Higher Koszul duality and connections with n-hereditary algebras
aa r X i v : . [ m a t h . R T ] J a n HIGHER KOSZUL DUALITY AND CONNECTIONS WITH n -HEREDITARY ALGEBRAS JOHANNE HAUGLAND AND MADS HUSTAD SANDØY
Abstract.
We generalize the notion of T -Koszul algebras and prove that ananalogue of classical Koszul duality still holds. Our approach is motivated byand has applications for n -hereditary algebras. In particular, we character-ize an important class of n - T -Koszul algebras of highest degree a in terms of ( na − -representation infinite algebras. As a consequence, we see that analgebra is n -representation infinite if and only if its trivial extension is ( n + 1) -Koszul with respect to its degree part. Furthermore, we show that whenan n -representation infinite algebra is n -representation tame, then the boundedderived categories of graded modules over the trivial extension and over the as-sociated ( n + 1) -preprojective algebra are equivalent. In the n -representationfinite case, we introduce the notion of almost n - T -Koszul algebras and obtainsimilar results. Contents
1. Introduction 21.1. Conventions and notation 42. Preliminaries 42.1. Graded algebras, modules and extensions 42.2. Graded algebras as dg-categories 52.3. Graded Frobenius algebras 63. Graded n -self-orthogonal modules and n - T -Koszul algebras 84. Tilting objects, equivalences and Serre functors 145. On n -hereditary algebras 166. Higher Koszul duality and n -representation infinite algebras 187. Higher almost Koszulity and n -representation finite algebras 27References 38 Mathematics Subject Classification.
Key words and phrases.
Generalized Koszul algebra, Auslander-Reiten theory, preprojec-tive algebra, n -representation finite algebra, n -representation infinite algebra, graded Frobeniusalgebra. Introduction
Global dimension is a useful measure for the objects one studies in representationtheory of finite dimensional algebras. However, while algebras of global dimension and are exceptionally well understood, it seems quite difficult to develop ageneral theory for algebras of higher global dimension. This is a backgroundfor studying the class of n -hereditary algebras [6, 10–12, 16–18], which plays animportant role in higher Auslander–Reiten theory [13–15, 20]. An n -hereditaryalgebra has global dimension less than or equal to n and is either n -representationfinite or n -representation infinite . As one might expect, these notions coincide withthe classical definitions of representation finite and infinite hereditary algebras inthe case n = 1 .Like in the classical theory, n -hereditary algebras have a notion of (higher) pre-projective algebras. If A is n -representation infinite and the ( n + 1) -preprojective Π n +1 A is graded coherent, there is an equivalence D b (mod A ) ≃ D b (qgr Π n +1 A ) ,where qgr Π n +1 A denotes the category of finitely presented graded modules modulofinite dimensional modules [26, Theorem 4.14]. On the other hand, the boundedderived category of a finite dimensional algebra of finite global dimension is al-ways equivalent to the stable category of finitely generated graded modules overits trivial extension [9]. Combining these two equivalences, and using the notation ∆ A for the trivial extension of A , one obtains(1.1) gr (∆ A ) ≃ D b (qgr Π n +1 A ) . The equivalence above brings to mind the acclaimed Bernstein-Gelfand-Gelfand-correspondence, which can be formulated as gr Λ ≃ D b (qgr Λ ! ) for a finite dimen-sional Frobenius Koszul algebra Λ and its graded coherent Artin-Schelter regularKoszul dual Λ ! [4]. The BGG-correspondence is known to descend from the Koszulduality equivalence between bounded derived categories of graded modules overthe two algebras, as indicated in the following diagram D b (gr Λ) D b (gr Λ ! ) gr Λ D b (qgr Λ ! ) . ≃≃ It is natural to ask whether something similar is true in the n -representation infinitecase. i.e. if the equivalence (1.1) is a consequence of some higher Koszul dualitypattern. This is a motivating question for this paper. Motivating question.
Is the equivalence (1.1) a consequence of some higherKoszul duality pattern?One reasonable approach to this question is to study generalizations of thenotion of Koszulity. A positively graded algebra Λ generated in degrees and IGHER KOSZUL DUALITY AND CONNECTIONS WITH n -HEREDITARY ALGEBRAS 3 with semisimple degree part is known as a Koszul algebra if Λ is a graded self-orthogonal module over Λ [3, 28]. This means that Ext i gr Λ (Λ , Λ h j i ) = 0 whenever i = j , where h−i denotes the graded shift. Using basic facts about Serre functorsand triangulated equivalences, one can show that a similar statement holds for ∆ A with respect to its degree part (∆ A ) = A in the case where A is n -representationinfinite. Here, the algebra A is clearly not necessarily semisimple, but it is of finiteglobal dimension.In [8] Green, Reiten and Solberg present a notion of Koszulity for more gen-eral graded algebras, where the degree part is allowed to be an arbitrary finitedimensional algebra. Their work provides a unified approach to Koszul dualityand tilting equivalence. Koszulity in this framework is defined with respect to amodule T , and thus the algebras are called T -Koszul. Madsen [24] gives a simpli-fied definition of T -Koszul algebras, which he shows to be a generalization of theoriginal one whenever the degree part is of finite global dimension.We generalize Madsen’s definition to obtain the notion of n - T -Koszul algebras ,where n is a positive integer and n = 1 returns Madsen’s theory. In Theorem 3.8we prove that an analogue of classical Koszul duality holds in this generality, andwe recover a version of the BGG-correspondence in Proposition 3.10. Moreover,Theorem 6.4 provides a characterization of an important class of n - T -Koszul alge-bras of highest degree a in terms of ( na − -representation infinite algebras. Moreprecisely, we show that a finite dimensional graded Frobenius algebra of highestdegree a ≥ is n - T -Koszul if and only if e T = ⊕ a − i =0 Ω − ni T h i i is a tilting object in theassociated stable category and the endomorphism algebra of this object is ( na − -representation infinite. As a consequence, we see in Corollary 6.6 that an algebrais n -representation infinite if and only if its trivial extension is ( n + 1) -Koszul withrespect to its degree part. Furthermore, we show in Corollary 6.9 that when A is n -representation infinite, then the higher Koszul dual of its trivial extension isgiven by the associated ( n + 1) -preprojective algebra. Combining this with ourversion of the BGG-correspondence, Corollary 6.10 gives an affirmative answer toour motivating question. In particular, we see that when an n -representation in-finite algebra A is n -representation tame, then the bounded derived categories ofgraded modules over ∆ A and over Π n +1 A are equivalent, and that this descendsto give an equivalence gr (∆ A ) ≃ D b (qgr Π n +1 A ) . Notice that in some sense, thetheory we develop is a generalized Koszul dual version of parts of [26].Having developed our theory for one part of the higher hereditary dichotomy,we ask and provide an answer to whether something similar holds in the higherrepresentation finite case. Inspired by and seeking to generalize the notion ofalmost Koszul algebras as developed by Brenner, Butler and King [5], we arriveat the definition of almost n - T -Koszul algebras . This enables us to show a similarcharacterization result as in the n - T -Koszul case, namely Theorem 7.13.This paper is organized as follows. In Section 2 we highlight relevant facts aboutgraded algebras, before the definition and general theory of n - T -Koszul algebras JOHANNE HAUGLAND AND MADS HUSTAD SANDØY is presented in Section 3. In Section 4 we give an overview of the notions of tiltingobjects and Serre functors, and construct an equivalence which will be heavilyused later on. As a foundation for the rest of the paper, Section 5 is devoted torecalling definitions and known facts about n -hereditary algebras. Note that thissection does not contain new results. In Section 6 we state and prove our resultson the connections between n - T -Koszul algebras and higher representation infinitealgebras. Finally, almost n - T -Koszul algebras are introduced in Section 7, and wedevelop their theory along the same lines as was done in Section 6.1.1. Conventions and notation.
Throughout this paper, let k be an algebraicallyclosed field and n a positive integer. All algebras are algebras over k . We denoteby D the duality D ( − ) = Hom k ( − , k ) .Notice that A and B always denote ungraded algebras, while the notation Λ and Γ is used for graded algebras. We work with right modules, homomorphisms act onthe left of elements, and we write the composition of morphisms X f −→ Y g −→ Z as g ◦ f . We denote by Mod A the category of A -modules and by mod A the categoryof finitely presented A -modules.We write the composition of arrows i α −→ j β −→ k in a quiver as αβ . In ourexamples, we use diagrams to represent indecomposable modules. This conventionis explained in more detail in Example 6.5.Given a set of objects U in an additive category A , we denote by add U the fullsubcategory of A consisting of direct summands of finite direct sums of objectsin U . If A is triangulated, we use the notation Thick A ( U ) for the smallest thicksubcategory of A which contains U . When it is clear in which category our thicksubcategory is generated, we will often omit the subscript A Moreover, note that we have certain standing assumptions given at the beginningof Section 3 and Section 6. 2.
Preliminaries
In this section we recall some facts about graded algebras which will be used laterin the paper. In particular, we observe how a graded algebra can be considered asa dg-category concentrated in degree . This plays an important role in our proofsin Section 3. We also provide an introduction to a class of algebras which will bestudied in Section 6 and Section 7, namely the graded Frobenius algebras.2.1. Graded algebras, modules and extensions.
Consider a graded k -algebra Λ = ⊕ i ∈ Z Λ i . The category of graded Λ -modules and degree morphisms is denotedby Gr Λ and the subcategory of finitely presented graded Λ -modules by gr Λ . Recallthat gr Λ is abelian if and only if Λ is graded right coherent, i.e. if every finitelygenerated homogeneous right ideal is finitely presented. IGHER KOSZUL DUALITY AND CONNECTIONS WITH n -HEREDITARY ALGEBRAS 5 Given a graded module M = ⊕ i ∈ Z M i , we define the j -th graded shift of M to bethe graded module M h j i with M h j i i = M i − j . The following basic result relatesungraded extensions to graded ones. Lemma 2.1. (See [27, Corollary 2.4.7] .) Let M and N be graded Λ -modules.If M is finitely generated and there is a projective resolution of M such that allsyzygies are finitely generated, then Ext i Λ ( M, N ) ≃ M j ∈ Z Ext i Gr Λ ( M, N h j i ) for all i ≥ . A non-zero graded module M = ⊕ i ∈ Z M i is said to be concentrated in degree m if M i = 0 for i = m . When Λ is finite dimensional and M finitely generated, thereis an integer h such that M h = 0 and M i = 0 for every i > h . We call h the highestdegree of M . In the same way, the lowest degree of M is the integer l such that M l = 0 and M i = 0 for every i < l .2.2. Graded algebras as dg-categories.
Recall that a dg-category is a k -linearcategory in which the morphism spaces are complexes over k and the compositionis given by chain maps. We refer to [22] for general background on dg-categories.In [23, Section 4] it is explained how one can encode the information of a gradedalgebra as a dg-category concentrated in degree . This is useful, as it enables usto apply known techniques developed for dg-categories to get information aboutthe derived category of graded modules. Let us briefly recall this construction,emphasizing the part which will be useful in Section 3.Given a graded algebra Λ = ⊕ i ∈ Z Λ i , we associate the category A , in which Ob( A ) = Z and the morphisms are given by Hom A ( i, j ) = Λ i − j . Multiplicationin Λ yields composition in A in the natural way. Observe that the Hom-sets of A behaves well with respect to addition in Z , namely that for any integers i and j ,we have(2.1) Hom A ( i, ≃ Hom A ( i + j, j ) . The category of right modules over A , meaning k -linear functors from A op into Mod k , is equivalent to Gr Λ . Similarly, as A is a dg-category concentrated indegree , dg-modules over A correspond to complexes of graded Λ -modules. Con-sequently, one obtain D ( A ) ≃ D (Gr Λ) , i.e. that the derived category of the dg-category A is equivalent to the usual derived category of Gr Λ .Instead of starting with a graded algebra, one can use this construction theother way around. Given a dg-category A concentrated in degree , for which theobjects are in bijection with the integers and the condition (2.1) is satisfied, wecan identify the category with the graded algebra Λ = M i ∈ Z Hom A ( i, , JOHANNE HAUGLAND AND MADS HUSTAD SANDØY in the sense that D ( A ) ≃ D (Gr Λ) . Notice that the fact that certain Hom-setscoincide is necessary in order to be able to use composition in our category todefine multiplication in Λ .2.3. Graded Frobenius algebras.
Recall that twisting by a graded algebra au-tomorphism φ of a graded algebra Λ yields an autoequivalence ( − ) φ on gr Λ . Given M in gr Λ , the module M φ is defined to be equal to M as a vector space with right Λ -action m · λ = mφ ( λ ) , while ( − ) φ acts trivially on morphisms.A finite dimensional positively graded algebra Λ will be called graded Frobenius if D Λ ≃ Λ h− a i as both graded left and graded right Λ -modules for some integer a . Notice that if Λ is concentrated in degree , we recover the usual notion of aFrobenius algebra. Observe also that the integer a in our definition must be equalto the highest degree of Λ , as ( D Λ) i = D (Λ − i ) . We will usually assume a ≥ .Being graded Frobenius is equivalent to being Frobenius as an ungraded algebraand having a grading such that the socle is contained in the highest degree. Lemma 2.2.
Let
Λ = ⊕ i ≥ Λ i be a finite dimensional algebra of highest degree a .The following are equivalent:(1) Λ is graded Frobenius;(2) There exists a graded automorphism µ of Λ such that Λ µ h− a i ≃ D Λ asgraded Λ -bimodules;(3) Λ is Frobenius as ungraded algebra and has a grading satisfying Soc Λ ⊆ Λ a .Proof. If Λ is graded Frobenius, [26, Lemma 2.9] implies that there exists a gradedautomorphism µ of Λ such that D Λ ≃ Λ µ h− a i ≃ µ − Λ h− a i as graded Λ -bimodules. It is hence clear that (1) is equivalent to (2) .To see that (1) is equivalent to (3) , use that graded lifts of finite dimensionalmodules are unique up to isomorphism and graded shift [3, Lemma 2.5.3] togetherwith the fact that Soc D Λ ⊆ ( D Λ) . (cid:3) The automorphism µ of a Frobenius algebra Λ as in the lemma above, isunique up to composition with an inner automorphism and is known as the gradedNakayama automorphism of Λ . We call Λ graded symmetric if µ can be chosen tobe trivial, and note that this notion also descends to the ungraded case.One class of examples which will be important for us, is that of trivial extensionalgebras. Recall that given a finite dimensional algebra A , the trivial extensionof A is ∆ A := A ⊕ DA as a vector space. The trivial extension is an algebrawith multiplication ( a, f ) · ( b, g ) = ( ab, ag + f b ) for a, b ∈ A and f, g ∈ DA . Weconsider ∆ A as a graded algebra by letting A be in degree and DA be in degree . Observe that ∆ A is graded symmetric as it is symmetric as an ungraded algebraand satisfies Soc ∆ A ⊆ (∆ A ) .The stable category of finitely presented graded modules over a graded algebra Λ is denoted by gr Λ . If Λ is self-injective, the category gr Λ is a Frobenius category, IGHER KOSZUL DUALITY AND CONNECTIONS WITH n -HEREDITARY ALGEBRAS 7 and gr Λ is triangulated with shift functor Ω − ( − ) . Notice that every Frobeniusalgebra is self-injective. Observe that twisting by a graded automorphism φ of Λ descends to an autoequivalence ( − ) φ on gr Λ . This functor commutes with takingsyzygies and cosyzygies, as well as with graded shift.We will often consider syzygies and cosyzygies of modules over self-injectivealgebras even when we do not work in a stable category. Whenever we do so,we assume having chosen a minimal projective or injective resolution, so that oursyzygies and cosyzygies do not have any non-zero projective summands. Becauseof our convention with respect to (representatives of) syzygies and cosyzygies, thenotions of highest and lowest degree make sense for these too.Throughout the paper, we often need to consider basic degree arguments, assummarized in the following lemma. We include a short proof for the convenienceof the reader. Lemma 2.3.
Let
Λ = ⊕ i ≥ Λ i be a finite dimensional self-injective graded algebraof highest degree a and Soc Λ ⊆ Λ a . The following statements hold:(1) Given any non-zero element x ∈ Λ , there exists λ ∈ Λ such that xλ ∈ Λ a is non-zero.(2) Let P be an indecomposable projective graded Λ -module of highest degree h . Then, given any non-zero element x ∈ P , there exists λ ∈ Λ such that xλ ∈ P h is non-zero.(3) Let M and P be finitely generated graded Λ -modules with P indecomposableprojective. Denote the highest degree of P by h . Then, for every non-zeromorphism f ∈ Hom gr Λ ( M, P ) , there exists an element x ∈ M such that f ( x ) ∈ P h is non-zero.(4) Let M be an non-projective finitely generated graded Λ -module of highestdegree h and lowest degree l . Then the highest degree of Ω i M is less thanor equal to h in the case i ≤ and greater than or equal to l + a in the case i > .(5) Assume a ≥ , and let M and N be modules concentrated in degree . Then Hom gr Λ ( M, N ) ≃ Hom gr Λ ( M, N ) . (6) Let M be a module concentrated in degree . Then Hom gr Λ ( M, Ω i M h j i ) = 0 for i, j < .(7) Let M be a module concentrated in degree . Then Hom gr Λ ( M, Ω i M h j i ) = 0 for i > and j ≥ − a .Proof. Combining the assumption
Soc Λ ⊆ Λ a with the facts that Rad Λ is nilpo-tent and
Soc Λ = { y ∈ Λ | y Rad Λ = 0 } , one obtains (1) .Part (2) follows from (1) , as projectives are direct summands of free modules. JOHANNE HAUGLAND AND MADS HUSTAD SANDØY
For (3) , let y ∈ M such that f ( y ) = 0 . By (2) , there exists an element λ ∈ Λ such that f ( y ) λ ∈ P h is non-zero. Consequently, the element x = yλ yields ourdesired conclusion.In order to prove (4) , let us first consider the case i ≤ . The statementclearly holds if i = 0 . Observe next that Soc M has highest degree h . Hence, theinjective envelope of M also has highest degree h . Since M is non-projective, thecosyzygi Ω − M is a non-zero quotient of this injective envelope, and consequentlyhas highest degree at most h . We are thus done by induction.For the case i > , note that each summand in the projective cover of M hashighest degree greater than or equal to l + a . As Ω M is a submodule of thisprojective cover, it follows from (3) that Ω M also has highest degree greater thanor equal to l + a . Moreover, the syzygy is itself non-projective of lowest degreegreater than or equal to l , so the claim follows by induction.To verify (5) , notice that there can be no non-zero homomorphism M → N factoring through a Λ -projective. Otherwise, one would have non-zero homomor-phisms M → Λ h i i and Λ h i i → N for some integer i . The former is possible only if i = − a by (3) . However, if i = − a , the latter is impossible as Λ h− a i is generatedin degree − a .Observe that (6) is immediate in the case where M is projective. Otherwise,note that the highest degree of Ω i M is at most by (4) . Hence, the highest degreeof Ω i M h j i is less than or equal to j . As j < , this yields our desired conclusion.For (7) , it again suffices to consider the case where M is non-projective. Apply-ing (4) , our assumptions yield that the highest degree of Ω i M h j i is greater than orequal to . By (3) , this gives Hom gr Λ ( M, Ω i M h j i ) = 0 , as syzygies are submodulesof projectives. (cid:3) Graded n -self-orthogonal modules and n - T -Koszul algebras Throughout the rest of this paper, let
Λ = ⊕ i ≥ Λ i be a positively graded algebra,where Λ is a finite dimensional algebra augmented over k × r for some r > . Weassume that Λ is locally finite dimensional, i.e. that Λ i is finite dimensional as avector space over k for each i ≥ .In this section we define more flexible notions of what it means for a module T to be graded self-orthogonal and an algebra to be T -Koszul than the ones Madsenintroduces in [24, Definition 3.1.1 and 4.1.1]. This enables us to talk about higher T -Koszul duality for a more general class of algebras. In particular, we obtain ahigher Koszul duality equivalence in Theorem 3.8 and we recover a version of theBGG-correspondence in Proposition 3.10. Note that the ideas in this section aresimilar to the ones in [24]. For the convenience of the reader, we nevertheless giveconcise proofs of this section’s main results, to show that the arguments work alsoin our generality.It should be noted that it is also possible to derive Theorem 3.8 by using [24,Theorem 4.3.4]. This strategy involves regrading the algebras so that they satisfy IGHER KOSZUL DUALITY AND CONNECTIONS WITH n -HEREDITARY ALGEBRAS 9 Madsen’s definition of graded self-orthogonality and tracking our original (derived)categories of graded modules through his equivalence. We spell this out in greaterdetail after our proof of Theorem 3.8. Proceeding in this way, one can recovergeneralized analogues of many of the results in [24]. We make no essential use ofthese results, but this approach could be relevant for future related work.We remark that we believe it to be undesirable to work with the regraded al-gebras throughout, since – as will become clear – the resulting graded modulecategories are in some sense too big. Moreover, we will consider endomorphismalgebras of tilting objects, and it is less convenient to study regraded versionsof these. In particular, as we want to relate our results to existing ones involvinggraded modules over trivial extensions or preprojective algebras, we cannot alwayswork directly with the regraded algebras.In order to state our main definitions, let us first recall the notion of a tiltingmodule.
Definition 3.1.
Let A be a finite dimensional algebra. A finitely generated A -module T is called a tilting module if the following conditions hold:(1) proj . dim A T < ∞ ;(2) Ext iA ( T, T ) = 0 for i > ;(3) There is an exact sequence → A → T → T → · · · → T l → with T i ∈ add T for i = 0 , . . . , l .We now define what it means for a module to be graded n -self-orthogonal. Definition 3.2.
Let T be a finitely generated basic graded Λ -module concentratedin degree . We say that T is graded n -self-orthogonal if Ext i gr Λ ( T, T h j i ) = 0 for i = nj .Usually, it will be clear from context what the parameter n is, so we often simplysay that a module satisfying the description above is graded self-orthogonal.Notice that this definition of graded self-orthogonality is more general than theone given in [24]. More precisely, the two definitions coincide exactly when n isequal to . In this case, examples of graded self-orthogonal modules are given by Λ in the classical Koszul situation or tilting modules if Λ = Λ . Moreover, we seein Section 6 that n -representation infinite algebras provide examples of moduleswhich are graded n -self-orthogonal for any choice of n .In general, a graded self-orthogonal module T might have syzygies which arenot finitely generated, so Lemma 2.1 does not apply. However, the followingproposition gives a similar result for graded self-orthogonal modules. This is ananalogue of [24, Proposition 3.1.2]. The proof is exactly the same, except that weuse our more general version of what it means for T to be graded self-orthogonal. Proposition 3.3.
Let T be a graded n -self-orthogonal Λ -module. Then Ext ni Λ ( T, T ) ≃ Ext ni gr Λ ( T, T h i i ) for all i ≥ . Using our definition of a graded self-orthogonal module T , we also get a moregeneral notion of what it means for an algebra to be Koszul with respect to T . Definition 3.4.
Assume gl . dim Λ < ∞ and let T be a graded Λ -module concen-trated in degree . We say that Λ is n - T -Koszul or n -Koszul with respect to T ifthe following conditions hold:(1) T is a tilting Λ -module;(2) T is graded n -self-orthogonal as a Λ -module.Like in the classical theory, we want a notion of a Koszul dual of a given n - T -Koszul algebra. Definition 3.5.
Let Λ be an n - T -Koszul algebra. The n - T -Koszul dual of Λ isgiven by Λ ! = ⊕ i ≥ Ext ni gr Λ ( T, T h i i ) .Note that while the notation for the n - T -Koszul dual is potentially ambiguous,it will in this paper always be clear from context which n - T -Koszul structure thedual is computed with respect to.By Proposition 3.3, we get the following equivalent description of the n - T -Koszuldual. Corollary 3.6.
Let Λ be an n - T -Koszul algebra. Then there is an isomorphismof graded algebras Λ ! ≃ ⊕ i ≥ Ext ni Λ ( T, T ) . Given a set of objects
U ⊆ D b (gr Λ) , let Thick h−i ( U ) denote the smallest thicksubcategory of D b (gr Λ) which contains U and is closed under graded shift. Usingthat Λ has finite global dimension and that T is a tilting Λ -module, one obtainsthat T generates the entire bounded derived category of gr Λ whenever Λ is an n - T -Koszul algebra. Lemma 3.7.
Let Λ be an n - T -Koszul algebra. Then Thick h−i ( T ) = D b (gr Λ) . The proof of Theorem 3.8 uses notions and techniques of dg-homological algebra.Since this is the only section where these are used, we refer the reader to [22] foran introduction. Notice that we have more or less adopted the notation of thatsource for the reader’s convenience. In particular, recall from [22] that given adg-category B , we define the category H B to have the same objects as B andmorphisms given by taking the -th cohomology of the morphism spaces in B .Similarly, also the category τ ≤ B has the same objects as B , and morphisms givenby taking subtle truncation.We are now ready to state and prove the main result of this section, namelyto show that we obtain a higher Koszul duality equivalence. This recovers [24, IGHER KOSZUL DUALITY AND CONNECTIONS WITH n -HEREDITARY ALGEBRAS 11 Theorem 4.3.4] in the case where n = 1 and is a version of [3, Theorem 2.12.6] inthe classical Koszul case. Theorem 3.8.
Let Λ be a finite dimensional n - T -Koszul algebra and assume that Λ ! is graded right coherent and has finite global dimension. Then there is anequivalence D b (gr Λ) ≃ D b (gr Λ ! ) of triangulated categories.Proof. Consider the full subcategory U = { T h i i [ ni ] | i ∈ Z } of D b (gr Λ) . Using astandard lift [22, Section 7.3], we replace U by a dg-category B which has objects { P h i i [ ni ] } , where P is some graded projective resolution of T , and Hom B ( P h i i [ ni ] , P h j i [ nj ]) k = Y m ∈ Z Hom gr Λ ( P m + ni h i i , P m + nj + k h j i ) . In other words, morphism spaces are given by all homogeneous maps of complexesthat are also homogeneous of degree with respect to the grading of Λ . Themorphism spaces are complexes with the standard super commutator differentialdefined by d ( f ) = d P h j i [ nj ] ◦ f − ( − k f ◦ d P h i i [ ni ] for f in Hom B ( P h i i [ ni ] , P h j i [ nj ]) k .Notice that Thick( U ) = Thick h−i ( T ) = D b (gr Λ) . Since we have used a stan-dard lift and idempotents split in D b (gr Λ) , we get that Thick( U ) = D b (gr Λ) isequivalent to D perf ( B ) , i.e. the subcategory of perfect objects.As T is graded n -self-orthogonal, the cohomology of each morphism space in B is concentrated in cohomological degree . Hence, we get a zigzag of dg-categories H B τ ≤ B B in which the dg-functors induce quasi-equivalences. Thus, we also get an equiva-lence D (H B ) ≃ D ( B ) [22, Sec. 7.1-7.2 and 9.1]. This equivalence descends to oneon the compact or perfect objects, and so we get D perf (H B ) ≃ D perf ( B ) .The dg-category H B is concentrated in degree , its objects are in naturalbijection with the integers and we can identify it with a graded algebra as describedin Section 2.2. As we wish this algebra to be positively graded, we let the object P h i i [ ni ] in H B correspond to the integer − i . This yields the algebra M i ≥ Hom H B ( P, P h i i [ ni ]) ≃ M i ≥ Ext ni gr Λ ( T, T h i i ) = Λ ! . It now follows that D (H B ) ≃ D (Gr Λ ! ) , which again yields an equivalence D perf (H B ) ≃ D perf (Gr Λ ! ) . As in the ungraded case, compact objects of D (Gr Λ ! ) coincides with perfect complexes, i.e. bounded complexes of finitely generatedgraded projective modules [22, Theorem 5.3]. Hence, as Λ ! is graded right coherentof finite global dimension, we also have the equivalence D perf (Gr Λ ! ) ≃ D b (gr Λ ! ) ,which completes our proof. (cid:3) Let us now provide more details on how to obtain the above theorem and gen-eralized analogues of other results in [24] using the equivalence constructed there.Observe first that given
Λ = ⊕ i ≥ Λ i satisfying the assumptions in Theorem 3.8,one can rescale the grading so that the regraded algebra Λ ρ is T -Koszul in thesense of [24, Definition 4.1.1]. To be precise, let Λ ρi = Λ j if i = nj for some in-teger j and Λ ρi = 0 otherwise. The category gr Λ embeds into gr Λ ρ as the fullsubcategory consisting of modules which are non-zero only in degrees multiplesof n . As the embedding is exact, it induces a triangulated functor between thecorresponding derived categories. By [31, Lemma 13.17.4], this functor yields anequivalence D b (gr Λ) ≃ −→ D b gr Λ (gr Λ ρ ) , where D b gr Λ (gr Λ ρ ) denotes the full subcate-gory of D b (gr Λ ρ ) consisting of objects with cohomology in gr Λ .Using that Λ ρ is T -Koszul and noticing that (Λ ! ) ρ ≃ (Λ ρ ) ! , we get by [24,Theorem 4.3.4] the equivalence in the upper row of the diagram D b (gr Λ ρ ) D b (gr(Λ ! ) ρ ) D b (gr Λ) D b gr Λ (gr Λ ρ ) D b gr Λ ! (gr(Λ ! ) ρ ) D b (gr Λ ! ) . ≃≃ ≃ ≃ In order to deduce Theorem 3.8 from this, we need to show that the equivalencerestricts as indicated by the dashed arrow. It is sufficient to show that objectswhich are non-zero only in degrees multiples of n are sent to objects satisfying thesame property. Examining the construction of the equivalence, we see that it isessentially the same as the one given in the proof of Theorem 3.8 in the case n = 1 .Consequently, we are done if the equivalences in the zig-zag and the equivalencefrom Thick( U ) to D perf ( B ) satisfy the desired condition.For the former equivalences, this is easily verified and is left to the reader,whereas for the latter, we begin by first recalling some necessary notions. Let A be the dg-category obtained by regarding the graded algebra Λ ρ as a categoryas outlined in Section 2.2, and recall that D ( A ) ≃ D (Gr Λ ρ ) . Moreover, see [22,Section 1.2] for the definition of the dg-category Dif A , and [22, Section 6.2] forthe definition of the triangulated functor R H X for X an A - B -dg-bimodule. If A is an ordinary algebra concentrated in cohomological degree , the objects of thecategory Dif A are complexes of modules over A and the morphisms are given byhomogeneous maps which do not necessarily respect the differentials. In this case,the functor R H X would be quasi-isomorphic to regular R Hom s. The theory ofstandard lifts [22, Section 7.3] implies that the equivalence
Thick( U ) → D perf ( B ) is the restriction of the functor R H X : D ( A ) → D ( B ) , where X is the A - B -dg-bimodule given by X ( j, k ) l = P l − kj + k , which has property (P) as defined in [22, IGHER KOSZUL DUALITY AND CONNECTIONS WITH n -HEREDITARY ALGEBRAS 13 Section 3.1]. Hence, we get R H X ( M ) lk = Hom Dif A ( X (? , k ) , M ) l = Y m ∈ Z Hom
Gr Λ ρ ( P m − k h− k i , M m + l ) ≃ R Hom
Gr Λ ρ ( P h− k i [ − k ] , M ) l . If n does not divide k , this is zero whenever M is non-zero only in degrees that aremultiples of n . Hence, one obtains that Madsen’s equivalence between D b (gr Λ ρ ) and D b (gr(Λ ! ) ρ ) restricts to yield an equivalence between D b (gr Λ) and D b (gr Λ ! ) as claimed.In our following two propositions, we will denote by K : D b (gr Λ) → D b (gr Λ ! ) the equivalence from Theorem 3.8. Since shifting by in gr Λ corresponds toshifting by n in gr Λ ρ , the argument above together with [24, Proposition 3.2.1]yield the following. Proposition 3.9.
Let Λ be a finite dimensional n - T -Koszul algebra and assumethat Λ ! is graded right coherent and has finite global dimension. We then have K ( M h i i ) = K ( M ) h− i i [ − ni ] for M ∈ D b (gr Λ) . We finish this section by showing that an analogue of the BGG-correspondenceholds in our generality. Recall that qgr Λ ! is defined as the localization of gr Λ ! atthe full subcategory of finite dimensional graded Λ ! -modules.We hence have a natural functor D b (gr Λ ! ) → D b (qgr Λ ! ) . In the case where Λ isgraded Frobenius, there is a well-known equivalence D b (gr Λ) / D perf (gr Λ) ≃ gr Λ [30, Theorem 2.1]. Note that we recall this result as Theorem 4.2 in our nextsection. One consequently obtains a functor D b (gr Λ) → D b (gr Λ) / D perf (gr Λ) ≃ −→ gr Λ . These two functors give the vertical arrows in the diagram in our propositionbelow.
Proposition 3.10.
Let Λ be a finite dimensional n - T -Koszul algebra and assumethat Λ ! is graded right coherent and has finite global dimension. If Λ is gradedFrobenius, then the equivalence K descends to yield gr Λ ≃ D b (qgr Λ ! ) , as indicatedin the following diagram D b (gr Λ) D b (gr Λ ! ) gr Λ D b (qgr Λ ! ) . K ≃ Proof.
Since D Λ is injective, we get that the k -th cohomology of R H X ( D Λ h i i ) j iszero unless k = ni = − nj , in which case it is isomorphic to Hom D b (gr Λ) ( T, D Λ) ≃ Hom gr Λ ( T, D Λ) ≃ Hom gr Λ op (Λ , DT ) ≃ DT.
Chasing this through the equivalences in the zig-zag in the proof of Theorem 3.8,we notice that this stalk complex has the Λ ! -action one expects, i.e. the actioninduced by letting Λ !0 ≃ End gr Λ ( T ) ≃ End Λ ( T ) act on T on the left by endo-morphisms. Our argument above hence yields that K restricts to an equivalence Thick h−i ( D Λ) ≃ −→ Thick h−i ( DT ) .Since tilting theory implies that DT is a tilting module over End Λ ( T ) , onededuces that Thick h−i ( DT ) is the full subcategory of D b (gr Λ ! ) of all objects withfinite dimensional cohomology. As qgr Λ ! is the localization of gr Λ ! at the Serresubcategory of finite dimensional Λ ! -modules and the quotient functor in this caseis known to have a left adjoint, we get that D b (gr Λ ! ) / Thick h−i ( DT ) ∼ −→ D b (qgr Λ ! ) is an equivalence by [31, Lemma 13.17.2-3].The triangulated quotient functor Q : D b (gr Λ ! ) → D b (gr Λ ! ) / Thick h−i ( DT ) haskernel Thick h−i ( DT ) ≃ K (Thick h−i ( D Λ)) , and hence composing it with K inducesa triangulated functor K : D b (gr Λ) / Thick h−i ( D Λ) → D b (qgr Λ ! ) satisfying K ◦ P = Q ◦ K by the universal property of quotient categories, in which P is the quotient functor P : D b (gr Λ) → D b (gr Λ) / Thick h−i ( D Λ) . As gr Λ ≃ D b (gr Λ) / Thick h−i ( D Λ) by [30, Theorem 2.1] and it is straightforwardto check that K is an equivalence, we are hence done. (cid:3) Tilting objects, equivalences and Serre functors
Tilting objects and the equivalences they provide play a crucial role throughoutthe rest of this paper. In this section we recall relevant notions and apply one ofYamaura’s ideas to give an explicit construction of an equivalence which will beheavily used in Section 6 and Section 7. We also describe the correspondence ofSerre functors induced by this equivalence.
Definition 4.1.
Let T be a triangulated category. An object T in T is a tiltingobject if the following conditions hold:(1) Hom T ( T, T [ i ]) = 0 for i = 0 ;(2) Thick T ( T ) = T . IGHER KOSZUL DUALITY AND CONNECTIONS WITH n -HEREDITARY ALGEBRAS 15 The first condition in the definition above is often referred to as rigidity .A triangulated category is called algebraic if it is triangle equivalent to the stablecategory of a Frobenius category. Recall that when Λ is a self-injective gradedalgebra, the category gr Λ is Frobenius, and consequently the stable category gr Λ is an algebraic triangulated category. By Keller’s tilting theorem [22, Theorem4.3], we hence know that if T is a tilting object in gr Λ and B = End gr Λ ( T ) has finite global dimension, then there is a triangle equivalence gr Λ ≃ D b (mod B ) .While Keller’s result is proved by applying general techniques from dg-homologicalalgebra, we need a more explicit description of this equivalence. Recall first thatgr Λ can be realized as the quotient category D b (gr Λ) / D perf (gr Λ) . Theorem 4.2. (See [30, Theorem 2.1] .) Let Λ be finite dimensional and self-injective. Then the canonical embedding gr Λ → D b (gr Λ) induces an equivalence gr Λ ≃ −→ D b (gr Λ) / D perf (gr Λ) of triangulated categories. Denote by G the quasi-inverse to the equivalence described in Theorem 4.2 andby P the projection functor D b (gr Λ) → D b (gr Λ) / D perf (gr Λ) . As T has a naturalstructure as a left B -module, we can consider the left derived tensor functor D b (mod B ) −⊗ L B T −−−−→ D b (gr Λ) . Note that when we think of the tilting object T in gr Λ as a graded Λ -module, wechoose a representative without projective summands.We now give an explicit description of the equivalence gr Λ ≃ D b (mod B ) . Thisconstruction and proof is essentially the same as [32, Proposition 3.14], but weshow that it also works in our more general setup. Proposition 4.3.
Let Λ be finite dimensional and self-injective and assume that gl . dim Λ < ∞ . Consider a tilting object T in gr Λ and denote its endomorphismalgebra by B = End gr Λ ( T ) . Then the composition F : D b (mod B ) −⊗ L B T −−−−→ D b (gr Λ) P −→ D b (gr Λ) / D perf (gr Λ) G −→ gr Λ is an equivalence of triangulated categories.Proof. Observe first that rigidity of T yields Hom D b (mod B ) ( B, B [ i ]) ≃ Hom gr Λ ( T, Ω − i T ) for every i ∈ Z . As F ( B ) is isomorphic to T in gr Λ , this means that the restrictionof F to the subcategory X = { B [ i ] | i ∈ Z } is fully faithful. As Λ has finite globaldimension, so has B by [32, Corollary 3.12]. Consequently, one obtain Thick( B ) =Thick( X ) = D b (mod B ) . Using that X is closed under translation, this impliesthat F is fully faithful. Since Thick( T ) = gr Λ and idempotents split in D b (mod B ) ,the functor F is also essentially surjective, and hence an equivalence. (cid:3) In the same way as B is the preimage of T under our equivalence above, wecan also describe projective B -modules in terms of summands of T . Given adecomposition T ≃ ⊕ ti =0 T i of T , let e i : T ։ T i ֒ → T denote the i -th projectionfollowed by the i -th inclusion. This yields a decomposition B ≃ ⊕ ti =0 P i of B into projectives P i = e i B . Notice that the projective P i is the preimage of thesummand T i under the equivalence F , as e i B ⊗ L B T ≃ e i T = T i .From Section 6 and on, the following notion will be crucial. Definition 4.4.
Let T be a k -linear Hom -finite triangulated category. An ad-ditive autoequivalence S on T is called a Serre functor provided there exists abi-functorial isomorphism
Hom T ( X, Y ) ≃ D Hom T ( X, S ( Y )) for all X, Y in Ob T .We want to compare the Serre functor on D b (mod B ) to that of gr Λ when Λ is a graded Frobenius algebra of highest degree a with Nakayama automorphism µ . In this case, it follows from Auslander–Reiten duality: see [2], [29, PropositionI.2.3], combined with the characterization in Lemma 2.2 that Ω( − ) µ h− a i is a Serrefunctor on gr Λ . As B is a finite dimensional algebra of finite global dimension,the derived Nakayama functor ν ( − ) = − ⊗ L B DB is a Serre functor on D b (mod B ) .By uniqueness of the Serre functor, the equivalence F from Proposition 4.3 yieldsa commutative diagram D b (mod B ) gr Λ D b (mod B ) gr Λ . Fν Ω( − ) µ h− a i F Note that throughout the rest of this paper, we will often use the equivalencefrom Proposition 4.3 and the correspondence of the Serre functors described in thediagram above without making the reference explicitly.5. On n -hereditary algebras The class of n -hereditary algebras was introduced in [12] and consists of thedisjoint union of n -representation finite and n -representation infinite algebras. Inthis section we recall some definitions and basic results from [12, 17, 18]. Thisforms a necessary background for exploring connections between the notion of n - T -Koszulity and higher hereditary algebras, which is the topic our next twosections. Note that Section 5 does not contain any new results.Throughout this section, let A be a finite dimensional algebra. Recall that if A has finite global dimension, then the derived Nakayama functor ν ( − ) = − ⊗ L A DA is a Serre functor on D b (mod A ) . We will use the notation ν n = ν ( − )[ − n ] . IGHER KOSZUL DUALITY AND CONNECTIONS WITH n -HEREDITARY ALGEBRAS 17 The algebra A is called n -representation finite if gl . dim A ≤ n and mod A con-tains an n -cluster tilting object. We have the following criterion for n -representationfiniteness in terms of the subcategory U = add { ν in A | i ∈ Z } ⊆ D b (mod A ) . Theorem 5.1. (See [18, Theorem 3.1] .) Assume gl . dim A ≤ n . The followingare equivalent:(1) A is n -representation finite;(2) DA ∈ U ;(3) ν U = U . In particular, an algebra A with gl . dim A ≤ n is n -representation finite if andonly if there for any indecomposable projective A -module P i , is an integer m i ≥ such that ν − m i n ( P i ) is indecomposable injective. We will need the following well-known property of n -representation finite algebras. Lemma 5.2. (See [12, Proposition 2.3] .) Let A be n -representation finite. Foreach indecomposable projective A -module P i , we then have H l ( ν − mn ( P i )) = 0 for l = 0 and ≤ m ≤ m i , where m i is given as above. Moving on to the second part of the n -hereditary dichotomy, recall that A iscalled n -representation infinite if gl . dim A ≤ n and H i ( ν − jn ( A )) = 0 for i = 0 and j ≥ .The following basic lemma will be needed in our next two sections. This factshould be well-known, but we include a proof as we lack an explicit reference.In the proof we abuse notation by letting ν denote both the derived Nakayamafunctor and the ordinary Nakayama functor, as context allows one to determinewhich one is intended. Lemma 5.3.
Let gl . dim A < ∞ and assume that for each indecomposable projec-tive A -module P , we have H i ( ν − n ( P )) = 0 for i / ∈ { , − n } . Then gl . dim A ≤ n .If there is at least one non-injective projective A -module, then gl . dim A = n .Proof. To show gl . dim A ≤ n , it is sufficient to check that inj . dim A ≤ n , as A hasfinite global dimension.Let P be an indecomposable projective A -module. Assume that in computing ν − n ( P ) we use a minimal injective resolution I • of P . As gl . dim A < ∞ , thisresolution is finite. If inj . dim P = m / ∈ { , n } , our assumption yields H m ( ν − ( P )) ≃ H m − n ( ν − n ( P )) = 0 . However, if there is no cohomology in degree m , this implies that the morphism ν − ( I m − → I m ) is an epimorphism. As ν − ( I m ) is projective, this morphism mustsplit. Since ν − is an equivalence when restricted to add DA , this contradicts theminimality of the resolution I • , and we can conclude that inj . dim P = 0 or n . Inparticular, one obtains inj . dim A ≤ n . If there exists P non-injective, we clearlyget the second claim. (cid:3) Like in the classical theory of hereditary algebras, the class of n -hereditary al-gebras also has an appropriate version of (higher) preprojective algebras whichis nicely behaved. Given an n -hereditary algebra A , we denote the ( n + 1) -preprojective algebra of A by Π n +1 A . Recall from [18, Lemma 2.13] that Π n +1 A ≃ M i ≥ Hom D b ( A ) ( A, ν − in ( A )) . If A is n -representation finite, the associated ( n + 1) -preprojective is finite di-mensional and self-injective, whereas in the n -representation infinite case, the ( n + 1) -preprojective is infinite dimensional graded bimodule ( n + 1) -Calabi–Yauof Gorenstein parameter . Remark 5.4.
Note that other authors refer to the classes of algebra we discusshere using different terms. For instance, an n -representation finite algebra is called‘ n -representation-finite n -hereditary’ in [19]. This terminology is very reasonable,but as we need to mention n -representation finite algebras frequently, we stick tothe notion from [17] for brevity.6. Higher Koszul duality and n -representation infinite algebras In this section we investigate connections between n -representation infinite al-gebras and the notion of higher Koszulity. Let us first present our standing as-sumptions. Setup.
Throughout the rest of this section, let
Λ = ⊕ i ≥ Λ i be a finite dimensionalgraded Frobenius algebra of highest degree a ≥ with gl . dim Λ < ∞ . Let T denote a basic graded Λ -module which is concentrated in degree and a tiltingmodule over Λ . Consider a decomposition T ≃ ⊕ ti =0 T i into indecomposablesummands and assume that twisting by the Nakayama automorphism µ of Λ onlypermutes these summands. This means that we have a permutation, for simplicityalso denoted by µ , on the set { , . . . , t } such that T iµ ≃ T µ ( i ) . For our fixed positiveinteger n , we consider the module e T = ⊕ a − i =0 Ω − ni T h i i . We denote the endomorphism algebra
End gr Λ ( e T ) by B .Note that, in the classical case, the Nakayama automorphism induces a permu-tation of the simples, i.e. the module corresponding to our T , and this is somehowthe justification for our assumption.One should note that as the Nakayama automorphism of Λ induces a permu-tation on the indecomposable summands of T , one immediately obtains T µ ≃ T ,and hence Ω T µ h− a i ≃ Ω T h− a i .Our first aim in this section is to describe the endomorphism algebra B as anupper triangular matrix algebra of finite global dimension. We start by recallingthe following lemma. IGHER KOSZUL DUALITY AND CONNECTIONS WITH n -HEREDITARY ALGEBRAS 19 Lemma 6.1. (See [7, Corollary 4.21 (4)] .) Let A and A ′ be finite dimensionalalgebras and M an A op ⊗ k A ′ -module. Then the algebra (cid:20) A M A ′ (cid:21) has finite global dimension if and only if both A and A ′ have finite global dimension. In Lemma 6.2 we describe B as an upper triangular matrix algebra associatedto the graded algebra Γ = ⊕ i ≥ Ext ni gr Λ ( T, T h i i ) . Notice that in the case where Λ is n - T -Koszul, the algebra Γ coincides with the n - T -Koszul dual Λ ! . Lemma 6.2.
The algebra B = End gr Λ ( e T ) is isomorphic to the upper triangularmatrix algebra B ≃ Γ Γ · · · Γ a − · · · Γ a − ... ... . . . ... · · · Γ , where Γ = ⊕ i ≥ Ext ni gr Λ ( T, T h i i ) . In particular, the global dimension of B is finite.Proof. For ≤ i, j ≤ a − , we consider Hom gr Λ (Ω − nj T h j i , Ω − ni T h i i ) ≃ Hom gr Λ ( T, Ω − n ( i − j ) T h i − j i ) . In the case i < j , we note that | i − j | ≤ a − and so Lemma 2.3 (7) applies.Consequently, Hom gr Λ ( T, Ω − n ( i − j ) T h i − j i ) ≃ Hom gr Λ ( T, Ω − n ( i − j ) T h i − j i ) = 0 . If i = j , one obtains End gr Λ ( T ) , which is isomorphic to End gr Λ ( T ) = Γ byLemma 2.3 (5) . For i > j , we get Hom gr Λ ( T, Ω − n ( i − j ) T h i − j i ) ≃ Ext n ( i − j )gr Λ ( T, T h i − j i ) = Γ i − j . Computing our matrix with respect to the decomposition e T = Ω − n ( a − T h a − i ⊕ · · · ⊕ Ω − n T h i ⊕ T, this yields our desired description.To see that B is of finite global dimension, notice that Γ ≃ End Λ ( T ) . As End Λ ( T ) is derived equivalent to Λ , which is of finite global dimension, Lemma 6.1applies and the claim follows. (cid:3) Note that we could also have deduced that B is of finite global dimension from[32, Corollary 3.12]. In the main result of this section, Theorem 6.4, we characterizewhen our algebra Λ is n - T -Koszul in terms of B being ( na − -representationinfinite. Our next lemma provides an important step in the proof of this result.Recall that given a graded Λ -module M = ⊕ i ∈ Z M i , each graded part M i is alsoa module over Λ . On the other hand, every Λ -module is trivially a graded Λ -module concentrated in degree . In the proof of Lemma 6.3, we repeatedly vary between thinking of graded Λ -modules concentrated in one degree and modulesover the degree part.We use the notation M ≥ i for the submodule of M with ( M ≥ i ) j = ( M j j ≥ i j < i, while the quotient module M (cid:30) M ≥ i +1 is denoted by M ≤ i . Note that M i is isomorphicto M ≥ i (cid:30) M ≥ i +1 . Lemma 6.3.
The module e T generates gr Λ as a thick subcategory, i.e. we have Thick gr Λ ( e T ) = gr Λ .Proof. We divide the proof into two steps. In the first part, we show that the setof objects { Λ h i i} i ∈ Z generates gr Λ as a thick subcategory. In the second part, weprove that this set is contained in Thick gr Λ ( e T ) , which yields our desired conclusion. Part 1:
Notice first that every graded Λ -module which is concentrated in degree i isnecessarily contained in the thick subcategory generated by Λ h i i . To see this,apply h i i to a finite Λ -projective resolution of the module, split up into short exactsequences and use that thick subcategories have the / -property on distinguishedtriangles.Let M be an object in gr Λ . Denote the highest and lowest degree of M by h and l , respectively. Observe that M ≥ h = M h . By the argument above, we know that M j is in Thick gr Λ ( { Λ h i i} i ∈ Z ) for every j . Considering the short exact sequences(6.1) M ≥ j +1 M ≥ j M j for j = l, . . . , h − , we can hence conclude that also M ≥ l = M is in our subcategory.This proves that Thick gr Λ ( { Λ h i i} i ∈ Z ) = gr Λ . Part 2:
As thick subcategories are closed under direct summands and translation, weimmediately observe that T h i i is in Thick gr Λ ( e T ) for i = 0 , . . . , a − . Since T isa tilting module over Λ , and Λ h i i thus has a finite coresolution in add T h i i , thisimplies that Λ h i i is in Thick gr Λ ( e T ) for i = 0 , . . . , a − . Note that by our argumentin Part 1 , we hence know that every module which is concentrated in degree i forsome i = 0 , . . . , a − , is contained in our subcategory.Consider the short exact sequences (6.1) for M = Λ , and recall that the module Λ ≥ = Λ is projective and hence zero in gr Λ . By a similar argument as before,this yields that Λ a is contained in Thick gr Λ ( e T ) . We next explain why this entailsthat also Λ h a i is in our subcategory. IGHER KOSZUL DUALITY AND CONNECTIONS WITH n -HEREDITARY ALGEBRAS 21 Since Λ is graded Frobenius, we have Λ h− a i ≃ D Λ as graded right Λ -modules,and thus D Λ ≃ Λ a as Λ -modules. As Λ has finite global dimension, this im-plies that Λ is contained in Thick D b (Λ ) (Λ a h− a i ) . Composing the equivalencefrom Theorem 4.2 with the associated quotient functor, one obtains a triangulatedfunctor Q : D b (gr Λ) → gr Λ . From the chain of subcategories Thick D b (Λ ) Λ a h− a i ⊆ Thick D b (gr Λ) Λ a h− a i ⊆ Q − (Thick gr Λ Λ a h− a i ) , we see that Λ h a i is in Thick gr Λ (Λ a ) , which again is contained in Thick gr Λ ( e T ) .Shifting the short exact sequences involved by positive integers and using thesame argument as above, one obtains that Λ h i i is in Thick gr Λ ( e T ) for all i ≥ .That Λ h i i is in Thick gr Λ ( e T ) for all i < is shown similarly using the short exactsequences j Λ ≤ j Λ ≤ j − for j = 1 , . . . , a . We can hence conclude that Λ h i i is in Thick gr Λ ( e T ) for everyinteger i , which finishes our proof. (cid:3) We are now ready to state and prove the main result of this section.
Theorem 6.4.
The following statements are equivalent:(1) Λ is n - T -Koszul;(2) e T is a tilting object in gr Λ and B = End gr Λ ( e T ) is ( na − -representationinfinite.Proof. We begin by proving (1) implies (2) . To see that e T is a tilting object, noticefirst that it generates gr Λ by Lemma 6.3. Thus, we need only check rigidity,i.e. that Hom gr Λ ( e T , Ω − l e T ) = 0 whenever l = 0 . Splitting up on summands of e T = ⊕ a − i =0 Ω − ni T h i i and reindexing appropriately, we see that it is enough to show(6.2) Hom gr Λ ( T, Ω − ( nk + l ) T h k i ) = 0 for l = 0 for any integer k with | k | ≤ a − .Assume nk + l = 0 . Now l = 0 implies k = 0 , so the condition above is satisfiedas our morphisms are homogeneous of degree .Let nk + l > . Now, Hom gr Λ ( T, Ω − ( nk + l ) T h k i ) ≃ Ext nk + l gr Λ ( T, T h k i ) , which is zero for l = 0 as Λ is n - T -Koszul.It remains to verify (6.2) in the case where nk + l < . As | k | ≤ a − , Lemma 2.3 (7) applies, and we hence see that (6.2) is satisfied also in this case, which meansthat e T is a tilting object in gr Λ .Recall from Lemma 6.2 that B has finite global dimension. To see that B is ( na − -representation infinite, we use that e T is a tilting object in gr Λ . Hence, the equivalence and correspondence of Serre functors described in Section 4 yields Hom gr Λ ( e T , Ω − ( nai + l ) e T h ai i ) ≃ Hom D b ( B ) ( B, ν − i ( B )[ nai − i + l ]) (6.3) ≃ Hom D b ( B ) ( B, ν − ina − ( B )[ l ]) ≃ H l ( ν − ina − ( B )) , where we have implicitly used that T µ ≃ T and that the functors Ω ± ( − ) , h± i and ( − ) µ commute.Splitting up on summands of e T and reindexing appropriately, we notice that Hom gr Λ ( e T , Ω − ( nai + l ) e T h ai i ) = 0 for l = 0 and i > if and only if (6.2) is satisfiedfor k > . The latter follows by the same argument as in our proof of rigidityabove, so we can conclude that H l ( ν − ina − ( B )) = 0 for i > and l = 0 . Note thatwhen i = 0 and l = 0 , we have H l ( ν − ina − ( B )) = H l ( B ) = 0 . Consequently, ouralgebra B is ( na − -representation infinite by Lemma 5.3.To show that (2) implies (1) , we verify that given any integer k , one obtains Ext nk + l gr Λ ( T, T h k i ) = 0 for l = 0 . If nk + l ≤ , this is immediately satisfied, soassume nk + l > . As before, we now have Ext nk + l gr Λ ( T, T h k i ) ≃ Hom gr Λ ( T, Ω − ( nk + l ) T h k i ) . If k < , this is zero by Lemma 2.3 (6) , so it remains to check the case where k isnon-negative.Observe that the isomorphism Hom gr Λ ( e T , Ω − ( nai + l ) e T h ai i ) ≃ H l ( ν − ina − ( B )) from (6.3) still holds, as e T is assumed to be a tilting object in gr Λ . As B is ( na − -representation infinite, we know that H l ( ν − ina − ( B )) = 0 for i ≥ and l = 0 . The isomorphism above hence yields that (6.2) is satisfied for k ≥ .This allows us to conclude that T is graded n -self-orthogonal. As T is a tiltingmodule over Λ by our standing assumptions, we have hence shown that Λ is n - T -Koszul. (cid:3) To illustrate our characterization result, we consider an example. As can beseen below, we use diagrams to represent indecomposable modules. The readershould note that in general one cannot expect modules to be represented uniquelyby such diagrams, but in the cases we will look at, they determine indecomposablemodules up to isomorphism.
Example 6.5.
Let A denote the path algebra of the quiver IGHER KOSZUL DUALITY AND CONNECTIONS WITH n -HEREDITARY ALGEBRAS 23
21 43 α α α α modulo the ideal generated by paths of length two. The trivial extension ∆ A isgiven by the quiver
21 43 α α ′ α α α ′ α ′ α α ′ with the trivial extension relations, i.e. all length two zero relations with the ex-ception of α i α ′ i and α ′ i α i . Instead, these latter paths satisfy all length two commu-tativity relations, i.e. α α ′ − α α ′ , α α ′ − α ′ α , α ′ α − α ′ α , and α ′ α − α α ′ .Moreover, we let ∆ A be graded with the trivial extension grading.The indecomposable projective injectives for ∆ A can be given as the diagrams , where the (non-subscript) numbers represent elements of a basis for the module,each of which is annihilated by all the idempotents except for e i with i equal tothe number. The subscript numbers represent the degree of the basis element.Let T be the tilting A -module given by the direct sum of the following modules . The initial two terms of the minimal injective ∆ A -resolution of the first summandof T as well as the first two cosyzygies can be given as − − ⊕ − − − − − − − − − − ⊕ − − − ⊕ − − − − − − − − − − − . Looking at this part of the resolution, it is not so obvious that T is graded -self-orthogonal as a ∆ A module, whereas by using the equivalence D b (mod A ) ≃ gr ∆ A or by degree arguments as we have done before, it is immediate that e T ≃ T isa tilting object in gr ∆ A . It is also easy to check that End gr ∆ A ( T ) is isomorphicto the hereditary algebra given by the path algebra of the quiver of A , which is representation infinite. Using Theorem 6.4, we can hence conclude that the algebra ∆ A is - T -Koszul.Note that this example also illustrates that, as has been remarked on in theliterature before, one cannot always expect nice minimal resolutions of T for (gen-eralized) T -Koszul algebras.As a consequence of Theorem 6.4, our next corollary shows that an algebra is n -representation infinite if and only if its trivial extension is ( n + 1) -Koszul withrespect to its degree part. This result is inspired by connections between n -representation infinite algebras and graded bimodule ( n + 1) -Calabi–Yau algebrasof Gorenstein parameter , as studied in [1, 12, 21, 26]. In some sense, Corollary 6.6is a T -Koszul dual version of [12, Theorem 4.36]. Corollary 6.6. If a = 1 , our algebra Λ is ( n + 1) -Koszul with respect to Λ ifand only if Λ is n -representation infinite. In particular, we obtain a bijectivecorrespondence isomorphism classesof n -representationinfinite algebras ⇄ isomorphism classes of graded symmetric finitedimensional algebras of highest degree which are ( n + 1) -Koszul with respect to their degree part , where the maps are given by A ∆ A and Λ ←− [ Λ .Proof. Notice that
End gr Λ (Λ ) ≃ End gr Λ (Λ ) ≃ Λ by Lemma 2.3 (5) . Observethat Hom gr Λ (Λ , Ω − i Λ ) ≃ Hom gr Λ (Ω i Λ , Λ ) = 0 for all i = 0 . This follows bydegree considerations similar to those used in the proof of Lemma 2.3 and usingthe fact that the syzygies of Λ are generated in degrees greater or equal to .Combining this with Lemma 6.3, one obtains that Λ is a tilting object in gr Λ ,and consequently our first statement follows from Theorem 6.4.We get the bijection as a special case of this, as ∆ A is a graded symmetric finitedimensional algebra of highest degree and Λ ≃ ∆Λ as graded algebras in thecase where Λ is symmetric. (cid:3) Our aim for the rest of this section is to use the theory we have developed toprovide an affirmative answer to our motivating question from the introduction.As in the case of the generalized AS-regular algebras studied by Minamoto andMori in [26], the notion of quasi-Veronese algebras is relevant.
Definition 6.7.
Let
Γ = ⊕ i ∈ Z Γ i be a Z -graded algebra and r a positive integer.The r -th quasi-Veronese algebra of Γ is a Z -graded algebra defined by Γ [ r ] = M i ∈ Z Γ ri Γ ri +1 · · · Γ ri + r − Γ ri − Γ ri · · · Γ ri + r − ... ... . . . ... Γ ri − r +1 Γ ri − r +2 · · · Γ ri . IGHER KOSZUL DUALITY AND CONNECTIONS WITH n -HEREDITARY ALGEBRAS 25 In Proposition 6.8 we show that if Λ is n - T -Koszul, then the na -th preprojectivealgebra of B = End gr Λ ( e T ) is isomorphic to a twist of the a -th quasi-Veronese of Λ ! .In order to make this precise, notice first that a graded algebra automorphism φ of a graded algebra Γ induces a graded algebra automorphism φ [ r ] of Γ [ r ] by letting φ [ r ] (( γ j,k )) = ( φ ( γ j,k )) . Here we use the notation ( γ j,k ) for the matrix with γ j,k inposition ( j, k ) . Recall also that we can define a possibly different graded algebra h φ i Γ with the same underlying vector space structure as Γ , but with multiplication γ · γ ′ = φ i ( γ ) γ ′ for γ ′ in Γ i .Recall that µ is the Nakayama automorphism of Λ , and denote our chosenisomorphism T µ ≃ T from before by τ . Note that twisting by µ might non-trivially permute the summands of T . In the case where Λ is n - T -Koszul, let µ bethe graded algebra automorphism of Λ ! defined on the i -th component Λ ! i = Ext ni gr Λ ( T, T h i i ) ≃ Hom gr Λ ( T, Ω − ni T h i i ) by the composition Hom gr Λ ( T, Ω − ni T h i i ) ( − ) µ −−→ Hom gr Λ ( T µ , Ω − ni T µ h i i ) ( − ) τ −−→ Hom gr Λ ( T, Ω − ni T h i i ) , where ( γ ) τ = Ω − ni ( τ ) h i i ◦ γ ◦ τ − for γ in Hom gr Λ ( T µ , Ω − ni T µ h i i ) .Before showing Proposition 6.8, recall that a decomposition of e T yields a de-composition of B = End gr Λ ( e T ) . In the proof below, we denote the summands of e T by X i = Ω − ni T h i i , while P i is the projective B -module which is the preimageof X i under the equivalence D b (mod B ) ≃ −→ gr Λ from Proposition 4.3. Proposition 6.8.
Let Λ be n - T -Koszul. Then Π na B ≃ h ( µ − ) [ a ] i (Λ ! ) [ a ] as gradedalgebras. In particular, we have Π na B ≃ (Λ ! ) [ a ] in the case where Λ is gradedsymmetric.Proof. As Λ is n - T -Koszul, we know from Theorem 6.4 that e T is a tilting objectin gr Λ and that B is ( na − -representation infinite. The i -th component of the na -th preprojective algebra of B is given by (Π na B ) i = Hom D b ( B ) ( B, ν − ina − B ) . For ≤ j, k ≤ a − , we hence consider Hom D b ( B ) ( P k , ν − ina − P j ) ≃ Hom gr Λ ( X k , Ω − ( na − i − i X jµ − i h ai i ) ≃ Hom gr Λ ( T, Ω − n ( ai + j − k ) T µ − i h ai + j − k i ) ( ∗ ) ≃ Ext n ( ai + j − k )gr Λ ( T, T µ − i h ai + j − k i ) ≃ Λ ! ai + j − k . Notice that the first isomorphism is a consequence of the equivalence and cor-respondence of Serre functors described in Section 4, while ( ∗ ) is obtained by ap-plying Lemma 2.3 (5) and (7) . The last isomorphism follows from the assumption T µ ≃ T . Computing our matrix with respect to the decomposition B ≃ P a − ⊕ · · · ⊕ P ⊕ P , this yields (Π na B ) i ≃ Λ ! ai Λ ! ai +1 · · · Λ ! ai + a − Λ ! ai − Λ ! ai · · · Λ ! ai + a − ... ... . . . ... Λ ! ai − a +1 Λ ! ai − a +2 · · · Λ ! ai , which shows that our two algebras are isomorphic as graded vector spaces.In order to see that the multiplications agree, consider the diagram ( P j , ν − i ′ na − P j ′ ) ⊗ ( P k , ν − ina − P j ) ( P k , ν − ( i + i ′ ) na − P j ′ )( ν − ina − P j , ν − ( i + i ′ ) na − P j ′ ) ⊗ ( P k , ν − ina − P j ) ( P k , ν − ( i + i ′ ) na − P j ′ )( X jµ − i ( ai ) , X j ′ µ − ( i + i ′ ) ( a ( i + i ′ ))) ⊗ ( X k , X jµ − i ( ai )) ( X k , X j ′ µ − ( i + i ′ ) ( a ( i + i ′ )))Λ ! ai ′ + j ′ − j ⊗ Λ ! ai + j − k Λ ! a ( i + i ′ )+ j ′ − k. For simplicity, we have here suppressed the
Hom -notation and denoted Ω − ni ( − ) h i i by ( − )( i ) . The horizontal maps are given by multiplication or composition, andthe vertical maps give our isomorphism of graded algebras. In particular, themiddle two horizontal maps are merely composition, whereas the top and bottomhorizontal maps are the multiplication of Π na B and h ( µ − ) [ a ] i (Λ ! ) [ a ] , respectively.Moreover, the bottom vertical maps are given by f ⊗ g i ′ − Y l =0 τ − µ l − i ′ ( ai ′ + j ′ − j ) ◦ f µ i ( − ai − j ) ⊗ i − Y l =0 τ − µ l − i ( ai + j − k ) ◦ g ( − k ) and f ◦ g i + i ′ − Y l =0 τ − µ l − i − i ′ ( a ( i + i ′ ) + j ′ − k ) ◦ ( f ◦ g )( − k ) . As the diagram commutes, we can conclude that Π na B ≃ h ( µ − ) [ a ] i (Λ ! ) [ a ] asgraded algebras. If Λ is assumed to be graded symmetric, the Nakayama au-tomorphism µ can be chosen to be trivial, so one obtains Π na B ≃ (Λ ! ) [ a ] . (cid:3) In the corollary below, we show that the ( n + 1) -th preprojective of an n -representation infinite algebra is isomorphic to the n - T -Koszul dual of its trivialextension. This is a T -Koszul dual version of [26, Proposition 4.20]. IGHER KOSZUL DUALITY AND CONNECTIONS WITH n -HEREDITARY ALGEBRAS 27 Corollary 6.9. If A is n -representation infinite, then Π n +1 A ≃ (∆ A ) ! as gradedalgebras.Proof. Let A be an n -representation infinite algebra. It then follows from Corol-lary 6.6 that ∆ A is ( n + 1) -Koszul with respect to A . By Lemma 2.3 part (5) ,one obtains End gr ∆ A ( A ) ≃ End gr ∆ A ( A ) ≃ A . Recall that ∆ A is graded symmet-ric of highest degree . Applying Proposition 6.8 to ∆ A hence yields our desiredconclusion. (cid:3) We are now ready to give an answer to our motivating question from the intro-duction, namely to see that we obtain an equivalence gr (∆ A ) ≃ D b (qgr Π n +1 A ) which descends from higher Koszul duality in the case where A is n -representationinfinite and Π n +1 A is graded right coherent.Recall that an n -representation infinite algebra A is called n -representation tame if the associated ( n + 1) -preprojective Π n +1 A is a noetherian algebra over its center[12, Definition 6.10]. Notice that a noetherian algebra is graded right coherent, soour result holds in this case. Corollary 6.10.
Let A be an n -representation infinite algebra for which Π n +1 A isgraded right coherent. Then there is an equivalence D b (gr ∆ A ) ≃ D b (gr Π n +1 A ) oftriangulated categories which descends to an equivalence gr (∆ A ) ≃ D b (qgr Π n +1 A ) .In particular, this holds if A is n -representation tame.Proof. It is well-known that Π n +1 A is of finite global dimension [26, Theorem 4.2].Hence, we get the equivalence D b (gr ∆ A ) ≃ D b (gr Π n +1 A ) by Theorem 3.8 com-bined with Corollary 6.6 and Corollary 6.9. By Proposition 3.10, this equivalencedescends to yield gr (∆ A ) ≃ D b (qgr Π n +1 A ) . (cid:3) Higher almost Koszulity and n -representation finite algebras In our previous section, we gave connections between higher Koszul duality and n -representation infinite algebras. Having developed our theory for one part ofthe higher hereditary dichotomy, it is natural to ask whether something similarholds in the n -representation finite case. To provide an answer to this question,we introduce the notion of higher almost Koszulity. As before, this should beformulated relative to a tilting module over the degree zero part of the algebra,which is itself assumed to be finite dimensional and of finite global dimension.Notice that after having presented the definitions, we prove our results given thesame standing assumptions as in Section 6.Our definition of what it means for an algebra to be almost n - T -Koszul is inspiredby and generalizes the notion of almost Koszulity, as introduced in [5]. Let us hencefirst recall the definition of an almost Koszul algebra. Definition 7.1. (See [5, Definition 3.1].) Assume that Λ is semisimple. We saythat Λ is (right) almost Koszul if there exist integers g, l ≥ such that (1) Λ i = 0 for all i > g ;(2) There is a graded complex → P − l → · · · → P − → P → of projective right Λ -modules such that each P − i is generated by its com-ponent P − ii and the only non-zero cohomology is Λ in internal degree and P − ll ⊗ Λ g in internal degree g + l .If Λ is almost Koszul for integers g and l , one also says that Λ is ( g, l ) -Koszul .Roughly speaking, by iteratively taking tensor products over the degree part,we see that if Λ is almost Koszul, then Λ has a somewhat periodic projectiveresolution which is properly piecewise linear for g > . This may remind one ofthe behaviour of the inverse Serre functor of an n -representation finite algebra onindecomposable projectives. However, note that for the latter the periods may bedifferent for different indecomposable projectives. This highlights one additionalarea in which we must generalize the notion of almost Koszulity, namely that thelength of the period of graded n -self-orthogonality can vary for different summandsof our tilting module.Motivated by our observations above, let us now define what it means for a mod-ule to be almost graded n -self-orthogonal. Note that throughout the rest of thissection, we always consider a fixed decomposition T ≃ ⊕ ti =1 T i into indecomposablesummands. Definition 7.2.
Let T ≃ ⊕ ti =1 T i be a finitely generated basic graded Λ -moduleconcentrated in degree . We say that T is almost graded n -self-orthogonal if foreach i ∈ { , . . . , t } , there exists an object T ′ ∈ add T and positive integers l i and g i such that the following conditions hold:(1) Ω − l i T i ≃ T ′ h− g i i ;(2) Ext j gr Λ ( T, T i h k i ) = 0 for j = nk and j < l i .This leads to our definition of what it means for an algebra to be almost n - T -Koszul. Definition 7.3.
Assume gl . dim Λ < ∞ and let T be a graded Λ -module concen-trated in degree . We say that Λ is almost n - T -Koszul or almost n -Koszul withrespect to T if the following conditions hold:(1) T is a tilting Λ -module;(2) T is almost graded n -self-orthogonal as a Λ -module.Whenever we work with an almost n - T -Koszul algebra, we use the notation l i and g i for integers given as in Definition 7.2.As a first class of examples, we verify that Definition 7.3 is indeed a generaliza-tion of Definition 7.1. IGHER KOSZUL DUALITY AND CONNECTIONS WITH n -HEREDITARY ALGEBRAS 29 Example 7.4.
Let Λ be a (right) almost Koszul algebra. We show that Λ is almost -Koszul with respect to Λ . It is immediate that gl . dim Λ < ∞ and that Λ is atilting module over itself. To see that Λ is almost -Koszul with respect to Λ , wemust hence check that Λ is almost graded -self-orthogonal as a Λ -module. Notethat by letting l i = l + 1 and g i = g + l , we get that condition (2) of Definition 7.1implies conditions (1) and (2) of Definition 7.3. To see this, we use the fact that analgebra is left ( g, l ) -Koszul if and only if it is right ( g, l ) -Koszul, i.e. [5, Proposition3.4]. Hence, a minimal left projective resolution of Λ can be dualized to yield aminimal right injective resolution of Λ .Trivial extensions of n -representation finite algebras provide another importantclass of examples of algebras satisfying Definition 7.3, as can be seen through thetheory we develop in the rest of this section. Our main result is Theorem 7.13,which is an almost n - T -Koszul analogue of the characterization result in Section 6,i.e. Theorem 6.4. We devide the proof of Theorem 7.13 into a series of smaller steps.In order to state our precise result, we need information about the relation betweenthe integers l i and g i of an almost n - T -Koszul algebra. As will become clear fromthe proof, the notion given in the definition below is sufficient. Recall that weconsider a fixed decomposition T ≃ ⊕ ti =1 T i into indecomposable summands. Definition 7.5.
An almost n -T-Koszul algebra Λ of highest degree a is called ( n, m i , σ i ) - T - Koszul or ( n, m i , σ i ) - Koszul with respect to T if for each i ∈ { , . . . , t } ,there exists non-negative integers m i and σ i with σ i ≤ a − such that(1) l i = nam i − nσ i + 1 ;(2) g i = a ( m i + 1) − σ i ;(3) There is no integer k satisfying < nk < l i and Ω − nk T i ≃ T ′ h− k i with T ′ ∈ add T .We say that an algebra is ( n, m, σ ) - T - Koszul if it is ( n, m i , σ i ) - T -Koszul with m i = m and σ i = σ for all i .Notice that when T is almost graded n -self-orthogonal, the third requirementin the definition above is equivalent to that there exists no positive integers l ′ i and g ′ i with l ′ i < l i satisfying Definition 7.2. One can hence think of part (3) inDefinition 7.5 as a minimality condition for each l i Setup.
Throughout the rest of this paper, we use the standing assumptions de-scribed at the beginning of Section 6.Given these assumptions, let us first show that the data of an ( n, m i , σ i ) - T -Koszul algebra determines a permutation on the set { , . . . , t } in a natural way. Lemma 7.6.
Let Λ be ( n, m i , σ i ) - T -Koszul. There is then a permutation π on theset { , . . . , t } such that Ω − l i T i ≃ T π ( i ) h− g i i for each i ∈ { , . . . , t } . Proof.
Let i ∈ { , . . . , t } . As T is almost graded n -self-orthogonal, there exists anobject T ′ ∈ add T such that Ω − l i T i ≃ T ′ h− g i i . Recall that T is concentrated in degree and that a ≥ . Since it follows fromLemma 2.2 that Soc Λ ⊆ Λ a , this implies that T i is not projective as a Λ -moduleby Lemma 2.3 (3) . As Ω − is an equivalence on the stable category, the object T ′ is indecomposable, and consequently T ′ ≃ T i ′ for some i ′ ∈ { , . . . , t } . Considerthe map π : { , . . . , t } → { , . . . , t } defined by setting π ( i ) = i ′ . If π ( i ) = π ( j ) , one has T i ≃ Ω l i − l j T j h g j − g i i . If l i = l j , this contradicts part (3) of Definition 7.5, so we must have T i ≃ T j . As T is basic, this means that π is injective and hence a permutation. (cid:3) Using our fixed decomposition T ≃ ⊕ ti =1 T i together with the definition of e T , wesee that the algebra B = End gr Λ ( e T ) decomposes as B ≃ t M i =1 a − M j =0 Hom gr Λ ( e T , X i,j ) , where X i,j = Ω − nj T i h j i . Hence, the indecomposable projective B -modules P i,j = Hom gr Λ ( e T , X i,j ) are indexed by the set J = { ( i, j ) | ≤ i ≤ t and ≤ j ≤ a − } . Notice that if e T is a tilting object in gr Λ , then X i,j is the image of P i,j under theequivalence D b (mod B ) ≃ gr Λ , which was explicitly constructed in Proposition 4.3.Given a permutation σ on the index set J , we let σ Lj and σ Ri be defined by σ ( i, j ) = ( σ Lj ( i ) , σ Ri ( j )) . We are now ready to state and prove the first part of our characterization result.Note that this direction in the proof of Theorem 7.13 explains and justifies thesomewhat technical definition of an ( n, m i , σ i ) - T -Koszul algebra. Theorem 7.7. If e T is a tilting object in gr Λ and B = End gr Λ ( e T ) is ( na − -representation finite, then there exist integers m i and σ i such that Λ is ( n, m i , σ i ) - T -Koszul.Proof. By [10, Proposition 0.2], there is a permutation σ on J such that for everypair ( i, j ) in J there is an integer m i,j ≥ with ν − m i,j na − P i,j ≃ I σ ( i,j ) , IGHER KOSZUL DUALITY AND CONNECTIONS WITH n -HEREDITARY ALGEBRAS 31 as B is ( na − -representation finite. Applying ν − na − on both sides, we get ν − m i,j − na − P i,j ≃ P σ ( i,j ) [ na − . Since e T is a tilting object in gr Λ , we have an equivalence D b (mod B ) ≃ gr Λ asdescribed in Proposition 4.3. Using that X i,j = Ω − nj T i h j i is the image of P i,j under this equivalence, combined with the correspondence of Serre functors, oneobtains Ω − ( na − m i,j +1) − ( m i,j +1) X i,jµ − mi,j − h a ( m i,j + 1) i ≃ Ω − ( na − X σ ( i,j ) . This again yields(7.1) Ω − nam i,j − X µ − mi,j − ( i ) ,j ≃ X σ ( i,j ) h− a ( m i,j + 1) i , as ( − ) µ commutes with cosyzygies and degree shifts and permutes the summandsof T . It follows that for each pair ( i, j ) in J , we get Ω − nam i,j − − n ( j − σ Ri ( j )) T µ − mi,j − ( i ) ≃ T σ Lj ( i ) h− a ( m i,j + 1) + σ Ri ( j ) − j i . Twisting by µ m i,j +1 and setting j = 0 , one obtains(7.2) Ω − ( nam i, − nσ Ri (0)+1) T i ≃ T µ mi, ( σ L ( i )) h− a ( m i, + 1) + σ Ri (0) i . Letting m i := m i, and σ i := σ Ri (0) , we hence see that l i and g i can be chosen sothat part (1) of the definition for being almost graded n -self-orthogonal is satisfiedfor T , and that parts (1) and (2) of being ( n, m i , σ i ) - T -Koszul is satisfied for Λ .Note that since g i of this form is always positive, so is l i , as can be seen by applyingLemma 2.3 (6) .In order to show part (3) of Definition 7.5, consider an integer k satisfying < nk < l i . Note that we can write k = qa − r with q ≥ and ≤ r ≤ a − .Aiming for a contradiction, assume that there is an integer j ∈ { , . . . , t } with Ω − n ( qa − r ) T i ≃ T j h− ( qa − r ) i . Twisting by ( − ) µ − q and using the equivalence D b (mod B ) ≃ gr Λ in a similar wayas in the beginning of this proof, we obtain ν − qna − P i, ≃ P µ − q ( j ) ,r . Applying ν na − on both sides yields(7.3) ν − ( q − na − P i, ≃ I µ − q ( j ) ,r [ − na + 1] . From the assumption nk < l i along with the description of l i , we deduce that ≤ q − ≤ m i . As long as na > , the expression (7.3) hence contradictsLemma 5.2, so we can conclude that the third condition of Definition 7.5 is satisfied.If na = 1 , the algebra B is semisimple. In particular, this implies that l i = 1 , sothe condition is trivially satisfied in this case. It remains to prove that T satisfies part (2) of Definition 7.2, i.e. that for each i ∈ { , . . . , t } , we have Ext nk + l gr Λ ( T, T i h k i ) = 0 for l = 0 and nk + l < l i . If nk + l ≤ ,this is immediately clear, so we can assume nk + l > . This yields Ext nk + l gr Λ ( T, T i h k i ) ≃ Hom gr Λ ( T, Ω − ( nk + l ) T i h k i ) . In the case k < , this is zero by Lemma 2.3 (6) , and we can thus assume k ≥ .As e T is a tilting object in gr Λ , a similar argument as in the proof of Theorem 6.4yields an isomorphism(7.4) Hom gr Λ ( e T , Ω − ( nam + l ) X µ − m ( i ) ,j h am i ) ≃ H l ( ν − mna − ( P i,j )) for every pair ( i, j ) in J . By Lemma 5.2, we know that H l ( ν − mna − ( P i,j )) = 0 for l = 0 and ≤ m ≤ m i,j as B is ( na − -representation finite. Using that ( − ) µ is an equivalence on gr Λ , that e T µ ≃ e T and splitting up on summands of e T = ⊕ a − s =0 Ω − ns T h s i , this yields(7.5) Hom gr Λ ( T, Ω − ( n ( am − s + j )+ l ) T i h am − s + j i ) = 0 for l = 0 and ≤ m ≤ m i,j . We simplify this by letting j = 0 . Hence, we have m i,j = m i . In the case k ≤ am i , we can write k = am − s for appropriate valuesof m and s , so (7.5) implies our desired conclusion in this case. If k > am i , we usethe isomorphism T i ≃ Ω l i T π ( i ) h− g i i to rewrite Hom gr Λ ( T, Ω − ( nk + l ) T i h k i ) ≃ Hom gr Λ ( T, Ω l i − ( nk + l ) T π ( i ) h k − g i i ) . When nk + l < l i , this is by Lemma 2.3 (7) . To see this, notice that theassumption k > am i combined with the definition of g i yields k − g i ≥ − a . Thisfinishes our proof. (cid:3) Before giving a result which explains why our choices of m i and σ i are reasonable,we need the following lemma. Lemma 7.8. If e T is a tilting object in gr Λ , then the algebra B = End gr Λ ( e T ) isbasic.Proof. As e T is a tilting object in gr Λ , it suffices to show that e T is basic. Notethat the indecomposable summands of e T are of the form Ω − nj T i h j i with ≤ i ≤ t and ≤ j ≤ a − . Assume that we have isomorphic summands Ω − nj T i h j i ≃ Ω − nl T k h l i . If j = l , it follows that i = k as T is basic. Without loss of generality, we henceassume j > l . Consider now Hom gr Λ ( T i , T i ) ≃ Hom gr Λ ( T i , Ω − n ( l − j ) T k h l − j i ) , which is non-zero as T i = 0 . This contradicts Lemma 2.3 (7) , as l − j ≥ − a and − n ( l − j ) > , so we can conclude that ( i, j ) = ( k, l ) . (cid:3) IGHER KOSZUL DUALITY AND CONNECTIONS WITH n -HEREDITARY ALGEBRAS 33 Recall from [10, Proposition 0.2] and the proof of Theorem 7.7 that when B is ( na − -representation finite, there is a permutation σ on J such that for everypair ( i, j ) in J there is an integer m i,j ≥ with ν − m i,j na − P i,j ≃ I σ ( i,j ) . As before, we use the notation σ ( i, j ) = ( σ Lj ( i ) , σ Ri ( j )) . The proposition below provides more information about how the permutation σ and the integers m i,j associated to B being ( na − -representation finite are relatedto the parameters m i and σ i . Proposition 7.9. If e T is a tilting object in gr Λ and B = End gr Λ ( e T ) is ( na − -representation finite, then Λ is ( n, m i , σ i ) - T -Koszul with m i = m i, and σ i = σ Ri (0) and we have σ Ri ( j ) = (cid:26) σ i + j if σ i + j ≤ a − σ i + j − a if σ i + j > a − and m i,j = (cid:26) m i if j ≤ σ Ri ( j ) m i − if j > σ Ri ( j ) . Additionally, if π is the permutation on { , . . . , t } induced by Λ being ( n, m i , σ i ) - T -Koszul, we have σ Lj ( i ) = µ − m i,j − ( π ( i )) . Proof.
We know that Λ is ( n, m i , σ i ) - T -Koszul with m i = m i, and σ i = σ Ri (0) by Theorem 7.7 and its proof. Note that in order to get the first equation in theformulation above, it is sufficient to show that σ Ri ( j ) = (cid:26) σ Ri (0) + j if j ≤ σ Ri ( j ) σ Ri (0) + j − a if j > σ Ri ( j ) . Consider a fixed integer i ∈ { , . . . , t } and let ≤ j ≤ a − . Assume first j ≤ σ Ri ( j ) . Observe that one obtains Ω − nam i,j − X µ − mi,j − ( i ) , ≃ X σ ( i,j ) − (0 ,j ) h− a ( m i,j + 1) i by applying Ω nj ( − ) h− j i to (7.1). Our assumption yields ≤ σ Ri ( j ) − j ≤ a − , sowe can run the argument at the beginning of the proof of Theorem 7.7 in reverseto get ν − m i,j na − P i, ≃ I σ ( i,j ) − (0 ,j ) . Recall that H ( ν − na − − ) ≃ τ − na − as endofunctors on mod B , where τ − na − denotesthe ( na − -Auslander–Reiten translation. Note that the τ − na − -orbit of a projective B -module contains precisely one injective [16, Proposition 1.3]. Compare ourexpression above with ν − m i, na − P i, ≃ I σ ( i, . If na > , we deduce that m i,j = m i, and I σ ( i,j ) − (0 ,j ) ≃ I σ ( i, . If na = 1 , then B is semisimple. This implies m i,j = m i, = 0 , and the same conclusion thus follows.In particular, this yields σ ( i, j ) − (0 , j ) = σ ( i, as B is basic, which shows this case of our statement once we have made thesubstitions m i = m i, and σ i = σ Ri (0) .For the second part of the proof, assume j > σ Ri ( j ) . Apply Ω − n ( a − j ) ( − ) h a − j i to (7.1) to get Ω − na ( m i,j +1) − X µ − ( mi,j +1) ( i ) , ≃ X σ ( i,j )+(0 ,a − j ) h− a (( m i,j + 1) + 1) i . Note that our assumption yields < σ Ri ( j ) + a − j ≤ a − . Twisting by ( − ) µ − and again reversing the argument at the beginning of the proof of Theorem 7.7,we hence obtain ν − ( m i,j +1) na − P i, ≃ I µ − ( σ Lj ( i )) ,σ Ri ( j )+ a − j . Similarly as in the previous case, we see that this implies the other case of ourstatement once the substitions m i = m i, and σ i = σ Ri (0) have been made. (cid:3) Our next aim is to prove the other direction of this section’s main result. Letus first give an overview of some useful observations.
Lemma 7.10.
Let Λ be ( n, m i , σ i ) - T -Koszul. The following statements hold for ≤ i ≤ t :(1) We have π ◦ µ = µ ◦ π , where π is the permutation on { , . . . , t } induced by Λ being ( n, m i , σ i ) - T -Koszul.(2) The constants l i and g i satisfy l i = l µ ( i ) and g i = g µ ( i ) .(3) The constants m i and σ i satisfy m i = m µ ( i ) and σ i = σ µ ( i ) .(4) We have g i ≥ a . Moreover, if m i = 0 , then σ i = 0 .Proof. For part (1) , recall that Ω ± and h± i both commute with ( − ) µ . Thisimplies that T µ ( i ) ≃ Ω l i T µ ( π ( i )) h− g i i and T µ ( i ) ≃ Ω l µ ( i ) T π ( µ ( i )) h− g µ ( i ) i both hold,and then an argument similar to that in Lemma 7.6 is sufficient. Observe that thisalso shows part (2) .Part (3) is a consequence of (2) .Part (4) is immediate from the definition of l i and g i . To be precise, the as-sumption that σ i ≤ a − implies the claim for g i , while it is clear that m i = 0 implies σ i = 0 as l i is positive. (cid:3) Compared to what was the case for n - T -Koszul algebras, it is somewhat moreinvolved to show that e T is a tilting object in gr Λ whenever Λ is ( n, m i , σ i ) - T -Koszul. We hence prove this as a separate result. Proposition 7.11. If Λ is ( n, m i , σ i ) - T -Koszul, then e T is a tilting object in gr Λ . IGHER KOSZUL DUALITY AND CONNECTIONS WITH n -HEREDITARY ALGEBRAS 35 Proof.
Since Lemma 6.3 yields
Thick gr Λ ( e T ) = gr Λ , we only need to check rigidity.As in the proof of Theorem 6.4, it is enough to verify that Hom gr Λ ( T, Ω − ( nk + l ) T h k i ) = 0 for l = 0 for any integer k with | k | ≤ a − . In the cases nk + l = 0 and nk + l < , theargument is exactly the same as in the proof of Theorem 6.4, so assume nk + l > .For each summand T i of T , one now obtains Hom gr Λ ( T, Ω − ( nk + l ) T i h k i ) ≃ Ext nk + l gr Λ ( T, T i h k i ) . In the case nk + l < l i , this is zero for l = 0 as T is almost graded n -self-orthogonal.Otherwise, we use the isomorphism T i ≃ Ω l i T π ( i ) h− g i i to rewrite the expressionabove. In the case nk + l = l i , we get Hom gr Λ ( T, Ω − ( nk + l − l i ) T π ( i ) h k − g i i ) = Hom gr Λ ( T, T π ( i ) h k − g i i ) . This is zero as | k | ≤ a − together with Lemma 7.10 (4) yields k − g i < . If nk + l > l i , one obtains Hom gr Λ ( T, Ω − ( nk + l − l i ) T π ( i ) h k − g i i ) ≃ Ext nk + l − l i gr Λ ( T, T π ( i ) h k − g i i ) . As nk + l − l i > and k − g i < , the first expression can not be written as an n -multiple of the second. If nk + l − l i < l π ( i ) , we are hence done. Otherwise, weiterate the argument until we reach our desired conclusion. (cid:3) We are now ready to show the other direction of Theorem 7.13.
Theorem 7.12. If Λ is ( n, m i , σ i ) - T -Koszul, then e T is a tilting object in gr Λ and B = End gr Λ ( e T ) is ( na − -representation finite.Proof. Since e T is a tilting object in gr Λ by Proposition 7.11, we only need to showthat B = End gr Λ ( e T ) is ( na − -representation finite. Let us first use the integers m i and σ i to define σ Ri ( j ) , m i,j and σ Lj ( i ) for ( i, j ) in J by the formulas in theformulation of Proposition 7.9. Note that this yields ≤ σ Ri ( j ) ≤ a − , as well as ≤ σ Lj ( i ) ≤ t and m i,j ≥ . The latter is a consequence of Lemma 7.10 (4) .Using that Λ is assumed to be ( n, m i , σ i ) - T -Koszul, we see that (7.2) is satisfied.Furthermore, we can run the argument at the beginning of the proof of Theorem 7.7in reverse, using that e T is a tilting object in gr Λ . Consequently, one obtains ν − m i,j na − P i,j ≃ I σ ( i,j ) for every indecomposable projective B -module P i,j , where σ ( i, j ) := ( σ Lj ( i ) , σ Ri ( j )) . Our next aim is to show that σ is a permutation on J . As J is a finite set, itis enough to check injectivity. Recall that µ and π are permutations, and henceinjective. Combining this with Lemma 7.10 (1) and (3) , notice that also σ L isinjective. Assume that σ ( i, j ) = σ ( k, l ) for ( i, j ) and ( k, l ) in J . If j ≤ σ Ri ( j ) and l ≤ σ Rk ( l ) ,we see that σ L ( i ) = σ Lj ( i ) = σ Ll ( k ) = σ L ( k ) , so i = k by injectivity of σ L . As we in this case also have σ Ri (0) + j = σ Ri ( j ) = σ Rk ( l ) = σ Rk (0) + l, it follows that j = l , so σ is injective. The argument in the case j > σ Ri ( j ) and l > σ Rk ( l ) is similar.By symmetry, it remains to consider the case where j ≤ σ Ri ( j ) and l > σ Rk ( l ) .Here, the assumption σ ( i, j ) = σ ( k, l ) yields σ L ( i ) = σ Lj ( i ) = σ Ll ( k ) = µ ( σ L ( k )) . Consequently, Lemma 7.10 (1) and (3) imply that i = µ ( k ) and σ Ri (0) = σ Rk (0) .As we in this case also have σ Ri (0) + j = σ Ri ( j ) = σ Rk ( l ) = σ Rk (0) + l − a, this means that j = l − a , contradicting the assumption ≤ j, l ≤ a − . Hence,this case is impossible, and we can conclude that σ is a permutation.It now follows that every indecomposable injective, and hence also DB , is con-tained in the subcategory U = add { ν lna − B | l ∈ Z } ⊆ D b (mod B ) . By Theorem 5.1, it thus remains to prove that gl . dim B ≤ na − . To show this,observe first that B has finite global dimension by Lemma 6.2. As e T is a tiltingobject in gr Λ , it follows from (7.4) in the proof of Theorem 7.7 that we have H l ( ν − na − ( P i,j )) ≃ Hom gr Λ ( e T , Ω − ( na + l ) X µ − ( i ) ,j h a i ) ≃ ⊕ a − s =0 Hom gr Λ ( T, Ω − ( n ( a + j − s )+ l ) T i h a + j − s i ) for every pair ( i, j ) in J . We want to show that this is zero whenever l
6∈ { − na, } .Note that the argument for this is similar to the proof of Proposition 7.11. Inparticular, it is enough to consider the case n ( a + j − s ) + l ≥ l i for each i , since theremaining cases are covered by our previous proof. Using that Ω − l i T i ≃ T π ( i ) h− g i i ,the summands in our expression above can be rewritten as Hom gr Λ ( T, Ω − n ( σ i + j − s − am i ) − ( na − l ) T π ( i ) h σ i + j − s − am i i ) . If n ( σ i + j − s − am i ) + na − l < l π ( i ) , this is non-zero only when l is as claimed.Otherwise, Lemma 7.10 (4) implies that we get a negative degree shift in the nextstep of the iteration, and we are done by the same argument as in the proof ofProposition 7.11. From this, one can see that the assumptions in Lemma 5.3 aresatisfied, and hence gl . dim B ≤ na − . Applying Theorem 5.1, we conclude that B is ( na − -representation finite, which finishes our proof. (cid:3) IGHER KOSZUL DUALITY AND CONNECTIONS WITH n -HEREDITARY ALGEBRAS 37 Altogether, combining Theorem 7.7 and Theorem 7.12, we have now proved thissection’s main result. Recall that we use the standing assumptions described atthe beginning of Section 6.
Theorem 7.13.
The following statements are equivalent:(1) There exist integers m i and σ i such that Λ is ( n, m i , σ i ) - T -Koszul;(2) e T is a tilting object in gr Λ and B = End gr Λ ( e T ) is ( na − -representationfinite.Moreover, the parameters m i , σ i and the permutation π obtained from Λ being ( n, m i , σ i ) - T -Koszul correspond to the parameter m i,j and the permutation σ ob-tained from B being ( na − -representation finite as described in Theorem 7.13. We now present some consequences of our characterization theorem similar tothe ones in Section 6. Note that unlike the corresponding result for n -representationinfinite algebras, the following corollary is not – as far as we know – an analogueof anything existing in the literature. Mutatis mutandis, the proof is the same asthat of Corollary 6.6 and is hence omitted. Note that the parameters of Λ and Λ in the statement correspond as described in Theorem 7.13. Corollary 7.14. If a = 1 , our algebra Λ is ( n + 1 , m i , σ i ) -Koszul with respect to Λ if and only if Λ is n -representation finite. In particular, we obtain a bijectivecorrespondence isomorphism classes of n -representataion finitealgebras A ⇄ isomorphism classes of graded symmetric finitedimensional algebras of highest degree whichare ( n + 1 , m i , σ i ) -Koszul with respect to theirdegree parts , where the maps are given by A ∆ A and Λ ←− [ Λ . Just like in Section 6, it is natural to consider the notion of an almost n - T -Koszuldual of a given almost n - T -Koszul algebra. Definition 7.15.
Let Λ be an almost n - T -Koszul algebra. The almost n - T -Koszuldual of Λ is given by Λ ! = ⊕ i ≥ Ext ni gr Λ ( T, T h i i ) .As before, note that while the notation Λ ! is potentially ambiguous, it is for usalways clear from context which structure the dual is computed with respect to.Our next proposition shows that if Λ is ( n, m i , σ i ) - T -Koszul, then the na -thpreprojective algebra of B = End gr Λ ( e T ) is isomorphic to a twist of the a -th quasi-Veronese of Λ ! . The proof is exactly the same as that of the corresponding resultin Section 6, namely Proposition 6.8. Proposition 7.16.
Let Λ be ( n, m i , σ i ) - T -Koszul. Then Π na B ≃ h ( µ − ) [ a ] i (Λ ! ) [ a ] as graded algebras. In particular, we have Π na B ≃ (Λ ! ) [ a ] in the case where Λ isgraded symmetric. The proof of our final corollary is similar to that of Corollary 6.9 and is henceomitted.
Corollary 7.17. If A is n -representation finite, then Π n +1 A ≃ (∆ A ) ! as gradedalgebras. Acknowledgements.
The authors would like to thank Steffen Oppermann forhelpful discussions and Louis-Philippe Thibault and Øyvind Solberg for carefulreading and helpful suggestions on a previous version of this paper.Parts of this work was carried out while the first author participated in the Ju-nior Trimester Program “New Trends in Representation Theory” at the HausdorffResearch Institute for Mathematics in Bonn. She would like to thank the Institutefor excellent working conditions.
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Department of mathematical sciences, NTNU, NO-7491 Trondheim, Norway
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