizing summandof e L and id F L ≤ id Γ e L ≤ id F L + 2. This gives [AS93c, Theorem 3.13]. The fact that the 4-tupleassignment restricts to an involution gives [AS93d, Theorem 2.8].8.2. Derived equivalence induced by an F-dualizing summand. Let [Λ , M, L, G ] be a 4-tuplesatisfying Λ ∈ add( G ), F = F G , L is F -cotilting and M is an F -dualizing summand of L . Thenby Theorem 8.9 the 4-tuple assignment gives a 4-tuple [Γ = End Λ ( M ) , Γ M, e L = ( G, M ) , e G = ( L, M )]satisfying Γ ∈ add( e G ), e F = F e G , e L is e F -cotilting and Γ M is an e F -dualizing summand of e L . We considerthe derived categories of exact categories D b F (Λ-mod) and D b e F (Γ-mod) and we will show the functors( − , Λ M ) and ( − , Γ M ) induce a duality between triangulated categories D b F (Λ-mod) and D b e F (Γ-mod). Proposition 8.12. Let [Λ , M, L, G ] be a -tuple such that Λ ∈ add( G ) , F = F G , L is F -cotilting and M is an F -dualizing summand of L and let [Γ , Γ M, e L, e G ] be the corresponding -tuple under the -tupleassignment. Then the functors ( − , Λ M ) and ( − , Γ M ) induce a triangle duality between D b F (Λ - mod) and D b e F (Γ - mod) .Proof. Let B = End Λ ( L ) and e B = End Γ ( e L ), then C := ( G, L ) is a cotilting B -module and e C :=( e G, e L ) is a cotilting e B -module. By [Bua01, Proposition 4.4.3] , the functor ( − , Λ L ) induces a triangleduality between D b F (Λ-mod) and D b ( B -mod) and the functor ( − , Γ e L ) induces a triangle duality between D b e F (Γ-mod) and D b ( e B -mod).We note that by Lemma 2.4 (1) the compositionEnd B ( C ) = (( G, L ) , ( G, L )) ∼ = −→ End Λ ( G ) op ∼ = −→ (( G, M ) , ( G, M )) = End Γ ( e L ) = e B is an isomorphism of algebras. Similarly, we have End e B ( e C ) ∼ = End Γ ( e G ) op ∼ = B . Since B C is cotilting, e B C is also cotilting and we have e B C = ( B, B C ) = (( L, L ) , ( G, L )) ∼ = −→ ( G, L ) ∼ = −→ (( L, M ) , ( G, M )) = ( e G, e L ) = e B e C by Lemma 2.4 (1). It follows that the functors ( − , B C ) and ( − , e B e C ) induce a triangle duality between D b ( B -mod) and D b ( e B -mod). The desired triangle duality follows by combining this duality and theabove triangle dualities. (cid:3) Remark 8.13. As the above proof suggests, there exist triangle equivalences D b F (Λ-mod) ≃ D b ( e B -mod)and D b F (Γ-mod) ≃ D b ( B -mod). The dual version of Proposition 8.12 shows that an F -codualizing sum-mand of an F -tilting module will induce a relative derived equivalence.8.3. F-Gorenstein algebra. Recall that an algebra Λ is called Gorenstein if id( Λ Λ) < ∞ andid(Λ Λ ) < ∞ . Define P ∞ (Λ) = { X ∈ Λ-mod | pd Λ X < ∞} and I ∞ (Λ) = { Y ∈ Λ-mod | id Λ Y < ∞} . Then Λ being Gorenstein is equivalent to P ∞ (Λ) = I ∞ (Λ). Let F = F G = F H be a subbifunctor ofExt and define P ∞ ( F ) = { X ∈ Λ-mod | pd F X < ∞} and I ∞ ( F ) = { Y ∈ Λ-mod | id F Y < ∞} . Following [AS93a] we call an algebra F - Gorenstein if P ∞ ( F ) = I ∞ ( F ), and F -Gorenstein algebrascan be chcaracterized as follows. Lemma 8.14. ([AS93a, Proposition 3.3])(1) An algebra Λ is F -Gorenstein if and only if there exists an F -cotilting F -tilting module. (2) An algebra Λ is F -Gorenstein if and only if every F -cotilting module is F -tilting and every F -tilting module is F -cotilting. Corollary 8.15. Let [Λ , M, L, G ] be a -tuple satisfying Λ ∈ add( G ) , F = F G , L is F -cotilting and M is an F -dualizing summand of L and let [Γ , Γ M, e L, e G ] be the corresponding -tuple under the -tupleassignment. Then Λ is an F -Gorenstein algebra if and only if Γ is an e F -Gorenstein algebra. roof. Consider the 6-tuple assignment [Λ , M, L, R, G, H ] [Γ , Γ M, e L, e R, e G, e H ] as in the proof ofTheorem 8.9. Then L, R are F -cotilting modules and e L, e R are e F -cotilting modules. By Lemma 8.14,Λ is F -Gorenstein if and only if L and R are F -tilting modules, if and only if e L, e R are e F -tilting modulesby the tilting version of Theorem 8.9, if and only if Γ is e F -Gorenstein by Lemma 8.14 again. (cid:3) Remark 8.16. (1) The tilting version of Theorem 8.9 implies that pd e F e L = pd F L , pd F L ≤ pd F R ≤ pd F L + 2 and pd e F e L ≤ pd e F e R ≤ pd e F e L + 2. Now by using [AS93a, Proposition 3.4],we see that pd e F e G = id e F e H ≤ pd e F e L + id e F e L ≤ pd F L + id F R + 2.(2) In particular, if we take M = L then the above result gives [AS93a, Proposition 3.1 andProposition 3.6]. 9. Special cotilting We assume throughout this section that F = F G = F H for a generator G and a cogenerator H .The easiest situation where relative dualizing summands appear in relative cotilting modules are whenthese summands are 1- F -faithful F -injective modules. Definition 9.1. Let C be an F -cotilting module of id F C ≤ r . We say that C is special if it has an F -injective ( r − F -dualizing summand I . This is equivalent to an F -injective summand I of C suchthat cogen r − F ( C ) = cogen r − F ( I ) by Lemma 5.19. We sometimes call C I -special if it is special withrespect to the F -injective I .Dually, we say an F -tilting module T of pd F T ≤ r is special if it has a F -projective summand P suchthat gen F r − ( T ) = gen F r − ( P ).We look at a minimal F -injective F -coresolution of G → G → I → I → I → · · · and define J n = L t ≤ n I t (so in particular we have G ∈ cogen n F ( J n ) Theorem 9.2. Let r ≥ . We consider the following three finite sets. (1) Isomorphism classes of basic special cotilting modules of id F ≤ r . (2) Isomorphism classes of basic F -injective modules I with G ∈ cogen r − F ( I ) . (3) Isomorphism classes of basic I ∈ add( H ) with J r − ∈ add( I ) .Then the sets (2) and (3) are equal. Mapping C to its maximal F -injective summand gives a bijectionbetween (1) and (2) . The inverse is given by mapping I to C I,r := I ⊕ Ω rI H .Proof. Assume J r − ∈ add( I ) ⊂ add( H ), then clearly G ∈ cogen r − F ( J r − ) ⊂ cogen r − F ( I ) and weconclude that (3) is a subset of (2). So assume I ∈ add( H ) with G ∈ cogen r − F ( I ). Since the minimal F -injective F -exact r -copresentation (of G ) must be a summand of any other F -injective F -exact r -copresentation, it follows that J r − ∈ add( I ) and therefore the sets (2) and (3) are equal.So let C be an I -special r - F -cotilting module and let J be its maximal injective summand - of course I ∈ add( J ) and clearly copres r − F ( I ) ⊆ copres r − F ( J ) ⊆ copres r − F ( C ). Since I, J are F -injective and C is r - F -cotilting we conclude that these inclusions of subcategories coincide with cogen r − F ( I ) ⊆ cogen r − F ( J ) ⊆ cogen r − F ( C ). Since C is I -special it follows that they are all equal, in particular J ∈ cogen F ( I ) implies J ∈ add( I ) and therefore add( I ) = add( J ). This means the map is well-defined. It follows from lemma 8.3 that the assignment I C I = I ⊕ Ω rI H is the inverse map. (cid:3) Let Σ r F (Λ) be the finite subposet of the poset of isomorphism classes of basic F -cotilting modulesof id F ≤ r , where the partial order is given by inclusion of perpendicular categories...Let add J r − ( H ) be the lattice given by isomorphism classes of basic summands I of H such that J r − ∈ add( I ) . The partial order is just given by inclusion of summands, the meet and join aredefined in the obvious way. In particular, if H = J r − ⊕ X with | X | = t , then the lattice add J r − ( H )is isomorphic to the power set P ( { , , . . . , t } ) which is a poset with respect to inclusion and a latticewith respect to intersection and union (sometimes also referred to as a t -dimensional cube). Corollary 9.3. The finite poset Σ r F (Λ) is a lattice and the bijection from the previous theorem givesa lattice isomorphism Σ r F (Λ) → add J r − ( H ) . e also observe that if an I -special r - F -cotilting module C has an ( r − F -dualizing summand M , then I ∈ add( M ).We give now several little applications, in particular connecting it with the other parts of the article.9.1. Examples and applications. (1) Non-relative special tilting has been defined in [PS17] and many special cases had been con-sidered before, as APR-tilting and BB-tilting [BGfP73], [BB80], [APR79], n -APR-tilts [IO11]or flip-flops for posets [Lad07]. Any endomorphism ring of a generator has a canonical specialcotilt, this has been used to define desingularizations of orbit closures and quiver Grassman-nians in [CIFR13], [CBS17], [PS18].(2) We explain that (non-relative) special cotilting naturally gives two recollements relating thecotilted algebras: Let I be a ( k − k ≥ C = C I,k = I ⊕ Ω kI D Λ the I -special k -cotilting module. Then Ω kI is an equivalence of categories add D Λ / add I → add C/ add I (for the definition of ideal quotients, see [ASS06, A.3]) with quasi-inverse Ω − kI (this follows from [AR91b, Theorem 5.2] with X = add( I )). Let B = End Λ ( C ) op , then B D C is special k -tilting module with respect to the ( k − P = ( C, I ).Let P = Bε and I = D( e Λ) for idempotents e ∈ Λ , ε ∈ B . Then the equivalence Ω kI inducedan isomorphism of algebras(Λ / ( e )) op = End add D Λ / add I (D Λ) ∼ = End add C/ add I ( C ) = ( B/ ( ε )) op Observe also e Λ e ∼ = End Λ ( I ) op ∼ = εBε , therefore we have two recollements with isomorphicends induced by the idempotents e, ε . Λ-mod F (cid:15) (cid:15) p u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ q u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ e ❙❙❙❙❙❙❙❙ ) ) ❙❙❙❙❙❙❙ Λ / ( e )- mod i ❦❦❦❦❦❦❦ ❦❦❦❦❦❦❦❦ j ❚❚❚❚❚❚❚ ) ) ❚❚❚❚❚❚❚ eΛe-mod ℓ i i ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ r i i ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ λ u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ ρ u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ B - mod π i i ❚❚❚❚❚❚❚❚❚❚❚❚❚❚ φ i i ❚❚❚❚❚❚❚❚❚❚❚❚❚❚ ε ❦❦❦❦❦❦❦ ❦❦❦❦❦❦❦❦ Furthermore, the cotilting functor F := D( − , C ) commutes with the following functors fromthe recollements ε ◦ F = e, F ◦ ℓ = λ .(3) The standard cogenerator correspondence says that the assignment [Λ , Λ E ] [Γ , Γ I ] definedby Γ = End( E ) , I = Γ E gives a bijection between(a) [Λ , Λ E ] with D Λ ∈ add E. (b) [Γ , Γ I ] with I injective and Γ ∈ cogen ( I ).Let us denote C I to be the special 2-cotilting Γ-module which exists in situation (b). Thenthe AS-assignment [Γ , Γ I, C I ] [Λ , Λ E, G ] with G = ( C I , I ) gives a natural extension of thecogenerator correspondence to a bijection between the following.(a’) [Λ , Λ E, G ] with D Λ ∈ add E and E is an F G -cotilting module.(b’) [Γ , Γ I, C I ] with I injective and Γ ∈ cogen ( I ), C I ( C I ) = cogen ( I ).This can be generalized to the 4-tuple assignment as follows:9.1.1. Example of the relative cotilting correspondence using special cotilting. This is our main exam-ple for theorem 8.9. Let us look at the 5-tuple assignment [Λ , I, L, G, H ] [Γ = End Λ ( I ) , Γ I, e L =( G, I ) , e G = ( L, I ) , e H = D( I, H )]. Then this gives a involution on the following 5-tuples [Λ , I, L, G, H ]with Λ ∈ add( G ) , D Λ ⊕ I ∈ add( H ), F = F H = F G and L is an I -special 2- F -cotilting module.The proof goes as follows: By Theorem 8.9 we know that e L is again an e F -cotilting module with e F = F e H and has an e F -dualizing summand Γ I . So we need to see that id e F e L ≤ 2, then e L is the(uniquely determined) Γ I -special 2- F -cotilting module. Recall that the assumption ensures that wehave an F -exact strong I -dualizing sequence 0 → L → I → I → H → I j ∈ add( I ), so wecan see R := H as the right end of it. This has been used to show that for e G := ( L, I ) , e H = D( I, H )we have e F = F e H = F e G . Now, apply ( − , I ) to a minimal projective presentation of G and D( I, − ) o a minimal injective copresentation of H to obtain an e F -exact, strong Γ I -dualizing sequence withleft end ( G, I ) = e L and right end D( I, H ) = e H . This ensures that id e F e L ≤ e L is an Γ I -special 2- e F -cotilting.We remark that special r -(co)tilting requires an F -injective ( r − F -dualizing summand. In ourpreviously considered assignments we looked only at 1- F -dualizing summands, that is why our exampleonly works for r = 2.9.1.2. Mutation and dualizing sequences induce special tilts on endomorphism rings. Lemma 9.4. Let → L → M → · · · → M k → R → be an F -exact strong k - M -dualizing sequencewith Ext j F ( L, R ) = 0 for j ≥ and L , R be F -selforthogonal. Let B = End( L ) and A = End( R ) .Then T = ( L, R ) is a special k -tilting A -module with respect to P = ( M, R ) and C = D( L, R ) is aspecial k -cotilting B -module with respect to I = D( L, M ) . Furthermore, we have End A ( T ) ∼ = B op and End B ( C ) ∼ = A op .Proof. Apply ( − , R ) to the strong dualizing sequence, setting P i = A ( M i , R ) ∈ add( P ), we get anexact sequence of A -modules 0 → A → P k → · · · → P → T → . This shows pd T ≤ k and A has an add( T )-resolution with all middle terms in add( P ) ( ⊆ add( T )).Since the dualizing sequence is strong and by assumption L ∈ F , ≤ ⊥ R ∩ cogen ∞ F ( R ), we can use Lemma2.4,(2) to get an isomorphism Ext j F ( L, L ) → Ext jA ( T, T ). Since L is F -selforthogonal, the module T is selforthogonal. This implies that T is a special k -tilting module with respect to P . Similarly, onecan show that C is a special k -cotilting module with respect to I . The last claim follows from Lemma2.4,(1). (cid:3) Passing to endomorphism rings of special cotilting modules. Recall, that in the non-relativecase the Brenner-Butler assignment ( BB ) : [Σ , J, L ′ ] [ B = End Σ ( L ′ ) , D( L ′ , J ) , B L ′ ] maps J -special t -cotilting Σ-modules L ′ to a D( L ′ , J )-special t -cotilting B -module and this assignment is an involutionon these triples.We explain how this relates to relative special cotilting: Let H be a basic cogenerator, Σ = End Λ ( H ) op and ε ∈ Σ the projection onto the summand D Λ, then we have a pair of adjoint functors ℓ = D( − , H ) : Λ-mod ⇄ Σ-mod : ε = (Σ ε, − )(cf. Appendix) with Im ℓ = gen (Σ ε ). As always we set F = F H . Then for I ∈ add( H ) we have ℓ ( I ) ∈ add(D Σ) and: H ∈ gen F t − ( I ) ⇔ D Σ ∈ gen t − ( ℓ ( I )) , L tj ≥ Ω jℓ ( I ) D Σ ∈ gen (Σ ε ).The assignment [Λ , I, L, H ] [Σ = End Λ ( H ) op , ℓ ( I ) , ℓ ( L )] injects an I -special t - F H -cotilting mod-ules L to an ℓ ( I )-special t -cotilting Σ-module ℓ ( L ). Any J -special t -cotilting Σ-module L ′ for some J ∈ add(D Σ) is in the image of this assignment if and only if L tj ≥ Ω jJ D Σ ∈ gen (Σ ε ). The as-signment [Λ , I, L, H ] [ B = End Λ ( L ) , D( L, I ) , D( L, H )] injects an I -special t - F H -cotilting modules L to an D( L, I )-special t -cotilting B -module D( L, H ). In fact, combining the assignments we get acommuting triangle as follows [Λ , I, L, H ] ℓ t t ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ D( L, − ) * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ [Σ = End Λ ( H ) op , ℓ ( I ) , ℓ ( L )] o o ( BB ) / / [ B = End Λ ( L ) , D( L, I ) , D( L, H )] Example 9.5. Here are the endomorphism rings of the special cotilts of the relative Auslander algebrasfor Λ = K (1 → → → → G = P ⊕ M, M = L i =1 L j ≥ τ − j P i , F = F G from Example 6.9,(4). We choose I = M ∈ add( H ), then the M -special tilting and cotilting modules conincide with: G, M ⊕ S , M ⊕ S , M ⊕ S , H . Their respective endomorphism ring is shown by the quiver with elations below. • ❅❅ • > > ⑦⑦ ❅❅ • ❅❅ • > > ⑦⑦ ❅❅ • ❅❅ > > ⑦⑦ • ❅❅ • > > ⑦⑦ • > > ⑦⑦ • > > ⑦⑦ •• > > ⑦⑦ • ❅❅ • > > ⑦⑦ ❅❅ • ❅❅ • > > ⑦⑦ ❅❅ • ❅❅ > > ⑦⑦ • ❅❅ • > > ⑦⑦ ❅❅ • > > ⑦⑦ • > > ⑦⑦ •• > > ⑦⑦ • ❅❅ • > > ⑦⑦ ❅❅ • ❅❅ • > > ⑦⑦ ❅❅ • ❅❅ > > ⑦⑦ • ❅❅ • > > ⑦⑦ • > > ⑦⑦ ❅❅ • > > ⑦⑦ •• > > ⑦⑦ • ❅❅ • > > ⑦⑦ ❅❅ • ❅❅ • > > ⑦⑦ ❅❅ • ❅❅ > > ⑦⑦ • ❅❅ • > > ⑦⑦ • > > ⑦⑦ • > > ⑦⑦ ❅❅ •• > > ⑦⑦ • ❅❅ • > > ⑦⑦ ❅❅ • ❅❅ • > > ⑦⑦ ❅❅ • ❅❅ > > ⑦⑦ • ❅❅ • > > ⑦⑦ • > > ⑦⑦ • > > ⑦⑦ • ❅❅ • Appendix: Embedding into an abelian category We fix ∆ = End Λ ( G ) op and e ∈ ∆ projection onto the summand Λ (resp. Σ = End Λ ( H ) op and ε ∈ Σ the projection onto D Λ), r = Hom Λ ( G, − ) then we have a pair ( e, r ) of adjoint functors (resp. ℓ = Σ ε ⊗ Λ − = D Hom Λ ( − , H ), then we have an adjoint pair ( ε, ℓ )) e : ∆-mod ⇄ Λ-mod : r (resp. ℓ : Λ-mod ⇄ Σ-mod : ε )with e is exact and r is fully faithful, maps F -exact sequences to exact sequences and add( G ) toadd(∆). In particular, it maps F -projective resolutions to projective resolutions and we get inducedisomorphisms Ext i F ( M, N ) → Ext i ∆ ( r ( M ) , r ( N )) , i ≥ . Dually, ε is exact, ℓ is fully faithful, maps F -exact sequences to exact sequences and add( H ) toadd(D Σ), it maps F -injective resolutions to injective resolutions and induces isomorphisms on theExt-groups Ext i F ( M, N ) → Ext i Σ ( ℓ ( M ) , ℓ ( N )) , i ≥ 0. We haveIm r = cogen (D( e ∆)) and Im ℓ = gen (Σ ε ) . It is also easy to see: If T is a relative tilting Λ-module, then r ( T ) is a tilting ∆-module: Conversely,every tilting ∆-module in cogen ( J ) restricts under e to a relative tilting module. This gives a bijection,respecting the partial order (given by inclusion of perpendicular categories).If C is a relative cotilting module then ℓ ( C ) is a cotilting Σ-module and every cotilting module inIm ℓ = gen (Σ ε ) restricts under ε to a relative cotilting module.Furthermore, in [AS93c] Auslander and Solberg showedgldim F Λ ≤ gldim ∆ ≤ gldim F Λ + 2 . Lemma 10.1. Let Λ and ∆ be as before and k ≥ . Then the following are equivalent: (1) pd F D Λ ≤ k and gldim ∆ ≤ k + 2 , (2) gldim F Λ ≤ k , (3) id F Λ ≤ k and gldim Σ ≤ k + 2 .Proof. (1) ⇒ (2) : Let J = D( e ∆). Clearly, gldim F Λ ≤ k if and only if Ext k +1∆ (cogen ( J ) , cogen ( J )) =0. We have J = D( e ∆) = r (D Λ) and it is easily seen that pd F D Λ ≤ k is equivalent to pd ∆ J ≤ k .We claim the stronger implication: gldim ∆ ≤ k +2 and pd J ≤ k implies Ext k +1 (cogen ( J ) , ∆-mod) =0 (i.e., pd X ≤ k for all X ∈ cogen ( J )).If we have an exact sequence 0 → A → J → B → J ∈ add( J ) and we apply a functor ( − , Y )then we get a dimension shift Ext i ( A, Y ) ∼ = Ext i +1 ( B, Y ) for all i ≥ k + 1. In particular, we havefor X ∈ cogen ( J ): Ext k +1 ( X, Y ) ∼ = Ext k +2 (Ω − X, Y ) ∼ = Ext k +3 (Ω − X, Y ) = 0 since we assume thatgldim ∆ ≤ k + 2.(2) ⇒ (1) Clearly, if gldim F Λ ≤ k , then pd F D Λ ≤ k . By Auslander-Solberg’s result (see before) wealso have gldim ∆ ≤ gldim F Λ + 2 ≤ k + 2 . The equivalence of (2) and (3) is proven analogously. (cid:3) xample 10.2. Let Λ = K (1 → → · · · → n ). Then there are 2 N with N = P n − k =1 k basic generators G . The minimal F -global dimension is 0 which is obtained if and only of G is the Auslander generator.The maximal F -global dimension is n − References [AB69] Maurice Auslander and Mark Bridger, Stable module theory , Memoirs of the American Mathematical Society,No. 94, American Mathematical Society, Providence, R.I., 1969.[AF92] F. W. Anderson and K. R. Fuller, Rings and categories of modules , second ed., Graduate Texts in Mathematics,vol. 13, Springer-Verlag, New York, 1992.[AI12] T. Aihara and O. Iyama, Silting mutation in triangulated categories , Journal of the London Mathematical Society (2012), no. 3, 633–668.[APR79] M. Auslander, M. Platzeck, and I. Reiten, Coxeter functors without diagrams , Trans. Amer. Math. Soc. (1979), 1–46. MR 530043[AR91a] M. Auslander and I. Reiten, Applications of contravariantly finite subcategories , Adv. Math. (1991), no. 1,111–152.[AR91b] , Cohen-Macaulay and Gorenstein Artin algebras , Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991), Progr. Math., vol. 95, Birkh¨auser, Basel, 1991, pp. 221–245.[ARS95] M. Auslander, I. Reiten, and Sverre O. Smalø, Representation theory of Artin algebras. , vol. 36, Cambridge:Cambridge University Press, 1995.[AS93a] M. Auslander and O. Solberg, Gorenstein algebras and algebras with dominant dimension at least 2, Comm.Algebra (1993), no. 11, 3897–3934.[AS93b] , Relative homology and representation theory. I. Relative homology and homologically finite subcategories ,Comm. Algebra (1993), no. 9, 2995–3031.[AS93c] , Relative homology and representation theory. II. Relative cotilting theory , Comm. Algebra (1993),no. 9, 3033–3079.[AS93d] , Relative homology and representation theory. III. Cotilting modules and Wedderburn correspondence ,Comm. Algebra (1993), no. 9, 3081–3097.[ASS06] I. Assem, D. Simson, and A. Skowro´nski, Elements of the representation theory of associative algebras. Vol. 1 ,London Mathematical Society Student Texts, vol. 65, Cambridge University Press, Cambridge, 2006, Techniquesof representation theory.[Aus99] M. Auslander, Representation dimension of artin algebras , Selected works of Maurice Auslander. Part 1, Amer-ican Mathematical Society, Providence, RI, 1999, Edited and with a foreword by I. Reiten, S. O. Smalo , andO. Solberg, pp. 505–575.[Azu66] G. Azumaya, Completely faithful modules and self-injective rings , Nagoya Math. J. (1966), 697–708.[BB80] S. Brenner and M. C. R. Butler, Generalizations of the Bernstein-Gel’fand-Ponomarev reflection functors , Rep-resentation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes inMath., vol. 832, Springer, Berlin-New York, 1980, pp. 103–169.[BGfP73] I. N. Bernstein, I. M. Gel’ fand, and V. A. Ponomarev, Coxeter functors, and Gabriel’s theorem , Uspehi Mat.Nauk (1973), no. 2(170), 19–33. MR 0393065[BS98] A. B. Buan and O. Solberg, Relative cotilting theory and almost complete cotilting modules , Algebras andmodules, II (Geiranger, 1996), CMS Conf. Proc., vol. 24, Amer. Math. Soc., Providence, RI, 1998, pp. 77–92.[Bua01] A. B. Buan, Closed subbifunctors of the extension functor , J. Algebra (2001), no. 2, 407–428.[CBS17] W. Crawley-Boevey and J. Sauter, On quiver Grassmannians and orbit closures for representation-finite alge-bras , Math. Z. (2017), no. 1-2, 367–395, arXiv:1509.03460 [math.RT].[CHU94] F. Coelho, D. Happel, and L. Unger, Complements to partial tilting modules , Journal of Algebra (1994),no. 1, 184 – 205.[CIFR13] G. Cerulli Irelli, E. Feigin, and M. Reineke, Desingularization of quiver Grassmannians for Dynkin quivers ,Adv. Math. (2013), 182–207.[CPS86] E. Cline, B. Parshall, and L. Scott, Derived categories and morita theory , Journal of Algebra (1986), no. 2,397–409.[CR72] R. S. Cunningham and E. A. Rutter, The double centralizer property is categorical , Rocky Mt. J. Math. (1972),627–629.[DR72] V. Dlab and C. M. Ringel, Rings with the double centralizer property , J. Algebra (1972), 480–501.[DRSS99] P. Dr¨axler, I. Reiten, S.O. Smalø, and O. Solberg, Exact categories and vector space categories. (with anappendix by b. keller). , Trans. Am. Math. Soc. (1999), no. 2, 647–682.[Hap88] D. Happel, Triangulated categories in the representation theory of finite-dimensional algebras , London Mathe-matical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988.[IO11] O. Iyama and S. Oppermann, n -representation-finite algebras and n -APR tilting , Trans. Amer. Math. Soc. (2011), no. 12, 6575–6614, arXiv:0909.0593 [math.RT].[IS18] O. Iyama and O. Solberg, Auslander-Gorenstein algebras and precluster tilting , Adv. Math. (2018), 200–240.[Iya07] O. Iyama, Auslander correspondence , Adv. Math. (2007), no. 1, 51–82.[Iya08] , Auslander-Reiten theory revisited , Trends in representation theory of algebras and related topics, EMSSer. Congr. Rep., Eur. Math. Soc., Z¨urich, 2008, arXiv:0803.2841 [math.RT], pp. 349–397. IZ18] O. Iyama and X. Zhang, Tilting modules over Auslander-Gorenstein Algebras , arXiv e-prints (2018),arXiv:1801.04738.[Kel96] B. Keller, Derived categories and their uses , Handbook of Algebra (M. Hazewinkel, ed.), Handbook of Algebra,vol. 1, North-Holland, 1996, pp. 671 – 701.[Lad07] S. Ladkani, Universal derived equivalences of posets of cluster tilting objects , arXiv e-prints (2007),arXiv:0710.2860.[Miy86] Y. Miyashita, Tilting modules of finite projective dimension , Math. Z. (1986), no. 1, 113–146.[Mor58] K. Morita, On algebras for which every faithful representation is its own second commutator , Math. Z. (1958),429–434.[Mor71] , Flat modules, injective modules and quotient rings , Math. Z. (1971), 25–40.[M¨ul68] B. J. M¨uller, The classification of algebras by dominant dimension , Canad. J. Math. (1968), 398–409.[Nee90] A. Neeman, The derived category of an exact category , Journal of Algebra (1990), no. 2, 388 – 394.[Pan16] S.Y. Pan, Relative derived equivalences and relative homological dimensions , Acta Math. Sin. (Engl. Ser.) (2016), no. 4, 439–456.[PS17] M. Pressland and J. Sauter, Special tilting modules for algebras with positive dominant dimension , arXiv e-prints(2017), arXiv:1705.03367.[PS18] , On quiver Grassmannians and orbit closures for gen-finite modules , arXiv e-prints (2018),arXiv:1802.01848.[Rin07] C. M. Ringel, Artin algebras of dominant dimension at least 2 , 2007, Seminar notes, Bielefeld.[SW09] Y. Sun and J. Wei, n - C -star modules and n - C -tilting modules , Comm. Algebra (2009), no. 7, 2457–2467.[Tac69] H. Tachikawa, On splitting of module categories , Math. Z. (1969), 145–150.[Tac70] , On left QF − rings , Pacific J. Math. (1970), 255–268.[Wak88] T. Wakamatsu, On modules with trivial self-extensions , J. Algebra (1988), no. 1, 106–114.[Wei10] J. Wei, A note on relative tilting modules , J. Pure Appl. Algebra (2010), no. 4, 493–500.[Wis00] R. Wisbauer, Static modules and equivalences , Interactions between ring theory and representations of algebras(Murcia), Lecture Notes in Pure and Appl. Math., vol. 210, Dekker, New York, 2000, pp. 423–449.[Xi00] C. Xi, The relative auslander-reiten theory of modules , 2000, unpublished preprint. Biao Ma, Department of Mathematics, Nanjing University, 22 Hankou Road, Nanjing 210093, People’sRepublic of China E-mail address : [email protected] Julia Sauter, Faculty of Mathematics, Bielefeld University, PO Box 100 131, D-33501 Bielefeld E-mail address : [email protected]@math.uni-bielefeld.de