EENHANCED FINITE TRIANGULATED CATEGORIES
FERNANDO MURO
Abstract.
We give a necessary and sufficient condition for the existence ofan enhancement of a finite triangulated category. Moreover, we show thatenhancements are unique when they exist, up to Morita equivalence.
Contents
Introduction 1Standing assumptions and notation 4Acknowledgements 51. Hochschild cohomology 52. The Gerstenhaber square in the kernel 83. The Euler derivation 84. Hochschild cohomology of weakly stable graded categories 115. Hochschild–Tate cohomology 136. Edge units 187. Lifting edge units 208. Enhancements 249. Enhanced triangulated structures and edge units 29References 38
Introduction
Let us fix a perfect ground field k . Recall that a field k is perfect if char( k ) = 0 orif char( k ) = p and any element in k has a p th root, e.g. finite and algebraically closedfields, but not function fields in positive characteristic. A (linear) additive category T is finite if it is idempotent complete, dim T ( X, Y ) < ∞ for any pair of objects X, Y ∈ T , and there are finitely many indecomposables up to isomorphism. Sucha category is essentially small and satisfies the Krull–Remak–Schmidt property,i.e. any object decomposes as a finite direct sum of indecomposables with localendomorphism algebra in an essentially unique way. If X ∈ T is a basic additivegenerator , consisting of a direct sum of one indecomposable for each isomorphismclass, then T ( X, − ) : T → proj(Λ) is an equivalence onto the category of finitelygenerated projective right Λ-modules for Λ = T ( X, X ), which is a basic algebra.
Mathematics Subject Classification.
Key words and phrases.
Triangulated category, A -infinity category, Hochschild cohomology,spectral sequence, obstruction theory.The author was partially supported by the Spanish Ministry of Economy under the grantMTM2016-76453-C2-1-P (AEI/FEDER, UE) and by the Andalusian Ministry of Economy andKnowledge and the Operational Program FEDER 2014–2020 under the grant US-1263032. a r X i v : . [ m a t h . K T ] M a y FERNANDO MURO
Conversely, proj(Λ) is finite for any finite-dimensional basic algebra Λ. Here, basic means that, as a right module, Λ decomposes as Λ = P ⊕ · · · ⊕ P n were each P i is indecomposable and P i (cid:29) P j for i (cid:54) = j . Any finite-dimensional algebra is Moritaequivalent to a basic one, which is unique up to isomorphism.Finite triangulated categories arise commonly in representation theory, and havebeen thoroughly studied from that viewpoint [62, 1, 29]. If T admits a triangulatedstructure, then Λ is a Frobenius algebra. Indeed, Freyd [21] showed that projectiveobjects in the category mod(Λ) of finitely presented right Λ-modules are injective.Therefore, the Baer criterion [30, Lemma 3.7] proves that Λ is self-injective, so itis Frobenius since it is basic [54, Proposition 3.9].In this paper, we care about enhancements. An enhanced triangulated category isjust a DG-category A . Its underlying triangulated category is D c ( A ), the derivedcategory of compact objects. An enhanced triangulated structure on T consists of aDG-category A and an equivalence D c ( A ) (cid:39) T . We may want to incorporate the suspension functor Σ : T → T to the picture. The pair ( T , Σ) has an enhancedtriangulated structure if there is an equivalence D c ( A ) (cid:39) T which commuteswith the suspension functors up to natural isomorphism. Recall that two DG-categories A and B are Morita equivalent if there is a DG-functor A → B suchthat the induced functor D c ( A ) → D c ( B ) is an equivalence, or a zig-zag of suchDG-functors connecting both.Our main result is the following theorem, where Λ e denotes the enveloping alge-bra of Λ. Theorem.
Let T (cid:39) proj(Λ) be a finite category over a perfect field k with Λ abasic Frobenius algebra:(1) T has an enhanced triangulated structure if and only if the third syzygy Ω e (Λ) is stably isomorphic to an invertible Λ -bimodule.(2) The possible suspension functors Σ : T → T such that ( T , Σ) admits anenhanced triangulated structure are Σ ∼ = − ⊗ Λ I , where I is an invertible Λ -bimodule whose inverse is stably isomorphic to Ω e (Λ) .(3) If Σ is as above, any two enhancements of ( T , Σ) are Morita equivalent. Most triangulated categories appearing in the literature are born with an en-hancement. This theorem is still interesting for finite triangulated categories forwhich an enhancement is known, since it shows uniqueness up to Morita equiv-alence. The stable category mod( A ) of finite-dimensional modules over a self-injective finite-dimensional algebra A of finite representation type fits in this frame-work, e.g. A = k [ x ] / ( x n ). Over an algebraically closed field, these algebras whereclassified by Riedtmann [44, 45, 46]. Over an arbitrary field, there are even moreexamples and the general picture is yet unknown, see e.g. [10]. Also the stablecategory MCM( A ) of maximal Cohen–Macaulay modules over a commutative com-plete local algebra A of finite Cohen–Macaulay representation type fits here [29, § A = k [[ x, y ]] / ( y + x n ) when n is odd and k is algebraically closed [63,Proposition 5.11].This theorem also applies to the non-standard finite 1-Calabi-Yau categoriesdefined in [1] from deformed preprojective algebras of generalized Dynkin type overan algebraically closed field of characteristic 2 [7, Theorem 1.3], [6, Corollary]. Sofar, these categories were only known to be triangulated in the ordinary sense, noenhancements were known (except for those of type L n [8, Theorems 2 and 3]) but(1) shows that an enhancement indeed exists. The simplest of these examples, of NHANCED FINITE TRIANGULATED CATEGORIES 3
Dynkin type D , is the algebra Λ obtained as the quotient of the path algebra ofthe quiver 0 2 31 a a ¯ a ¯ a ¯ a a over an algebraically closed field of characteristic 2 by the two-sided ideal generatedby the following five elements¯ a a , ¯ a a , a ¯ a , a ¯ a + a ¯ a + ¯ a a + a ¯ a a ¯ a , a ¯ a a ¯ a + a ¯ a a ¯ a . There are infinitely many known examples of this kind associated to the Dynkinquivers D n , n ≥ E , and E . There may actually be many more, even in charac-teristic (cid:54) = 2, since deformed preprojective algebras associated to these quivers arenot yet classified.If Λ is connected, in the sense that it is not a product of two non-trivial algebras,and not separable, we prove that, if T has an enhanced triangulated structure, thenthere is an essentially unique suspension functor in (2) so, by (3), the enhancementof T is unique up to Morita equivalence (Proposition 9.8). Deformed preprojectivealgebras of generalized Dynkin type fit in this family, e.g. the example above.If k is algebraically closed, Λ is connected, T is standard, i.e. equivalent to themesh category of its Auslander–Reiten quiver, and we know that an enhancementexists, then (3) was established in [28], and was already implicit in [1], where allpossible such Λ are classified. The simplest of these examples is the algebra of dualnumbers Λ = k [ x ] / ( x ), and more generally, the algebra whose representations areperiodic chain complexes of any fixed period.Non-standard examples are not classified, even over algebraically closed fields.Apart from the aforementioned ones, we have the deformed mesh algebras of type B n , n ≥
3, also in characteristic 2, see [18, Example 9.1]. We do not know anyexamples in characteristic (cid:54) = 2. The claimed examples in characteristic 3 in [9]are flawed, since the deformations do not satisfy the admissibility condition forthe quivers E n , n = 6 , ,
8, described in [18]. Nevertheless, there are non-standardFrobenius algebras in arbitrary characteristic [53, § k n satisfies (1) for all n ≥ T is the n -fold power of the category mod( k ) of finite-dimensional vector spaces. Moreover, the enveloping algebra of Λ is k n , whichis semi-simple, hence Σ can be the tensor product with an arbitrary invertibleΛ-bimodule, by (2). In this case this is the same as saying that Σ can be anypermutation of n elements, regarded as an automorphism of T = mod( k ) n in theobvious way, see [11, Proposition 3.8]. By (3), the pair ( T , Σ) has a unique k -linearenhancement up to Morita equivalence. If Σ is the cyclic permutation then ( T , Σ)cannot be decomposed as a non-trivial product, i.e. it is connected as a pair. Thisis actually the triangulated category considered in [47, Example] when n > k = (cid:96) ( x , . . . , x n +1 ) is a function field on n + 1 variables (which is perfectin characteristic 0). In that paper the authors show that this particular ( T , Σ) has
FERNANDO MURO non-Morita equivalent enhancements over (cid:96) . We deduce that only one of them canbe defined over k as per (3).Hanihara [22, Theorem 1.2] proved recently that, under the assumptions of theprevious theorem, T admits an ordinary triangulated structure if and only if Ω e (Λ)is stably isomorphic to an invertible Λ-bimodule, see also [11, Proposition 3.8]. Heactually did not use the octahedral axiom at all. Hence, we deduce that T ad-mits an ordinary triangulated structure in the sense of Puppe [43] if and only if itadmits an enhanced triangulated structure. Beware that the underlying ordinarytriangulated structure of the latter need not coincide with the former, thereforethere may be finite triangulated categories over a perfect field without enhance-ments (there may even be finite Puppe triangulated categories not satisfying theoctahedral axiom!). It would be interesting to know whether this can really happen.There are many Frobenius algebras which do not satisfy (1) in the previoustheorem. Nakayama algebras yield a whole family of examples (Proposition 9.9).Among these, the simplest are Λ = k [ x ] / ( x n ), n > § Standing assumptions and notation.
Throughout this paper, we will workover a fixed ground field k , which acts everywhere. In particular, all categories areassumed to be k -linear, except for few clear exceptions. The tensor product over k will simply be denoted by ⊗ , unless otherwise indicated. In our main results,we need k to be perfect. We will make explicit this assumption when needed.It will be a standing assumption in Sections 6, 7, and 9. Whenever we have afinite category T , Λ will denote the endomorphism algebra of a basic additivegenerator, as above. We will often work with graded objects, such as graded vectorspaces, graded algebras, graded categories, etc. This means Z -graded, and theKoszul sign rule will be in place. We will also work with bigraded and, occasionally,trigraded objects. Algebras will be regarded as categories with only one object. Wewill often work with modules over small categories C , rather than just algebras.We refer to [55, 2] for basic homological algebra in this context. We will denotethe Grothendieck abelian category of right C -modules by Mod( C ), and mod( C )will be the full subcategory of finitely presented objects. If C is graded, then wewill actually consider the graded categories of graded modules. For C ungraded,modules will also be ungraded, except when explicitly stated otherwise. For aFrobenius algebra Λ, we will also consider the stable module categories Mod(Λ) andmod(Λ), obtained by quotienting out maps factoring through a projective-injectiveobject. Left modules are the same as right modules over the opposite category NHANCED FINITE TRIANGULATED CATEGORIES 5 C op , and bimodules are right modules over the enveloping category C e = C ⊗ C op .Further notation and conventions will be introduced the first time we use them. Acknowledgements.
This paper has benefited from valuable conversations, emailexchange, and MathOverflow interaction with many colleagues, including JerzyBia(cid:32)lkowski, Marco Farinati, Ram´on Flores, Norihiro Hanihara, Dolors Herbera,Bernhard Keller, Henning Krause, Georges Maltsiniotis, Victor Ostrik, JeremyRickard, Antonio Rojas, Manuel Saor´ın, Stefan Schwede, and Mariano Su´arez-´Alvarez (in surname alphabetical order). The author wishes to express his gratitudeto all of them and to an anonymous referee.1.
Hochschild cohomology
The
Hochschild cohomology of a graded category C with coefficients in a C -bimodule M is HH (cid:63), ∗ ( C , M ) = Ext (cid:63), ∗ C e ( C , M ) . Here (cid:63) is the
Hochschild or horizontal degree , the length of the extension, and ∗ is the internal or vertical degree , coming from the fact that the category of C -bimodules is graded. The total degree is the sum of both. We will denote by | x | thedegree of an element x in a singly graded object, or the total degree of an elementin a bigraded object; horizontal and vertical degrees will be denoted by | x | h and | x | v , respectively.Hochschild cohomology can be computed as the cohomology of the Hochschildcochain complex C (cid:63), ∗ ( C , M ), given byC n, ∗ ( C , M ) = (cid:89) X ,...,X n ∈ C Hom ∗ k ( C ( X , X ) ⊗ · · · ⊗ C ( X n , X n − ) , M ( X n , X )) , with bidegree (1 ,
0) differential d ( ϕ )( f , . . . , f n +1 ) = ( − | ϕ | v | f | f · ϕ ( f , . . . , f n +1 )+ n (cid:88) i =1 ( − i ϕ ( . . . , f i f i +1 , . . . ) + ( − n +1 ϕ ( f , . . . , f n ) · f n +1 . This complex arises from the bar projective resolution B (cid:63) ( C ) of C as a C -bimodule,called standard complex in [33, § B n ( C ) = (cid:77) X ,...,X n ∈ C C ( X , − ) ⊗ C ( X , X ) ⊗ · · · ⊗ C ( X n , X n − ) ⊗ C ( − , X n ) , with differential d ( f ⊗ · · · ⊗ f n +1 ) = n (cid:88) i =0 ( − i · · · ⊗ f i f i +1 ⊗ · · · and augmentation ε : B (cid:63) ( C ) −→ C , ε ( f ⊗ f ) = f f . We will now describe the algebraic structure on Hochschild cohomology, followingthe sign conventions in [39, § C by using the endomorphism operad of the suspension of the graded vector space(with several objects) underlying C . In this section, we do not suspend (nor weuse the language of operads, although it is implicit). Therefore, some formulas here FERNANDO MURO have extra signs coming from the well known operadic suspension as recalled in [36,Definition 2.4 and Remark 2.5].Given two C -bimodules M and N , we have the cup-product operation, (cid:94) : C p,q ( C , M ) ⊗ C s,t ( C , N ) −→ C p + s,q + t ( C , M ⊗ C N ) , which is a map of bigraded complexes defined by the following formula,( ϕ (cid:94) ψ )( f , . . . , f p + s ) = ( − t (cid:80) pi =1 | f i | ϕ ( f , . . . , f p ) ⊗ ψ ( f p +1 , . . . , f p + s ) . This operation comes from the differential graded comonoid structure of B (cid:63) ( C ) inthe category of C -bimodules defined by the comultiplication∆ : B (cid:63) ( C ) −→ B (cid:63) ( C ) ⊗ C B (cid:63) ( C ) , ∆( f ⊗ · · · ⊗ f n +1 ) = n (cid:88) i =0 ( f ⊗ · · · ⊗ f i ⊗ X i ) ⊗ (1 X i ⊗ f i +1 ⊗ · · · ⊗ f n +1 ) . The counit is the augmentation ε . The induced cup-product on Hochschild co-homology will be denoted in the same way. If M is a monoid in the category of C -bimodules then the cup-product induces a differential bigraded algebra structureon C (cid:63), ∗ ( C , M ) and a bigraded algebra structure on HH (cid:63), ∗ ( C , M ). The main exam-ple is M = C itself, the tensor unit in C -bimodules. The cup-product is bigradedcommutative in HH (cid:63), ∗ ( C , C ), satisfying the Koszul sign rule with respect to bothdegrees separately. If we tweak the differential and the cup product in the followingway d (cid:48) ( ϕ ) = ( − | ϕ | v d ( ϕ ) ,ϕ · ψ = ( − tp ϕ (cid:94) ψ, then the product becomes a map of complexes at the cochain level, and gradedcommutative in HH (cid:63), ∗ ( C , C ), all with respect to the total degree. Actually, thisdot product and d (cid:48) correspond to the cup product and the differential used in[39]. Therefore, the bigraded commutativity of the cup product in HH (cid:63), ∗ ( C , C ), asdefined here, follows formally from the graded commutativity of the dot productwith respect to the total degree, checked in [39], by purely formal reasons, compare[40, Remark 2.8].The Gerstenhaber bracket endows C (cid:63), ∗ ( C , C ) with a DG-Lie algebra structureof degree − d (cid:48) ,[ − , − ] : C p,q ( C , C ) ⊗ C s,t ( C , C ) −→ C p + s − ,q + t ( C , C ) . It is defined as the commutator[ ϕ, ψ ] = ϕ • ψ − ( − ( p + q − s + t − ψ • ϕ of the pre-Lie product (1.1) • : C p,q ( C , C ) ⊗ C s,t ( C , C ) −→ C p + s − ,q + t ( C , C )given by( ϕ • ψ )( f , . . . , f p + s − ) = p (cid:88) i =1 ( − ( s − p − i )+ t ( p − (cid:80) i − j =1 | f j | ) ϕ ( f , . . . , f i − ,ψ ( f i , . . . , f i + s − ) , f i + s , . . . , f p + s − ) . NHANCED FINITE TRIANGULATED CATEGORIES 7 If m ∈ C , ( C , C ) denotes the composition in C , then d (cid:48) = [ m , − ]. Thispre-Lie product corresponds to the brace operation ϕ { ψ } in [39].The square of the pre-Lie productSq( ϕ ) = ϕ • ϕ induces in cohomology an operation, called Gerstenhaber square ,Sq : HH p,q ( C , C ) −→ HH p − , q ( C , C )whenever p + q is even or char k = 2.The relations satisfied by the product, the Lie bracket, and the Gerstenhabersquare in HH (cid:63), ∗ ( C , C ) are:( x · y ) · z = x · ( y · z ) ,x · y = ( − | x || y | y · x, [ x, y ] = − ( − ( | x |− | y |− [ y, x ] , [ x, x ] = 0 , | x | odd , [ x, [ y, z ]] = [[ x, y ] , z ] + ( − ( | x |− | y |− [ y, [ x, z ]] , [ x, [ x, x ]] = 0 , | x | even , [ x, y · z ] = [ x, y ] · z + ( − ( | x |− | y | y · [ x, z ] , Sq( x + y ) = Sq( x ) + Sq( y ) + [ x, y ] , | x | , | y | even or char k = 2 , Sq( x · y ) = Sq( x ) · y + x · [ x, y ] · y + x · Sq( y ) , idem , [Sq( x ) , y ] = [ x, [ x, y ]] , | x | even or char k = 2 . We call this algebraic structure a
Gerstenhaber algebra (usually only the product,the Lie bracket, and a subset of relations are required). The relations show that2 · Sq( x ) = [ x, x ]whenever the Gerstenhaber square is defined. The Gerstenhaber square and therelations where it appears are redundant if char k (cid:54) = 2.The bivariant functoriality of Hochschild cohomology can be described as in[34]. A graded functor F : D → C and a C -bimodule morphism τ : N → M inducemorphisms on Hochschild cochain complexes that we denote by F ∗ : C (cid:63), ∗ ( C , M ) −→ C (cid:63), ∗ ( D , M ( F, F )) , F ∗ ( ϕ )( g , . . . , g n ) = ϕ ( F ( g ) , . . . , F ( g n )); τ ∗ : C (cid:63), ∗ ( C , N ) −→ C (cid:63), ∗ + | τ | ( C , M ) , τ ∗ ( ϕ )( f , . . . , f n ) = τ ( ϕ ( f , . . . , f n )) . The induced morphisms on Hochschild cohomology, denoted in the same way, F ∗ : HH (cid:63), ∗ ( C , M ) −→ HH (cid:63), ∗ ( D , M ( F, F )) ,τ ∗ : HH (cid:63), ∗ ( C , N ) −→ HH (cid:63), ∗ + | τ | ( C , M ) , satisfy F ∗ τ ∗ = τ ( F, F ) ∗ F ∗ . These induced morphisms are compatible with the cup-product in the obvious way, actually at the cochain level. Hochschild cohomologyis Morita invariant , i.e. F ∗ is an isomorphism for any coefficient bimodule provided F induces an equivalence of categories Mod( C ) (cid:39) Mod( D ).Ungraded objects are graded objects concentrated in degree 0. An ungradedcategory C has graded and ungraded modules and bimodules. If C is an ungraded FERNANDO MURO category and M is a graded C -bimodule, then graded and ungraded Hochschildcohomology are related by the following obvious formula,HH p,q ( C , M ) = HH p ( C , M q ) . The Gerstenhaber square in the kernel
In this short section we prove that the Gerstenhaber square operation restrictsto the kernel of any morphism induced by a graded functor.
Proposition 2.1.
Let F : D → C be a graded functor. If x ∈ HH (cid:63), ∗ ( C , C ) is inthe kernel of F ∗ : HH (cid:63), ∗ ( C , C ) → HH (cid:63), ∗ ( D , C ( F, F )) and | x | is even or char k = 2 ,then F ∗ (Sq( x )) = 0 too. Given a graded functor F : D → C , we can define an operation • : C p,q ( C , C ) ⊗ C s,t ( D , C ( F, F )) −→ C p + s − ,q + t ( D , C ( F, F ))by essentially the same formula as the pre-Lie product in (1.1),( ϕ • ψ )( g , . . . , g p + s − ) = p (cid:88) i =1 ( − ( s − p − i )+ t ( p − (cid:80) i − j =1 | g j | ) ϕ ( F ( g ) , . . . , F ( g i − ) ,ψ ( g i , . . . , g i + s − ) , F ( g i + s ) , . . . , F ( g p + s − )) . We recover (1.1) for F the identity. This new operation and the pre-Lie productare clearly related by the formula F ∗ ( ϕ • ϕ (cid:48) ) = ϕ • F ∗ ( ϕ (cid:48) ) . The proof of the previous proposition is based on the following formula, whichis a slight generalization of [39, (1.4)].
Lemma 2.2.
Given ϕ ∈ C (cid:63), ∗ ( C , C ) and ψ ∈ C (cid:63), ∗ ( D , C ( F, F )) , the following for-mula holds, F ∗ ( ϕ ) · ψ − ( − | ϕ || ψ | ψ · F ∗ ( ϕ ) = ( − | ϕ | ( d (cid:48) ( ϕ • ψ ) − d (cid:48) ( ϕ ) • ψ + ( − | ϕ | ϕ • d (cid:48) ( ψ )) . The proof of this lemma is straightforward but rather tedious, hence we omit it.
Proof of Proposition 2.1.
Let ϕ ∈ C (cid:63), ∗ ( C , C ) be a representative of x , so d (cid:48) ( ϕ ) = 0.Since F ∗ ( x ) = 0, there exists ψ ∈ C (cid:63), ∗ ( D , C ( F, F )) with F ∗ ( ϕ ) = d (cid:48) ( ψ ). If x haseven total degree, so does ϕ . Therefore | ψ | is odd and the Leibniz rule for d (cid:48) together with the formula in Lemma 2.2 yield, d (cid:48) ( ψ · ψ ) = F ∗ ( ϕ ) · ψ − ψ · F ∗ ( ϕ )= d (cid:48) ( ϕ • ψ ) + ϕ • F ∗ ( ϕ ) . Hence, the cohomology class of ϕ • F ∗ ( ϕ ) = F ∗ ( ϕ • ϕ ) is trivial. The same formulaproves the result in char k = 2, where signs do not matter. (cid:3) The Euler derivation
In this section we recall the Euler class in Hochschild cohomology and we studyits interaction with the Gerstenhaber algebra structure.
Definition 3.1.
Given a graded category C , the Euler derivation δ ∈ C , ( C , C )is the cochain defined as δ ( f ) = | f | · f, f ∈ C ( X , X ) . NHANCED FINITE TRIANGULATED CATEGORIES 9 If C is ungraded then δ = 0, so everything in this section pertains to the gradedsetting. Proposition 3.2.
Given a Hochschild cochain ϕ ∈ C p,q ( C , C ) , [ δ, ϕ ] = q · ϕ. Proof.
The result is a consequence of the following equations,[ δ, ϕ ] = δ • ϕ − ϕ • δ, ( δ • ϕ )( f , . . . , f p ) = δ ( ϕ ( f , . . . , f p ))= (cid:32) q + p (cid:88) i =1 | f i | (cid:33) · ϕ ( f , . . . , f p ) ϕ • δ = p (cid:88) i =1 ϕ ( f , . . . , f i − , δ ( f i ) , f i +1 , . . . , f p )= p (cid:88) i =1 | f i | ϕ ( f , . . . , f p ) . (cid:3) Corollary 3.3.
The Euler derivation δ is a Hochschild cocycle.Proof. Simply note that [ m , δ ] = − [ δ, m ] = 0 · m = 0 since m ∈ C , ( C , C ). (cid:3) The
Euler class is the Hochschild cohomology class of the Euler derivation { δ } ∈ HH , ( C , C ) . Corollary 3.4.
Given x ∈ HH ∗ ,q ( C , C ) , [ { δ } , x ] = q · x . Proposition 3.5. If char k = 2 , then Sq( δ ) = δ • δ = δ , hence Sq( { δ } ) = { δ } . It suffices to notice that any integer is congruent to its square modulo 2.
Proposition 3.6.
Given y ∈ HH n, − ( C , C ) with n odd, and x ∈ HH p,q ( C , C ) ,then y · x = [ y, { δ } · x ] + { δ } · [ y, x ] . Proof.
Using Proposition 3.2 and the laws of a Gerstenhaber algebra,[ y, { δ } ] = − [ { δ } , y ] [ y, { δ } · x ] = [ y, { δ } ] · x − { δ } · [ y, x ]= y, = y · x − { δ } · [ y, x ] . (cid:3) Remark . The formula in the previous proposition shows that { δ } · − is a chainnull-homotopy for y · − , if we think of [ y, − ] as a differential. If Sq( y ) = 0, then[ y, − ] is a differential and multiplication by y is a chain map since[ y, [ y, x ]] = [Sq( y ) , x ] = 0 , [ y, y · x ] = [ y, y ] · x + y · [ y, x ] = 2 · Sq( y ) · x + y · [ y, x ] = y · [ y, x ] . This observation is crucial for the proof of Theorem 9.1.
Proposition 3.8.
The square of the Euler class vanishes, { δ } = 0 ∈ HH , ( C , C ) . Proof.
Consider the cochain β ∈ C , ( C , C ) defined by β ( f ) = | f | (1 − | f | )2 · f, f ∈ C ( X , X ) . On the one hand, ( δ · δ )( f , f ) = δ ( f ) · δ ( f )= | f || f | f · f . On the other hand, d ( β )( f , f ) = f · β ( f ) − β ( f · f ) + β ( f ) · f = (cid:18) | f | (1 − | f | )2 − ( | f | + | f | )(1 − ( | f | + | f | ))2 + | f | (1 − | f | )2 (cid:19) f · f = | f || f | f · f . Hence, d ( β ) = δ . (cid:3) Proposition 3.9.
Given x ∈ HH p,q ( C , C ) and y ∈ HH s,t ( C , C ) , [ { δ } · x, { δ } · y ] = ( t − q ) · { δ } · x · y. In particular, if char k (cid:54) = 2 and p + q is odd then Sq( { δ } · x ) = 0 . Proof.
The following equations, which are consequences of the laws of a Gersten-haber algebra and Propositions 3.2 and 3.8, prove the claim,[ { δ } · x, { δ } · y ] = [ { δ } · x, { δ } ] · y + ( − p + q { δ } · [ { δ } · x, y ]= − q { δ } · x · y + ( − p + q { δ } · { δ } · [ x, y ]+ ( − p + q +( p + q )( s + t − { δ } · [ { δ } , y ] · x = − q { δ } · x · y + ( − ( p + q )( s + t ) t { δ } · y · x = − q { δ } · x · y + t { δ } · x · y. (cid:3) Proposition 3.10. If char k = 2 and x ∈ HH (cid:63),q ( C , C ) then Sq( { δ } · x ) = ( q + 1) · { δ } · x . Proof.
Using the Gerstenhaber algebra relations together with Propositions 3.5 and3.8 and Corollary 3.4 we obtainSq( { δ } · x ) = Sq( { δ } ) · x + { δ } · [ { δ } , x ] · x + { δ } · Sq( x )= { δ } · x + { δ } · ( q · x ) · x = ( q + 1) · { δ } · x . (cid:3) NHANCED FINITE TRIANGULATED CATEGORIES 11 Hochschild cohomology of weakly stable graded categories
Given an ungraded category T equipped with an automorphism Σ : T → T , wecan form a graded category T Σ with the same object set, morphism objects givenby T n Σ ( X, Y ) = T ( X, Σ n Y ) , n ∈ Z , and composition T p Σ ( Y, Z ) ⊗ T q Σ ( X, Y ) −→ T p + q Σ ( X, Z ) ,f ⊗ g (cid:55)→ (Σ q f ) g. Here, on the right, we have a composition in T . The degree 0 part of T Σ is precisely T , and we denote the inclusion by i : T ⊂ T Σ . This can also be done if Σ is justa self equivalence, after choosing an adjoint inverse.We can extend Σ to an automorphism Σ : T Σ → T Σ , defined as in T on objects,and on morphisms as ( − n Σ in each degree n ∈ Z . In this way, the graded Σ isequipped with a natural isomorphism(4.1) ı X : X ∼ = Σ X of degree − X . The sign in the extension of Σ is necessaryfor graded naturality, because of Koszul’s sign rule. Graded categories equivalentto some T Σ are called weakly stable [55]. They are characterized by the fact thatshifts of representable functors are representable, or equivalently, each object hasan isomorphism of any given degree, i.e. for any object X and any n ∈ Z thereexists an object Y and an isomorphism X → Y of degree n .In the special case of an ungraded algebra Λ equipped with an automorphism σ : Λ → Λ, the graded algebra given by the previous construction will rather bedenoted by Λ( σ ), so as not to confuse it with a twisted bimodule. It can bedescribed as Λ( σ ) = Λ (cid:104) ı ± (cid:105) ( ıx − σ ( x ) ı ) x ∈ Λ , | ı | = − . The extension of σ to Λ( σ ) is given by σ ( ı ) = − ı . The degree 0 part of Λ( σ ) isΛ, and the degree n part is isomorphic to the twisted Λ-bimodule σ n Λ . Recallthat, given two automorphisms φ, ψ : Λ → Λ and a Λ-bimodule M , the twisted Λ -bimodule φ M ψ has underlying k -vector space M and bimodule structure · givenby a · x · b = φ ( a ) xψ ( b ). Here, a, b ∈ Λ, x ∈ M , and the product on the right ofthe equation is given by the Λ-bimodule structure of M . Up to isomorphism, Λ( σ )only depends on the class of σ in the outer automorphism group Out(Λ) of Λ. It isworth to notice that the twisted Λ-bimodule φ Λ ψ is isomorphic to Λ both as a leftand as a right Λ-module, but in general not as a bimodule. Remark . The isomorphism between the degree n part of Λ( σ ) and σ n Λ is givenby the generator ı − n . Therefore, σ : Λ( σ ) → Λ( σ ) in degree n corresponds to theΛ-bimodule isomorphism ( − n σ : σ n Λ → σ n +1 Λ σ .If T is finite and Λ is the endomorphism algebra of a basic additive generator X ∈ T , then Λ( σ ) is the endomorphism algebra of X in T Σ for σ an automorphisminduced by Σ. Here we use that Σ preserves the basic additive generator up toisomorphism since Σ is an equivalence and the basic additive generator is a directsum of one representative for each isomorphism class of indecomposables in T . Theclass of the automorphism σ in the outer automorphism group is well defined. In the following statement we use shifted modules over a graded algebra. If A ∗ is a graded algebra and M ∗ is a left A ∗ -module, then the shifted object M ∗− isa left A ∗ -module, and the product am in M ∗− is defined as ( − | a | am in M ∗ . If M ∗ is a right A ∗ -module, then there is no sign twisting in the definition of the rightaction of A ∗ on M ∗− . Proposition 4.3.
Let i : T ⊂ T Σ be the inclusion of the degree part. Then thereis a long exact sequence · · · HH n, ∗ ( T Σ , T Σ ) HH n, ∗ ( T , T Σ ) HH n, ∗ ( T , T Σ )HH n +1 , ∗ ( T Σ , T Σ ) · · · i ∗ id − Σ − ∗ Σ ∗ ∂ where i ∗ and Σ − ∗ Σ ∗ are graded algebra morphisms, the map ∂ : HH (cid:63) − , ∗ ( T , T Σ ) → HH (cid:63), ∗ ( T Σ , T Σ ) is an HH (cid:63), ∗ ( T Σ , T Σ ) -bimodule morphism whose image is a square-zero ideal, and the composite ∂i ∗ is left multiplication by { δ } .Proof. The maps i ∗ , Σ ∗ , and Σ ∗ are graded algebra morphisms by functoriality, ac-tually differential graded algebra morphisms on cochains. The long exact sequencewas constructed in [38, Proposition 2.3], where we show thatC (cid:63), ∗ ( T Σ , T Σ ) C (cid:63), ∗ ( T , T Σ ) C (cid:63), ∗ ( T , T Σ ) i ∗ id − Σ − ∗ Σ ∗ can be extended to an exact triangle of complexes. An explicit null-homotopy for(id − Σ − ∗ Σ ∗ ) i ∗ is given by h : C n, ∗ ( T Σ , T Σ ) −→ C n − , ∗ ( T , T Σ ) ,h ( ϕ )( f , . . . , f n − ) = n − (cid:88) i =0 ( − i ı − X ϕ (Σ f , . . . , Σ f i , ı X i , f i +1 , . . . , f n − ) . The equation (id − Σ − ∗ Σ ∗ ) i ∗ = d (cid:48) h + hd (cid:48) follows easily from the definitions and the naturality of ı .The standard exact triangle completion of id − Σ − ∗ Σ ∗ , that we will denote byC (cid:63), ∗ ( T , Σ), is C (cid:63), ∗ ( T , T Σ ) ⊕ C (cid:63) − , ∗ ( T , T Σ )endowed with the differential (cid:18) d (cid:48) − Σ − ∗ Σ ∗ − d (cid:48) (cid:19) . This is the desuspension of the mapping cone of id − Σ − ∗ Σ ∗ . The previous null-homotopy defines an explicit quasi-isomorphismC (cid:63), ∗ ( T Σ , T Σ ) −→ C (cid:63), ∗ ( T , Σ) ,ϕ (cid:55)→ ( i ∗ ( ϕ ) , h ( ϕ )) . (4.4)Indeed, C (cid:63), ∗ ( T , Σ) can be obtained by applying Hom ∗ T eΣ ( − , T Σ ) to the resolu-tion of T Σ constructed in the proof of [38, Proposition 2.3], in the same way asC (cid:63), ∗ ( T Σ , T Σ ) is built from the bar resolution, and (4.4) is defined by a map ofresolutions. NHANCED FINITE TRIANGULATED CATEGORIES 13
The complex C (cid:63), ∗ ( T , Σ) has a semi-direct product differential graded algebrastructure given by( ϕ, ϕ (cid:48) ) · ( ψ, ψ (cid:48) ) = ( ϕ · ψ, ϕ (cid:48) · ψ + ( − | ϕ | Σ − ∗ Σ ∗ ( ϕ ) · ψ (cid:48) ) . This is indeed the very definition of the semi-direct product of the graded algebraC (cid:63), ∗ ( T , T Σ ) and the twisted and then shifted C (cid:63), ∗ ( T , T Σ )-bimodule Σ − ∗ Σ ∗ C (cid:63) − , ∗ ( T , T Σ ) . It is straightforward to check that the differential of C (cid:63), ∗ ( T , Σ) satisfies the Leibnizrule with respect to this product.The quasi-isomorphism (4.4) is also a differential graded algebra map since thenull-homotopy h satisfies the following kind of Leibniz rule twisted by Σ − ∗ Σ ∗ , h ( ϕ · ψ ) = h ( ϕ ) · i ∗ ψ + ( − | ϕ | Σ − ∗ Σ ∗ i ∗ ( ϕ ) · h ( ψ ) . This is a tedious but straightforward checking.The long exact sequence in the statement is therefore also defined by the followingshort exact sequence of complexesC (cid:63) − , ∗ ( T , T Σ ) (cid:44) → C (cid:63), ∗ ( T , Σ) (cid:16) C (cid:63), ∗ ( T , T Σ ) . Here, the map on the right is the projection onto the first coordinate, which is adifferential graded algebra morphism. The map on the left is the inclusion of the sec-ond factor, which is a C (cid:63), ∗ ( T , Σ)-bimodule morphism if we first regard the source asthe twisted and then shifted C (cid:63), ∗ ( T , T Σ )-bimodule Σ − ∗ Σ ∗ C (cid:63) − , ∗ ( T , T Σ ) and thenwe pull it back along the second map. The cohomology of Σ − ∗ Σ ∗ C (cid:63) − , ∗ ( T , T Σ ) isthe HH (cid:63), ∗ ( T , T Σ )-bimodule Σ − ∗ Σ ∗ HH (cid:63) − , ∗ ( T , T Σ ) . If we restrict coefficients toHH (cid:63), ∗ ( T Σ , T Σ ) along i ∗ , then Σ − ∗ Σ ∗ HH (cid:63) − , ∗ ( T , T Σ ) coincides with the bimoduleHH (cid:63) − , ∗ ( T , T Σ ), with no twisting, because i ∗ maps HH (cid:63), ∗ ( T Σ , T Σ ) to the equalizerof Σ − ∗ Σ ∗ and the identity in HH (cid:63), ∗ ( T , T Σ ). Therefore, ∂ is an HH (cid:63), ∗ ( T Σ , T Σ )-bimodule morphism. Moreover, the second direct summand in C (cid:63), ∗ ( T , Σ) is asquare-zero two-sided ideal by definition, hence the image of ∂ is a square-zeroideal in HH (cid:63), ∗ ( T Σ , T Σ ).In order to conclude this proof, we must check that the map in cohomologyinduced by the endomorphism of C (cid:63), ∗ ( T , Σ) given by ( ϕ, ψ ) (cid:55)→ (0 , − ϕ ) coincideswith left multiplication by { δ } . The image of the Euler derivation in C (cid:63), ∗ ( T , Σ)along (4.4) is (0 , −
1) since i ∗ ( δ ) = 0 ,h ( δ )( X ) = ı − X · δ ( ı X )= | ı X | ı − X · ı X = − , and (0 , − · ( ϕ, ψ ) = (0 , − ϕ ), hence we are done. (cid:3) Hochschild–Tate cohomology
In this section we recall stable Hochschild cohomology, as introduced in [19].We emphasize some explicit constructions and perform computations which will beuseful for our purposes.The stable module category
Mod(Λ) of a Frobenius algebra Λ is the quotient ofMod(Λ) by the ideal of maps which factor through a projective-injective object.
The syzygy functor
Ω : Mod(Λ) → Mod(Λ), which is an equivalence, is defined bythe choice of short exact sequencesΩ( M ) (cid:44) → P (cid:16) M with projective-injective middle term. Moreover, Mod(Λ) is triangulated with sus-pension functor Ω − , the cosyzygy functor , an inverse equivalence of Ω which issimilarly defined by the choice of short exact sequences M (cid:44) → P (cid:48) (cid:16) Ω − ( M )with projective-injective middle term. Graded morphisms in Mod(Λ) Ω − are called Tate or stable Ext functors , n ∈ Z ,Ext n Λ ( M, N ) = Mod(Λ) n Ω − ( M, N ) = Hom Λ ( M, Ω − n ( N )) ∼ = Hom Λ (Ω n ( M ) , N ) ∼ = H n Hom Λ ( P (cid:63) , N ) . Here, Hom Λ denotes morphism vector spaces in Mod(Λ) and P (cid:63) is a complete or two-sided resolution of M . The obvious projection P (cid:63) (cid:16) P ≥ defines natural comparison maps , n ≥
0, Ext n Λ ( M, N ) −→ Ext n Λ ( M, N )which are isomorphisms for n > n = 0. Composition in Mod(Λ) Ω − extends the Yoneda product.The Hochschild–Tate or stable Hochschild cohomology of Λ with coefficients in aΛ-bimodule M is defined asHH (cid:63) (Λ , M ) = Ext (cid:63) Λ e (Λ , M ) . In particular, we have comparison maps HH (cid:63) (Λ , M ) −→ HH (cid:63) (Λ , M )which are isomorphisms for n > n = 0. Hochschild–Tate cohomology,as a functor on the coefficients, is functorial in Mod(Λ e ).The category Mod(Λ e ) is monoidal for the tensor product ⊗ Λ and Λ e is Frobe-nius [5, Lemmas 3.1 and 3.2], but Mod(Λ e ) does not inherit the monoidal structurebecause the functor M ⊗ Λ − is not exact and it does not preserve projective-injectiveobjects in general. Nevertheless, the full subcategory Mod p (Λ e ) ⊂ Mod(Λ e ) spannedby those Λ-bimodules which are left and right projective is a Frobenius exact cate-gory containing all projective-injective Λ-bimodules. Moreover, the tensor product ⊗ Λ restricts to Mod p (Λ e ) and the functors M ⊗ Λ − and − ⊗ Λ M are exact andpreserve projective-injective objects for M in Mod p (Λ e ). Therefore, the stable cat-egory Mod p (Λ e ), which is a full triangulated subcategory of Mod(Λ e ), inherits themonoidal structure ⊗ Λ .From now on, in this and in the following two sections, we exceptionally use thesymbol ⊗ to denote ⊗ Λ . The k -linear tensor product will therefore be denoted by ⊗ k here.The tensor product functor in Mod p (Λ e ) is triangulated in both variables, sothere are natural isomorphisms in Mod p (Λ e )(5.1) Ω( M ) ⊗ N ∼ = Ω( M ⊗ N ) ∼ = M ⊗ Ω( N ) . NHANCED FINITE TRIANGULATED CATEGORIES 15
These isomorphisms satisfy coherence conditions that can be easily deduced fromthe explicit construction below. Moreover, the square(5.2) Ω( M ) ⊗ Ω( N ) Ω( M ⊗ Ω( N ))Ω(Ω( M ) ⊗ N ) Ω ( M ⊗ N )commutes up to a − p (Λ e ) fits in the general framework described in [56].In order to fix a syzygy functor Ω in Mod p (Λ e ), we first define Ω(Λ) via anyshort exact sequence in Mod p (Λ e )Ω(Λ) j (cid:44) → P p (cid:16) Λwith P projective-injective, and then Ω( M ) viaΩ(Λ) ⊗ M (cid:44) → P ⊗ M (cid:16) Λ ⊗ M ∼ = M. For M = Λ, both definitions are equivalent via the canonical isomorphism Ω(Λ) ⊗ Λ ∼ = Ω(Λ). With this choice, Ω = Ω(Λ) ⊗ − , the first isomorphism in (5.1) isthe identity, and the second one is given by a coherent natural isomorphism inMod p (Λ e )(5.3) ζ M : Ω(Λ) ⊗ M ∼ = M ⊗ Ω(Λ)defined by any choice of map of extensions in Mod p (Λ e ) of the following formΩ(Λ) ⊗ M P ⊗ M Λ ⊗ M ∼ = MM ⊗ Ω(Λ) M ⊗ P M ⊗ Λ ∼ = M j ⊗ p ⊗ ⊗ j ⊗ p Coherence here means that the following diagrams commuteΩ(Λ) ⊗ M ⊗ NM ⊗ Ω(Λ) ⊗ N M ⊗ N ⊗ Ω(Λ) ζ M ⊗ N ζ M ⊗ ⊗ ζ N Ω(Λ) ⊗ Λ Λ ⊗ Ω(Λ)Ω(Λ) ζ Λ ∼ = ∼ = Here, and elsewhere, we omit associativity constraints.
Lemma 5.4.
The automorphism ζ Ω(Λ) : Ω(Λ) ⊗ Ω(Λ) ∼ = Ω(Λ) ⊗ Ω(Λ) in Mod p (Λ e ) is − . In particular, ζ − = ζ Ω(Λ) .Proof.
This result does not depend on the short exact sequence chosen to defineΩ(Λ). Therefore, we can take the following one,Ω(Λ) j (cid:44) → Λ ⊗ k Λ µ (cid:16) Λ , where µ : Λ ⊗ k Λ → Λ is the product in Λ and j is the inclusion of Ω(Λ) = Ker µ .With these choices, we have the following commutative diagramΩ(Λ) ⊗ Ω(Λ) Λ ⊗ k Λ ⊗ Ω(Λ) Λ ⊗ Ω(Λ) ∼ = Ω(Λ)Ω(Λ) ⊗ Ω(Λ) Ω(Λ) ⊗ Λ ⊗ k Λ Ω(Λ) ⊗ Λ ∼ = Ω(Λ) j ⊗ − µ ⊗ f ⊗ j ⊗ µ where f is defined as f ( a ⊗ b ⊗ ( (cid:80) i c i ⊗ d i )) = (cid:80) i ( abc i ⊗ − a ⊗ bc i ) ⊗ ⊗ d i . This proves the claim. (cid:3)
We can similarly choose an adjoint inverse cosyzygy functor by taking a shortexact sequence in Mod p (Λ e ) Λ j (cid:48) (cid:44) → P (cid:48) p (cid:48) (cid:16) Ω − (Λ)with P (cid:48) projective-injective and setting Ω − ( M ) = Ω − (Λ) ⊗ M . The counit isdetermined by an isomorphism in Mod p (Λ e ) ξ : Ω(Λ) ⊗ Ω − (Λ) ∼ = Λgiven by any choice of map of extensions in Mod p (Λ e ) as followsΩ(Λ) ⊗ Ω − (Λ) P ⊗ Ω − (Λ) Λ ⊗ Ω − (Λ) ∼ = Ω − (Λ)Λ P (cid:48) Ω − (Λ) j ⊗ p ⊗ j (cid:48) p (cid:48) Once ξ is chosen, there is only one possible isomorphism in Mod p (Λ e ) ξ (cid:48) : Λ ∼ = Ω − (Λ) ⊗ Ω(Λ)defining the unit. It is easy to check, using the coherence of ζ and Lemma 5.4, that ξ (cid:48) = − ζ Ω − (Λ) ξ − . There are also coherent isomorphisms in Mod p (Λ e )(5.5) ζ (cid:48) M : M ⊗ Ω − (Λ) ∼ = Ω − (Λ) ⊗ M, like those in (5.3). The isomorphisms ζ M and ζ (cid:48) M are compatible in the sense thatthe two isomorphisms Ω(Λ) ⊗ M ⊗ Ω − (Λ) ∼ = M which can be constructed by usingthese isomorphisms and ξ coincide.Now, if we define Ω n (Λ), n ∈ Z , as a tensor power of Ω(Λ) or Ω − (Λ), accordingto the sign of n , the isomorphisms ξ and ξ (cid:48) define associative isomorphisms Ω p (Λ) ⊗ Ω q (Λ) ∼ = Ω p + q (Λ), p, q ∈ Z .Given two objects M and N in Mod p (Λ e ), there is a natural associative cup-product operationHH p (Λ , M ) ⊗ k HH q (Λ , N ) −→ HH p + q (Λ , M ⊗ N )defined by(Ω p (Λ) f → M ) (cid:94) (Ω q (Λ) g → N ) = (Ω p + q (Λ) ∼ = Ω p (Λ) ⊗ Ω q (Λ) f ⊗ g −→ M ⊗ N ) . In particular, HH (cid:63) (Λ , M ) is a graded algebra if M is a monoid in Mod p (Λ e ), even abigraded algebra if M is graded. The comparison maps from Hochschild to stableHochschild cohomology preserve the cup-product.Let σ : Λ → Λ be an algebra automorphism. We will need some knowledge onunits in the bigraded algebra HH (cid:63), ∗ (Λ , Λ( σ )). Proposition 5.6.
The comparison map HH (Λ , Λ) (cid:16) HH (Λ , Λ) (co)restricts toa surjection between groups of units HH (Λ , Λ) × (cid:16) HH (Λ , Λ) × . This follows from [15, Corollary 2.3] since HH (Λ , Λ) is the center of Λ, andhence a finite-dimensional commutative algebra.
NHANCED FINITE TRIANGULATED CATEGORIES 17
Proposition 5.7.
An element f ∈ HH p,q (Λ , Λ( σ )) = HH p (Λ , σ q Λ ) is a unit in HH (cid:63), ∗ (Λ , Λ( σ )) if and only if f : Ω p (Λ) → σ q Λ is an isomorphism in Mod p (Λ e ) .Proof. Given g : Ω p (cid:48) (Λ) → σ q (cid:48) Λ , if we use the isomorphisms Ω p (Λ) ⊗ Ω p (cid:48) (Λ) ∼ =Ω p + p (cid:48) (Λ) as identifications, the cup-product f (cid:94) g above coincides with ( f ⊗ σ q (cid:48) Λ )(Ω p (Λ) ⊗ g ) = ( σ q Λ ⊗ g )( f ⊗ Ω p (cid:48) (Λ)).Assume f is invertible. Then 1 Λ = f (cid:94) f − = ( f ⊗ σ − q Λ )(Ω p (Λ) ⊗ f − ) =( σ q Λ ⊗ f − )( f ⊗ Ω − p (Λ)). The functor Ω p = Ω p (Λ) ⊗ − is an equivalence, and so is −⊗ Ω − p (Λ), which is naturally isomorphic to Ω − p via ζ or ζ (cid:48) . Moreover, σ q Λ ⊗− isalso an equivalence of categories since σ q Λ is an invertible Λ-bimodule with inverse σ − q Λ , and similarly − ⊗ σ − q Λ . Therefore, f is both a split epimorphism and asplit monomorphism in Mod p (Λ e ), and hence an isomorphism.Conversely, if f is an isomorphism in Mod p (Λ e ), then so is f ⊗ σ − q Λ , and theinverse isomorphism Ω − p ( f ⊗ σ − q Λ ) − is a cup-product inverse for f . (cid:3) Remark . For any p > f ∈ HH p,q (Λ , Λ( σ )) = Ext p Λ (Λ , σ q Λ ), we can takea representing extension σ q Λ (cid:44) → P p − → · · · → P (cid:16) Λwith P p − , . . . , P projective-injective. Using Proposition 5.7, we see that f is aunit in HH (cid:63), ∗ (Λ , Λ( σ )) if and only if P p − is also projective-injective.We can also use the previous choice of Ω(Λ) to fix a syzygy functor in the stablecategory of right modules Mod(Λ), namely Ω( M ) = M ⊗ Ω(Λ), and similarly forthe cosyzygy functor Ω − . The extended Yoneda product in Mod(Λ),Ext p Λ ( M, N ) ⊗ k Ext q Λ ( L, M ) −→ Ext p + q Λ ( L, N ) , can be computed as follows,Hom Λ (Ω p ( M ) , N ) ⊗ k Hom Λ (Ω q ( L ) , M ) −→ Hom Λ (Ω p + q ( L ) , N ) , ( M ⊗ Ω p (Λ) f → N ) ⊗ ( L ⊗ Ω q (Λ) g → M ) (cid:55)→ ( L ⊗ Ω p + q (Λ) ∼ = L ⊗ Ω q (Λ) ⊗ Ω p (Λ) g ⊗ −→ M ⊗ Ω p (Λ) f → N ) . Remark . As above, f ∈ Ext p Λ ( M, N ) is a Yoneda unit if and only if f : Ω p ( M ) → N is an isomorphism in Mod(Λ). For p >
0, Ext p Λ ( M, N ) = Ext p Λ ( M, N ) so we cantake an extension N → P p − → · · · → P (cid:16) M representing f with P p − , . . . , P projective-injective, and f is a Yoneda unit if andonly if P p − is also projective-injective.For any Λ-bimodule M , there is a natural mapHH n (Λ , M ) −→ HH (mod(Λ) , Ext n Λ ( − , − ⊗ M ))defined as (Ω n (Λ) f → M ) (cid:55)→ (cid:16) L (cid:55)→ (cid:16) Ω n ( L ) = L ⊗ Ω n (Λ) L ⊗ f −→ L ⊗ M (cid:17)(cid:17) . Let σ : Λ → Λ be an automorphism. These maps assemble to a map(5.10) ε : HH (cid:63), ∗ (Λ , Λ( σ )) −→ HH , ∗ (mod(Λ) , Ext (cid:63), ∗ Λ ( − , − ⊗ Λ( σ ))) . The coefficient mod(Λ)-bimodule on the right is a monoid for the Yoneda compo-sition. Indeed, since each σ q Λ is left and right projective, the functor − ⊗ σ q Λ
18 FERNANDO MURO is exact (actually, an exact equivalence since σ q Λ is invertible) and the productof g ∈ Ext p,q Λ ( L, M ⊗ Λ( σ )) = Ext p Λ ( L, M ⊗ σ q Λ ) and f ∈ Ext s,t Λ ( M, N ⊗ Λ( σ )) =Ext s Λ ( M, N ⊗ σ t Λ ) is the Yoneda product of g and f ⊗ σ q Λ . In particular, anelement in the target is a unit if and only if it is pointwise a Yoneda unit. Remark . We can regard mod(Λ) as the degree 0 part of mod(Λ( σ )), compare[38, § σ )) is a graded Frobenius abelian category and gradedTate Ext n, ∗ Λ( σ ) functors are defined for all n ∈ Z (also the extended Yoneda product).The inclusion mod(Λ) ⊂ mod(Λ( σ )) corresponds to the extension of scalars alongΛ ⊂ Λ( σ ). Therefore, Ext n, ∗ Λ ( − , − ⊗ Λ( σ )) coincides with Ext n, ∗ Λ( σ ) as a monoid inmod(Λ)-bimodules (also in the non-Tate case), and we can exchange them in thetarget of (5.10). Proposition 5.12. If k is perfect, then the morphism ε in (5.10) preserves andreflects units for (cid:63) > .Proof. Recall from Remark 5.8 that any x ∈ HH n,s (Λ , Λ( σ )) with n > σ s Λ (cid:44) → P n − → · · · → P (cid:16) Λwith P , . . . , P n − projective. Moreover, x is a unit if and only if P n − is alsoprojective. All bimodules in this sequence are projective as left or right Λ-modules.For each finitely presented right Λ-module M , ε ( x )( M ) ∈ Ext n Λ ( M, M ⊗ σ s Λ )is represented by the extension M ⊗ σ s Λ (cid:44) → M ⊗ P n − → · · · → M ⊗ P (cid:16) M ⊗ Λ ∼ = M obtained by applying M ⊗− to the bimodule extension. All right Λ-modules M ⊗ P i are projective for 0 ≤ i ≤ n −
2. Moreover, ε ( x ) is a unit if and only if each ε ( x )( M )is a Yoneda unit, and this happens if and only if M ⊗ P n − is also a projective rightΛ-module for all M , see Remark 5.9. This clearly holds if P n − is projective. Theconverse is also true when k is perfect by [3, Theorem 3.1]. (cid:3) Edge units
Throughout this section, we will assume that the ground field k is a perfect field,since we will derive consequences of Proposition 5.12. Moreover, Λ is a Frobeniusalgebra and σ : Λ → Λ is an algebra automorphism. Recall also that, in this section,like in the previous one, ⊗ stands for ⊗ Λ .We defined in [38, Proposition 5.1] a first quadrant spectral sequence(6.1) E p,q = HH p, ∗ (mod(Λ( σ )) , Ext q, ∗ Λ( σ ) ) = ⇒ HH p + q, ∗ (Λ( σ ) , Λ( σ )) . It has edge morphismsHH (cid:63), ∗ (Λ( σ ) , Λ( σ )) −→ HH , ∗ (mod(Λ( σ )) , Ext (cid:63), ∗ Λ( σ ) ) . Definition 6.2.
An element in HH (cid:63), ∗ (Λ( σ ) , Λ( σ )) is an edge unit if its image alongHH (cid:63), ∗ (Λ( σ ) , Λ( σ )) edge −→ HH , ∗ (mod(Λ( σ )) , Ext (cid:63), ∗ Λ( σ ) ) comp. −→ HH , ∗ (mod(Λ( σ )) , Ext (cid:63), ∗ Λ( σ ) )is a unit. NHANCED FINITE TRIANGULATED CATEGORIES 19
Note that it is really crucial to get to Tate Ext coefficients. Otherwise, therewould only be edge units in horizontal degree (cid:63) = 0.There is another spectral sequence(6.3) E p,q = HH p, ∗ (mod(Λ) , Ext q, ∗ Λ( σ ) ) = ⇒ HH p + q, ∗ (Λ , Λ( σ ))with ungraded first variables, see [38, Remark 5.4]. It has edge morphismsHH (cid:63), ∗ (Λ , Λ( σ )) −→ HH , ∗ (mod(Λ) , Ext (cid:63), ∗ Λ( σ ) ) . Definition 6.4.
An element in HH (cid:63), ∗ (Λ , Λ( σ )) is an edge unit if its image alongHH (cid:63), ∗ (Λ , Λ( σ )) edge −→ HH , ∗ (mod(Λ) , Ext (cid:63), ∗ Λ( σ ) ) comp. −→ HH , ∗ (mod(Λ) , Ext (cid:63), ∗ Λ( σ ) )is a unit.These edge units have the following smarter characterization in positive Hochschilddegree. Proposition 6.5.
For n > , an element in HH n, ∗ (Λ , Λ( σ )) is an edge unit if andonly if its image along the comparison morphism HH n, ∗ (Λ , Λ( σ )) −→ HH n, ∗ (Λ , Λ( σ )) is a unit in HH (cid:63), ∗ (Λ , Λ( σ )) .Proof. By [38, Remark 5.7], the following square commutes,HH n, ∗ (Λ , Λ( σ )) HH , ∗ (mod(Λ) , Ext n, ∗ Λ( σ ) )HH n, ∗ (Λ , Λ( σ )) HH , ∗ (mod(Λ) , Ext n, ∗ Λ( σ ) ) edgecomparison comparison ε Here, the bottom horizontal arrow is the horizontal degree n part of (5.10) (seeRemark 5.11), which preserves and reflects units for n > (cid:3) Proposition 6.6.
The morphism induced by the inclusion i : Λ ⊂ Λ( σ ) , i ∗ : HH n, ∗ (Λ( σ ) , Λ( σ )) −→ HH n, ∗ (Λ , Λ( σ )) , preserves and reflects edge units.Proof. By [38, Remark 7.4], there is a morphism of spectral sequences from (6.1)to (6.3) induced by the inclusion of the degree 0 part Λ ⊂ Λ( σ ), hence we have acommutative diagramHH (cid:63), ∗ (Λ( σ ) , Λ( σ )) HH , ∗ (mod(Λ( σ )) , Ext (cid:63), ∗ Λ( σ ) ) HH , ∗ (mod(Λ( σ )) , Ext (cid:63), ∗ Λ( σ ) )HH (cid:63), ∗ (Λ , Λ( σ )) HH , ∗ (mod(Λ) , Ext (cid:63), ∗ Λ( σ ) ) HH , ∗ (mod(Λ) , Ext (cid:63), ∗ Λ( σ ) ) edge comp.edge comp. whose vertical arrows are induced by Λ ⊂ Λ( σ ). The two last vertical arrows areinjective because they are in Hochschild degree 0, see Proposition 4.3. Moreover,the last one preserves and reflects units by the first paragraph of the proof of [38,Corollary 3.4]. This suffices. (cid:3) Corollary 6.7.
For n > , an element in HH n, ∗ (Λ( σ ) , Λ( σ )) is an edge unit if andonly if its image along HH n, ∗ (Λ( σ ) , Λ( σ )) i ∗ −→ HH n, ∗ (Λ , Λ( σ )) comp. −→ HH n, ∗ (Λ , Λ( σ )) is a unit in HH (cid:63), ∗ (Λ , Λ( σ )) . In order to conclude this section, we record a property of edge units which iscrucial for later computations.
Lemma 6.8. If x ∈ HH , − (Λ( σ ) , Λ( σ )) is an edge unit, then the map x · − : HH p,q (Λ( σ ) , Λ( σ )) −→ HH p +3 ,q − (Λ( σ ) , Λ( σ )) given by left multiplication by x is an isomorphism for all p ≥ and q ∈ Z and anepimorphism for p = 1 and all q ∈ Z .Proof. By the previous corollary i ∗ ( x ) ∈ HH , − (Λ , Λ( σ )) = HH , − (Λ , Λ( σ )) isa unit in HH (cid:63), ∗ (Λ , Λ( σ )). Since the Hochschild–Tate cohomology coincides withHochschild’s in positive Hochschild degrees, and the latter surjects onto the formerin Hochschild degree 0, the map i ∗ ( x ) · − : HH p,q (Λ , Λ( σ )) −→ HH p +3 ,q − (Λ , Λ( σ ))is an isomorphism for all p ≥ q ∈ Z and an epimorphism for p = 0 and all q ∈ Z .By Proposition 4.3, we have a map of long exact sequences... ...HH p,q (Λ( σ ) , Λ( σ )) HH p +3 ,q − (Λ( σ ) , Λ( σ ))HH p,q (Λ , Λ( σ )) HH p +3 ,q − (Λ , Λ( σ ))HH p,q (Λ , Λ( σ )) HH p +3 ,q − (Λ , Λ( σ ))HH p +1 ,q (Λ( σ ) , Λ( σ )) HH p +4 ,q − (Λ( σ ) , Λ( σ ))... ... i ∗ x ·− i ∗ id − σ − ∗ σ ∗ i ∗ ( x ) ·− id − σ − ∗ σ ∗ ∂ i ∗ ( x ) ·− ∂x ·− Hence, the proposition follows from the previous paragraph and the five lemma. (cid:3) Lifting edge units
The goal of this section is to lift edge units along the morphism in Proposition6.6, mainly in bidegree (3 , − k will be a perfect field. We start with some consequencesof results from Section 5.Let Λ be a Frobenius algebra and σ : Λ → Λ an automorphism. Recall that i : Λ ⊂ Λ( σ ) denotes the inclusion of the degree 0 part. In the rest of this section, NHANCED FINITE TRIANGULATED CATEGORIES 21 we will work in the monoidal triangulated category Mod p (Λ e ) introduced in Section5, and ⊗ will exceptionally stand for ⊗ Λ .Recall the natural isomorphism ζ M from (5.3). The following result is an imme-diate consequence of Lemma 5.4. Corollary 7.1.
For n ≥ , the automorphism ζ Ω n (Λ) of Ω(Λ) ⊗ n +1 is ( − n . Corollary 7.2. If f : Ω n (Λ) → M is a map, n ≥ , then the following diagramcommutes, Ω(Λ) ⊗ Ω(Λ) ⊗ n Ω(Λ) ⊗ n ⊗ Ω(Λ)Ω(Λ) ⊗ M M ⊗ Ω(Λ) ( − n ⊗ f f ⊗ ζ M This follows by naturality.
Corollary 7.3. If f : Ω n (Λ) → M is a map, n ≥ , then the following diagramcommutes in Mod p (Λ e ) , Ω(Λ) ⊗ n ⊗ Ω(Λ) ⊗ n Ω(Λ) ⊗ n ⊗ Ω(Λ) ⊗ n Ω(Λ) ⊗ n ⊗ M M ⊗ Ω(Λ) ⊗ n ( − n ⊗ f f ⊗ (cid:81) ni =1 ⊗ i − ⊗ ζ M ⊗ ⊗ n − i We use again naturality here.
Corollary 7.4. If f : Ω n (Λ) → M is an isomorphism in Mod p (Λ e ) , n ≥ , thenthe following diagram commutes in this category, Ω(Λ) ⊗ n ⊗ M M ⊗ Ω(Λ) ⊗ n M ⊗ M M ⊗ M (cid:81) ni =1 ⊗ i − ⊗ ζ M ⊗ ⊗ n − i f ⊗ ⊗ f ( − n Proof.
It suffices to consider the following commutative diagram of isomorphisms,Ω(Λ) ⊗ n ⊗ Ω(Λ) ⊗ n Ω(Λ) ⊗ n ⊗ Ω(Λ) ⊗ n Ω(Λ) ⊗ n ⊗ M M ⊗ Ω(Λ) ⊗ n M ⊗ M M ⊗ M ( − n ⊗ ff ⊗ f f ⊗ f ⊗ f (cid:81) ni =1 ⊗ i − ⊗ ζ M ⊗ ⊗ n − i f ⊗ ⊗ f (cid:3) The functor σ Λ ⊗ − ⊗ Λ σ : Mod p (Λ e ) −→ Mod p (Λ e )is an exact equivalence, so it gives rise to a triangulated equivalence σ Λ ⊗ − ⊗ Λ σ : Mod p (Λ e ) −→ Mod p (Λ e ) . Part of this triangulated equivalence is a natural isomorphism in Mod p (Λ e ), whichshows that this functor commutes with the syzygy functor, ω M : Ω( σ Λ ⊗ M ⊗ Λ σ ) ∼ = σ Λ ⊗ Ω( M ) ⊗ Λ σ . It is simply given by ω M = ζ σ Λ ⊗ M ⊗ Λ σ : Ω(Λ) ⊗ σ Λ ⊗ M ⊗ Λ σ ∼ = σ Λ ⊗ Ω(Λ) ⊗ M ⊗ Λ σ . We also have an exact equivalenceMod p (Λ e ) −→ Mod p (Λ e ) : M (cid:55)→ σ M σ , which is exactly the same thing as the restriction of scalars along σ : Λ → Λ. It isnaturally isomorphic to the previous one via the map ρ M : σ Λ ⊗ M ⊗ Λ σ ∼ = σ M σ , ρ M ( a ⊗ x ⊗ b ) = axb. In addition, we obtain an induced triangulated equivalenceMod p (Λ e ) −→ Mod p (Λ e ) : M (cid:55)→ σ M σ , also naturally isomorphic to the previous one in the same way, as triangulatedfunctors. Again, part of this triangulated equivalence is a natural isomorphism inMod p (Λ e ), which shows that this functor commutes with the syzygy functor,¯ ω M : Ω( σ M σ ) ∼ = σ Ω( M ) σ . The natural isomorphism of triangulated equivalences is compatible with the nat-ural isomorphisms ω and ¯ ω , in the sense that the following square of natural iso-morphisms commutes,Ω( σ Λ ⊗ M ⊗ Λ σ ) σ Λ ⊗ Ω( M ) ⊗ Λ σ Ω( σ M σ ) σ Ω( M ) σω M Ω( ρ M ) ρ Ω( M ) ¯ ω M Let us denote ω ( n ) M = n (cid:89) i =1 Ω i − ( ω Ω n − i ( M ) ) , ¯ ω ( n ) M = n (cid:89) i =1 Ω i − (¯ ω Ω n − i ( M ) ) . These are isomorphisms fitting in a similar diagram,(7.5) Ω n ( σ Λ ⊗ M ⊗ Λ σ ) σ Λ ⊗ Ω n ( M ) ⊗ Λ σ Ω n ( σ M σ ) σ Ω n ( M ) σω ( n ) M Ω n ( ρ M ) ρ Ω n ( M ) ¯ ω ( n ) M Proposition 7.6. If f : Ω(Λ) ⊗ n = Ω n (Λ) → σ Λ is an isomorphism in Mod p (Λ e ) , n ≥ , then the following diagram commutes in this category, Ω n (Λ) Ω n ( σ Λ σ ) σ Ω n (Λ) σσ Λ σ Λ σ Ω n ( σ ) ¯ ω n Λ σ f σ f ( − n σ Proof.
By the previous corollaries and various naturality properties, we have thefollowing commutative diagram of isomorphisms,
NHANCED FINITE TRIANGULATED CATEGORIES 23
It therefore suffices to check that the bottom composite is ( − n σ . This is veryeasy to check. Actually, if we recall that ρ is defined by multiplication, we readilysee that, starting with a ∈ σ Λ in the bottom left corner,1 ⊗ σ ( a ) ⊗ − n ⊗ σ ( a ) ⊗ a ⊗ a ⊗ σ ( a ) ( − n σ ( a ) (cid:3) Corollary 7.7.
For n > , if n is odd or char k = 2 , then i ∗ : HH n, (Λ( σ ) , Λ( σ )) −→ HH n, (Λ , Λ( σ )) induces a surjection on edge units.Proof. This morphism induces a map on edge units by Proposition 6.6. Considerthe long exact sequence in Proposition 4.3 for T Σ = Λ( σ ) and ∗ = 1. Proposition7.6 shows that any edge unit in the target is in the kernel of id − σ − ∗ σ ∗ . Here weuse that n is odd and that the graded σ is defined on the degree 1 part of Λ( σ ),which is σ Λ , as the ungraded − σ , see Remark 4.2. Therefore, any edge unit in thetarget has a preimage, which must also be an edge unit by Proposition 6.6. (cid:3) Since Λ( σ ) is the same as Λ( σ − ) reversing the degrees, we also deduce thefollowing result. Corollary 7.8.
For n > , if n is odd or char k = 2 , then i ∗ : HH n, − (Λ( σ ) , Λ( σ )) −→ HH n, − (Λ , Λ( σ )) induces a surjection on edge units. Proposition 7.9.
Given an edge unit u ∈ HH , − (Λ( σ ) , Λ( σ )) , there exists y ∈ HH , − (Λ( σ ) , Λ( σ )) such that Sq( u + { δ } · y ) = 0 .Proof. For any y we haveSq( u + { δ } · y ) = Sq( u ) + Sq( { δ } · y ) + [ u, { δ } · y ]= Sq( u ) + u · y − { δ } · [ u, y ]Here we use the Gerstenhaber algebra laws and Propositions 3.9, 3.10 and 3.6. Since u · − : HH , − (Λ( σ ) , Λ( σ )) → HH , − (Λ( σ ) , Λ( σ )) is an isomorphism by Lemma y such that Sq( u ) + u · y = 0. Let us fix this y . If weprove that [ u, y ] = 0, then we will be done. Again, u · − : HH , − (Λ( σ ) , Λ( σ )) → HH , − (Λ( σ ) , Λ( σ )) is an isomorphism, so it is enough to check that u · [ u, y ] =0. We have that [Sq( u ) , u ] = [ u, [ u, u ]] = 0 and u · y = − Sq( u ), therefore 0 =[ u, u · y ] = [ u, u ] · y + u · [ u, y ]. Once again, since u · − : HH , − (Λ( σ ) , Λ( σ )) → HH , − (Λ( σ ) , Λ( σ )) is an isomorphism, it suffices to prove that u · [ u, u ] · y = 0.This follows from u · [ u, u ] · y = [ u, u ] · ( u · y )= 2 · Sq( u ) · ( − Sq( u ))= − · Sq( u ) = 0 . Here we use that | Sq( u ) | = 5 − (cid:3) Proposition 7.10.
The map i ∗ : HH , − (Λ( σ ) , Λ( σ )) −→ HH , − (Λ , Λ( σ )) induces a bijection between the set of edge units u in the source satisfying Sq( u ) = 0 and the set of edge units in the target with no extra condition.Proof. The map i ∗ restricts to edge units by Proposition 6.6. Let us first checksurjectivity. Any edge unit in HH , − (Λ , Λ( σ )) has a preimage u by Corollary 7.8.The element u + { δ } · y in Proposition 7.9 is another preimage since Λ is ungradedso i ∗ ( { δ } ) = 0 and therefore i ∗ ( u + { δ } · y ) = i ∗ ( u ). Moreover, u + { δ } · y is also anedge unit by Proposition 6.6.We now check injectivity. Let u, u (cid:48) ∈ HH , − (Λ( σ ) , Λ( σ )) be edge units withSq( u ) = Sq( u (cid:48) ) = 0 and i ∗ ( u ) = i ∗ ( u (cid:48) ). The element z = u (cid:48) − u is in the kernel of i ∗ , hence Sq( z ) too by Proposition 2.1. Using the Gerstenhaber algebra laws andProposition 3.6 we obtain0 = Sq( u (cid:48) ) uz = [ u, { δ } z ] + { δ } [ u, z ]= Sq( u + z ) = { δ } [ u, z ]= Sq( u ) + [ u, z ] + Sq( z ) = −{ δ } Sq ( z )= [ u, z ] + Sq( z ) , = 0 . On the right, we use twice that the kernel of i ∗ is a square zero ideal, see Proposition4.3, and i ∗ ( { δ } ) = 0. By Lemma 6.8, uz = 0 implies that z = 0, so u = u (cid:48) . (cid:3) Enhancements
Recall that a DG-functor between DG-categories A → B is a Morita equivalence if it induces an equivalence between their derived categories D ( A ) → D ( B ), orequivalently between their full subcategories of compact objects. Definition 8.1.
Let T be a finite category and Σ : T → T an automorphism.The set of enhanced triangulated categories with underlying suspended category( T , Σ), denoted by ETC( T , Σ) , NHANCED FINITE TRIANGULATED CATEGORIES 25 is the set of Morita equivalence classes of DG-categories A such that the derivedcategory of compact objects D c ( A ) is equivalent to ( T , Σ) as a suspended cate-gory. A suspended category is a category equipped with a self-equivalence, and twosuspended categories ( T , Σ), ( T (cid:48) , Σ (cid:48) ) are equivalent if there is an equivalence ofcategories F : T → T (cid:48) such that F Σ ∼ = Σ (cid:48) F . This is equivalent to the existence ofa graded category equivalence T Σ (cid:39) T (cid:48) Σ (cid:48) .The aim of this paper is to show that, under mild conditions, ETC( T , Σ) iseither empty or a singleton. In this section, we give alternative descriptions of thisset which will help in proving that.Let DGCat
Mor be the category of DG-categories endowed with Tabuada’s Moritamodel structure [57, 59, 60]. Let DGCat
Mor ( T , Σ) be the full subcategory of DG-categories A with D c ( A ) equivalent to ( T , Σ). The set ETC( T , Σ) has beenimplicitly defined as the set of connected components of the classification space | DGCat
Mor ( T , Σ) | in the sense of [16]. This space is the nerve of the categoryof Morita weak equivalences between DG-categories A with D c ( A ) equivalent to( T , Σ). This space contains much more information than its mere set of connectedcomponents. In a sense, it knows about the homotopy symmetries of each DG-category in the Morita model structure [61, Corollary A.0.4].Let DGCat Eq be the category of DG-categories endowed with the model struc-ture in [58], whose weak equivalences are quasi-equivalences. Let us denote byDGCat Eq ( T , Σ) be the full subcategory of pre-triangulated DG-categories A , inthe sense of [12], such that H ( A ) is equivalent to ( T , Σ) as a suspended cat-egory. A DG-category A is pre-triangulated if the canonical inclusion functor H ( A ) (cid:44) → D ( A ) identifies H ( A ) with a triangulated subcategory of the tar-get. A DG-category A is fibrant in DGCat Mor precisely if it is pre-triangulatedand H ( A ) is idempotent complete. This is equivalent to saying that the previouscanonical inclusion induces an equivalence H ( A ) (cid:39) D c ( A ) with the full sub-category of compact objects in D ( A ). In particular, objects in DGCat Eq ( T , Σ)are Morita fibrant. Since DGCat
Mor is a left Bousfield localization of DGCat Eq ,Morita equivalences between Morita fibrant DG-categories are the same as quasi-equivalences. Proposition 8.2.
There is a homotopy equivalence | DGCat Eq ( T , Σ) | (cid:39) | DGCat
Mor ( T , Σ) | . Proof.
We have an inclusion functor j : DGCat Eq ( T , Σ) ⊂ DGCat
Mor ( T , Σ). A fi-brant replacement functor R on DGCat Mor (co)restricts to R : DGCat Mor ( T , Σ) → DGCat Eq ( T , Σ). Both functors j and R preserve weak equivalences. Moreover,both composites jR and Rj are equipped with natural weak equivalences to thecorresponding identity functor. Hence, we are done. (cid:3) Let Λ be the endomorphism algebra of a basic additive generator of T , thatwe assume to be Frobenius as a necessary condition for ETC( T , Σ) to be non-empty, σ : Λ → Λ an automorphism induced by Σ, and DGCat Eq (Λ( σ )) the fullsubcategory of DGCat Eq spanned by the DG-categories B satisfying the followingproperties:(1) The category H ( B ) has a unique object up to isomorphism.(2) The graded endomorphism algebra of some (and hence any) object in H ∗ ( B ) is isomorphic to Λ( σ ). (3) The functor D c ( B ) → Mod(Λ) defined by evaluating at an object inducesan equivalence onto the full subcategory of finitely presented projectiveΛ-modules.Clearly, (3) is equivalent to the following condition:(3 (cid:48) ) The functor H ( B ) (cid:44) → D c ( B ) induces an equivalence from the completionof the source by direct sums and idempotents.This completion, which will also be used below, consists of first formally adding allfinite direct sums of objects and then all retracts of idempotents. By the Moritainvariance of the derived category, (3 (cid:48) ) is equivalent to:(3 (cid:48)(cid:48) ) The completion of B by direct sums and idempotents in H ( B ) is pre-triangulated.Moreover, after (2), (3) is also equivalent to(3 (cid:48)(cid:48)(cid:48) ) The functor D c ( B ) Σ → Mod(Λ( σ )) defined by evaluating at an objectinduces an equivalence onto the full graded subcategory of finitely presentedprojective Λ( σ )-modules.This is because the source and target of the functor in (3 (cid:48)(cid:48)(cid:48) ) are weakly stable, andits degree 0 part is the functor in (3). Definition 8.3. A minimal A -infinity algebra structure ( A, m , . . . , m n , . . . ) ona graded algebra A consists of Hochschild cochains m n ∈ C n, − n ( A, A ), n ≥ m is acocycle, whose cohomology class is called universal Massey product , { m } ∈ HH , − ( A, A ) . Remark . Actually, the universal Massey product is well defined for any minimal A -algebra structure on A which admits an A -extension. Using Kadeishvili’s the-orem [24, 31], we can also define the universal Massey product of a DG-algebra asthe universal Massey product of any minimal model. The universal Massey producthas been previously studied in e.g. [25, 4].By [38, Corollary 6.2] and the Morita invariance of Hochschild cohomology (i.e. itdoes not change if we complete by direct sums and idempotents), we can also replace(3) in the previous list with:(3 (cid:48)(cid:48)(cid:48)(cid:48) ) The universal Massey product of B is an edge unit. Proposition 8.5.
There is a homotopy equivalence | DGCat Eq (Λ( σ )) | (cid:39) | DGCat Eq ( T , Σ) | . Proof.
We define functorsDGCat Eq (Λ( σ )) R (cid:29) j DGCat Eq ( T , Σ)in the following way. The functor j takes A to the full sub-DG-category j ( A )spanned by the objects which become basic additive generators in H ( A ). There-fore j ( A ) satisfies (1), (2), and (3 (cid:48)(cid:48) ), since the completion of j ( A ) mentionedtherein is quasi-equivalent to A . In particular, the full inclusion j ( A ) ⊂ A is aMorita equivalence. The functor R is the (co)restriction of a fibrant replacementfunctor in DGCat Mor . NHANCED FINITE TRIANGULATED CATEGORIES 27
In order for R to be well defined we must check that, for each B in the source, D c ( R ( B )) is equivalent to ( T , Σ). The natural cofibration B (cid:26) R ( B ) is a Moritaequivalence, so it induces a suspended equivalence D c ( B ) (cid:39) D c ( R ( B )), and D c ( B )is equivalent to ( T , Σ) by (3 (cid:48)(cid:48)(cid:48) ).Both j and R preserve quasi-equivalences. There is an obvious natural quasi-equivalence from the identity in DGCat Eq (Λ( σ )) to jR . Hence, both functors inducehomotopic maps on the classification space.We will finish the proof as soon as we show that the functor Rj induces a maphomotopic to the identity in the classification space of DGCat Eq ( T , Σ). For this,we choose a functorial factorization in DGCat
Mor , which functorially sends eachDG-functor f : A → B to a factorization A ∼ (cid:26) R (cid:48) ( f ) (cid:16) B of f consisting of a trivial cofibration followed by a fibration. Therefore, here,and also below in this proof, ∼ stands for weak equivalence in the Morita modelstructure. We can suppose that R ( A ) = R (cid:48) ( A → e ), where e is the terminal DG-category. Given A in DGCat Eq ( T , Σ), we can apply the functorial factorizationto the commutative square j ( A ) A j ( A ) e f ∼ where the top arrow is the inclusion which, as we pointed out above, is a Moritaequivalence. This yields j ( A ) R (cid:48) ( f ) A j ( A ) Rj ( A ) e ∼∼ Here, the top right horizontal arrow and the middle vertical arrow are Moritaequivalences between fibrant objects in DGCat
Mor . Here we use the 2-out-of-3property and the fact that f is a Morita equivalence. Therefore, those two mapsare quasi-equivalences. This shows that the maps induced by the functors Rj and A (cid:55)→ R (cid:48) ( j ( A ) → A ) are homotopic to the identity map in | DGCat Eq ( T , Σ) | . (cid:3) Let DGAlg be the usual model category of (unital) DG-algebras [51, § σ ))the full subcategory of DG-algebras A with H ∗ ( A ) ∼ = Λ( σ ) whose universal Masseyproduct is an edge unit. Proposition 8.6.
There is a bijection π | DGAlg(Λ( σ )) | (cid:39) π | DGCat Eq (Λ( σ )) | . Proof.
We can regard any DG-algebra as a DG-category with only one object. Thisdefines an inclusion DGAlg ⊂ DGCat Eq preserving weak equivalences. Using thecharacterization of DGCat Eq (Λ( σ )) in terms of universal Massey products, givenby (1), (2), and (3 (cid:48)(cid:48)(cid:48)(cid:48) ) above, we see that the previous inclusion (co)restricts to DGAlg(Λ( σ )) ⊂ DGCat Eq (Λ( σ )). This is where the universal Massey productcomes into play.Using [61, Corollary A.0.5], and arguing as in the proof of [61, Proposition2.3.3.5], it is easy to see that the homotopy fiber of the map | DGAlg(Λ( σ )) | −→ | DGCat Eq (Λ( σ )) | induced by the previous inclusion at a DG-category B in the target category is themapping space Map DGCat Eq ( k, B ) , where k is the ground field regarded as a DG-category with only one object withendomorphism algebra k . In general, π of such mapping space is the set of iso-morphism classes of objects in H ( B ). In our case, this is a singleton, hence thestatement follows. (cid:3) Definition 8.7. An enhanced triangulated structure on ( T , Σ) is a minimal A -infinity algebra structure on Λ( σ ) whose universal Massey product is an edge unit.Two enhanced triangulated structures are gauge equivalent if there is an A -infinitymorphism with identity linear part between them.The gauge equivalence relation is an honest equivalence relation by well-knownproperties of A -infinity morphisms. The quotient set will be denoted byETS( T , Σ) . Definition 8.7 is explained by the characterization of pre-triangulated DG- and A -infinity categories, in the sense of Bondal-Kapranov [12], in terms of edge units,used above, compare [38, Corollary 6.2].We now relate the sets ETC( T , Σ) and ETS( T , Σ).
Remark . Notice that the automorphism group Aut(Λ( σ )) of the graded algebraΛ( σ ) acts by conjugation on the right of the set of minimal A -infinity algebrastructures (Λ( σ ) , m , . . . , m n , . . . ) on Λ( σ ). More precisely, given g ∈ Aut(Λ( σ )),(Λ( σ ) , m , . . . , m n , . . . ) g = (Λ( σ ) , g − m g ⊗ , . . . , g − m n g ⊗ n , . . . ) . This action passes to ETS( T , Σ).
Theorem 8.9.
There is a bijection
ETS( T , Σ) / Aut(Λ( σ )) ∼ = ETC( T , Σ) . Proof.
Let Chain be the category of chain complexes with the projective modelstructure. The forgetful functor DGAlg → Chain induces a map | DGAlg | −→ |
Chain | . Forgetting the product, we can regard Λ( σ ) as a fibrant-cofibrant object in Chainwith trivial differential. By [35, Theorem 4.6], [37, Theorem 1.2 and Remark 6.5],and [36, Corollary 2.3], the set of connected components π F (cid:48) of the homotopyfiber F (cid:48) of the previous map at Λ( σ ) is a quotient of the set of minimal A -infinityalgebra structures (Λ( σ ) , m (cid:48) , m , . . . , m n , . . . ) with underlying graded vector spaceΛ( σ ). Here, m (cid:48) endows Λ( σ ) with a unital graded associative algebra structurewhich may be different from the given one m . The equivalence relation is given bythe existence of an A -infinity morphism with identity linear part, as in the definitionof ETS( T , Σ).
NHANCED FINITE TRIANGULATED CATEGORIES 29
Let DGAlg(Λ( σ )) (cid:48) be the full subcategory of DGAlg spanned by all the DG-algebras A with such that H ∗ ( A ) is isomorphic to Λ( σ ) as graded vector spaces. Theset π | DGAlg(Λ( σ )) (cid:48) | is by definition the image of the map π F (cid:48) → π | DGAlg | ,hence it is the quotient of π F (cid:48) by the action of π ( | Chain | , Λ( σ )). This group isthe automorphism group of Λ( σ ) as a graded vector space. The action on π F (cid:48) ,can be described as follows. The automorphism group of Λ( σ ) acts on the rightof the previous set of A -infinity algebra structures (Λ( σ ) , m (cid:48) , m , . . . , m n , . . . ) byconjugation, and this induces the action on the quotient set π F (cid:48) .The category DGAlg(Λ( σ )) is the full subcategory of DGAlg(Λ( σ )) (cid:48) spannedby the objects A with H ∗ ( A ) ∼ = Λ( σ ) as graded algebras whose universal Masseyproduct is an edge unit. Let F ⊂ F (cid:48) be the full subspace spanned by the connectedcomponents such that the graded algebra (Λ( σ ) , m (cid:48) ) is isomorphic to (Λ( σ ) , m )and the universal Massey product { m } is an edge unit. Note that all these condi-tions are preserved by gauge equivalence. Then, we have that π | DGAlg(Λ( σ )) | ∼ = π F/π ( | Chain | , Λ( σ )). Any element in the quotient has a representative with m = m (cid:48) and π F can be alternatively described as the quotient of those represen-tatives by the subgroup of automorphisms of the graded vector space Λ( σ ) whichfix m . This is precisely the automorphism group of Λ( σ ) as a graded algebra. Thistheorem now follows from the previous results in this section. (cid:3) Enhanced triangulated structures and edge units
In this section, the ground field k is required to be perfect. Consider a fi-nite category T equipped with an automorphism Σ : T → T . Let Λ be theendomorphism algebra of a basic additive generator. We assume that Λ is Frobe-nius, as per Freyd’s necessary condition for the existence of triangulated structures.Moreover, let σ : Λ → Λ be an automorphism induced by Σ. Recall from Defini-tions 8.3 and 8.7 the notions of universal Massey product, enhanced triangulatedstructure, and gauge equivalence. The group Aut(Λ( σ )) acts on the right of theHochschild cohomology of Λ( σ ) by conjugation, more precisely, given g ∈ Aut(Λ( σ ))and x ∈ HH (cid:63), ∗ (Λ( σ ) , Λ( σ )), x g = g ∗ ( g − ) ∗ ( x ). This is action is compatible withthe Gerstenhaber algebra structure. Theorem 9.1.
Universal Massey products define a bijection between
ETS( T , Σ) and the set of edge units u ∈ HH , − (Λ( σ ) , Λ( σ )) satisfying Sq( u ) = 0 . Thisbijection is Aut(Λ( σ )) -equivariant.Proof. The map in the statement is clearly Aut(Λ( σ ))-equivariant, see Remark 8.8.The set of gauge equivalence classes of arbitrary minimal A -infinity algebra struc-tures is π Map dgOp ( A ∞ , E (Λ( σ ))) . Here A ∞ is the A -infinity operad, E (Λ( σ )) is the endomorphism operad of the gradedvector space Λ( σ ), and Map dgOp stands for the mapping space in the model categoryof differential graded operads. The universal Massey product is invariant by gaugeequivalences, i.e. if two minimal A -infinity algebra structures on the graded vectorspace Λ( σ ) are quasi-isomorphic by an A -infinity quasi-isomorphism with identitylinear part then they have the same product and the same universal Massey product[31, Lemme B.4.2]. Therefore the property of being an enhanced triangulatedstructure on ( T , Σ) is also invariant by gauge equivalences.
The cofibrant DG-operad A ∞ is the union of a sequence of cofibrations A n ⊂ A n +1 , n ≥
2, where A n is the A n operad. In [39], we extend the Bousfield–Kan spectralsequence of the tower of fibrations { X n } n ≥ , with X n = Map dgOp ( A n +2 , E (Λ( σ )))and bonding maps defined by restriction, for the computation of the homotopygroups of X ∞ = lim n X n = Map dgOp ( A ∞ , E (Λ( σ ))). The Bousfield–Kan terms E p,qr are only defined for q ≥ p ≥ E p,q ∞ contributes to π q − p X ∞ . The differentialslook like d r : E p,qr −→ E p + r,q + r − r and they are only defined for q > p ≥
0, so the elements of the so-called fringedline , i.e. E p,pr , p ≥
0, are not defined as the homology of differentials. Actually,these terms are plain pointed sets for general towers of spaces, and moreover theterms E p,p +1 r , p ≥
0, are possibly non-abelian groups.We extend the range of definition of the previous spectral sequence to all q ∈ Z for p ≥ r − p ≥
0. Most new and old terms are endowed with k -vector spacestructures. The only pointed sets are E p,pr for 0 ≤ p ≤ r −
2, and a finite amountof remaining terms are abelian groups (non-abelian groups do not show up for ourparticular tower of spaces). Moreover, on each page E r we define differentials d r likeabove out of all terms except for 0 ≤ p = q ≤ r − d , whichis defined on E , ), and the term E p,qr +1 is given by the homology of d r whenever E p,qr has an incoming and an outgoing differential (the incoming differential is taken tobe d r = 0 if q > p < r ). The new terms do not contribute to the homotopy groupsof X ∞ , but they help in the computation of the Bousfield–Kan terms, and theyalso contain obstructions, as we explain below. Moreover, we fully compute the E terms and the differential d of our extended spectral sequence, in particular in theBousfield–Kan part, where this was not previously known either.The spectral sequence is defined if a base point in X ∞ is given. If we only havea base point x n ∈ X n , then the spectral sequence is defined up to the terms of page (cid:98) n +32 (cid:99) , hence we call it truncated . Moreover, there is an obstruction in E n +1 ,nr ,1 ≤ r ≤ n +32 , which vanishes if and only if there exists a vertex x n +1 ∈ X n +1 whichhas the same image in X n − r +1 as x n .The universal Massey product { m } of an enhanced triangulated structure is anedge unit and satisfies Sq( { m } ) = 0 since this is the obstruction for n = r = 2, see[39, Proposition 6.7]. Tautologically, any element u ∈ HH , − (Λ( σ ) , Λ( σ )) is theuniversal Massey product of some minimal A -algebra structure x ∈ X definedby a representing cocycle, that we fix, which extends to A . Let x (cid:48) ∈ X be anextension. If Sq( u ) = 0, the obstruction vanishes and there is some A -algebrastructure x (cid:48) ∈ X which restricts to x ∈ X .Since we have x (cid:48) ∈ X , the truncated spectral sequence is defined up to the E terms, and the obstructions living therein are also defined. We will prove belowthat, if u is an edge unit, then E p,q = 0 for all p ≥
2. Hence, the possibly non-trivialpart of the E page looks like st NHANCED FINITE TRIANGULATED CATEGORIES 31
Let us show now that this implies that the map in the statement is bijective.In order to check surjectivity, we prove that there is an A -infinity algebra struc-ture x ∞ ∈ X ∞ which restricts to x ∈ X . More precisely, we prove by inductionthat, given elements x i ∈ X i , 1 ≤ i ≤ n −
2, and x (cid:48) n ∈ X n , with x the fixedelement above, compatible by restriction, we can obtain a similar collection of ele-ments x i ∈ X i , 1 ≤ i ≤ n −
1, and x (cid:48) n +1 ∈ X n +1 . Here, the only new elements are x n − and x (cid:48) n +1 , and we have forgot x (cid:48) n . The initial case is n = 3, defined above. Foreach n ≥
3, it suffices to show the existence of some x (cid:48) n +1 with the same image in X n − as x (cid:48) n , i.e. x n − , since we can then take x n − as the image of x (cid:48) n +1 in X n − .The obstruction to this lives in E n +1 ,n = 0, hence we are done with surjectivity.Let us check injectivity. For this, we fix x ∞ ∈ X ∞ restricting to x ∈ X . Inparticular, the whole spectral sequence is defined. We want to prove that any other x (cid:48)∞ ∈ X ∞ with universal Massey product u lies in the same connected componentas x ∞ . Since the universal Massey products agree, the restriction of x (cid:48)∞ to X liesin the same component as x . This implies that the restrictions of x ∞ and x (cid:48)∞ to X n lie in the same connected component for all n ≥
1, since E nn is a singleton forany n ≥ E n,n = Ker[ π X n → π X n − ] ∩ Im[ π X n +2 → π X n ] . This does not directly imply that x ∞ and x (cid:48)∞ lie in the same component of X ∞ .For this, we need to know that lim n π X n = 0, see [13, IX.3.1]. This follows from[13, IX.5.4] since the vanishing on E implies that E p,q = E p,qr for all q − p ≥ r ≥ E claimed above. Recall from [39] that the E terms (those which are defined) are E p,q = (cid:26) HH p +2 , − q (Λ( σ ) , Λ( σ )) , p > , q ∈ Z ;Z , − q (Λ( σ ) , Λ( σ )) , p = 0 , q > . Here, Z (cid:63), ∗ (Λ( σ ) , Λ( σ )) ⊂ C (cid:63), ∗ (Λ( σ ) , Λ( σ )) denotes the Hochschild cocycles. Thereis a remaining E term, namely E , which is the pointed set of graded algebrastructures with the same underlying graded vector space as Λ( σ ), based at Λ( σ ).The second differential d : E p,q −→ E p +2 ,q +12 , is defined except for ( p, q ) = (0 , d = ± [ u, − ] , except for ( p, q ) = (0 , p = 0 and q >
1, we understand that we first projectthe Hochschild cocycles onto the Hochschild cohomology and then apply ± [ u, − ].For ( p, q ) = (0 , α : HH , − (Λ( σ ) , Λ( σ )) −→ HH , − (Λ( σ ) , Λ( σ )) ,x (cid:55)→ x + [ u, x ] . (9.2)The first quadratic summand vanishes if char k (cid:54) = 2, since HH (cid:63), ∗ (Λ( σ ) , Λ( σ )) isgraded commutative and x has odd total degree. If char k = 2, the quadraticsummand is not k -linear unless k = F , hence this differential, unlike the rest, isnot a k -vector space morphism, but a plain abelian group morphism. Excluding E , where d is not defined, the second page of the truncated spectralsequence splits into two families of cochain complexes C ∗ n and D ∗ n , n ∈ Z , C ∗− C ∗ C ∗ C ∗ C ∗ C ∗− C ∗− D ∗ D ∗ D ∗− D ∗− D ∗ D ∗ with the following descriptions, C mn = (cid:26) E m,n + m , m ≥ , or m = 0 and n ≥ , elsewhere; D mn = (cid:26) E m +1 ,n + m , m ≥ , elsewhere . The differential is of course d in all cases. The E p,q terms are defined for q ≥ p ≥ p ≥ q ∈ Z . All of them are given by the homology of d , except for E and E , i.e. H m C ∗ n = E m,n + m , m ≥ , or m ≥ n ≥ H m D ∗ n = E m +1 ,n + m , m ≥ , or m ≥ n ≥ C ∗ n and ¯ D ∗ n , n ∈ Z , which agree with the former almost everywhere, andcan be depicted as follows NHANCED FINITE TRIANGULATED CATEGORIES 33¯ C ∗ ¯ C ∗ ¯ C ∗ ¯ C ∗ ¯ C ∗− ¯ C ∗− ¯ C ∗− ¯ D ∗ ¯ D ∗ ¯ D ∗− ¯ D ∗− ¯ D ∗ ¯ D ∗ Here, on any coordinate ( p, q ), p ≥ q ∈ Z , we place the Hochschild cohomologygroup HH p +2 , − q (Λ( σ ) , Λ( σ )). Therefore, for m ≥ n ∈ Z ,¯ C mn = HH m +2 , − n − m (Λ( σ ) , Λ( σ )) , ¯ D mn = HH m +3 , − n − m (Λ( σ ) , Λ( σ )) , and ¯ C mn = ¯ D mn = 0 for m <
0. The differential is [ u, − ] in all non-trivial cases.The cup product with the universal Massey product u · − induces cochain maps f : ¯ C ∗ n −→ ¯ D ∗ +1 n , g : ¯ D ∗ n −→ ¯ C ∗ +2 n − , for n ∈ Z , depicted below in red and blue, respectively, ¯ C ∗ ¯ C ∗ ¯ C ∗ ¯ C ∗ ¯ C ∗− ¯ C ∗− ¯ C ∗− ¯ D ∗ ¯ D ∗ ¯ D ∗− ¯ D ∗− ¯ D ∗ ¯ D ∗ By Lemma 6.8, f and g are injective, the cokernel of f is concentrated in degree ∗ = −
1, and the cokernel of g is concentrated in degrees ∗ = − , −
1. Therefore, theassociated long exact sequences induce the following isomorphisms in cohomologyfor m ≥ n ∈ Z , f ∗ : H m ¯ C ∗ n ∼ = H m +1 ¯ D ∗ n , g ∗ : H m ¯ D ∗ n ∼ = H m +2 ¯ C ∗ n . By Proposition 3.6 and Remark 3.7, f and g are null-homotopic, so, for m ≥ n ∈ Z , H m ¯ C ∗ n = 0 = H m ¯ D ∗ n . Clearly, D ∗ n = ¯ D ∗ n , hence E m +1 ,n + m = H m D ∗ n = 0 , m ≥ , n ∈ Z . For n ≤ C ∗ n is the naive truncation of ¯ C ∗ n at ∗ ≥
1, therefore E m,n + m = H m C ∗ n = H m ¯ C ∗ n = 0 , m ≥ n ≤ . For n ≥
2, there is an obvious surjective cochain map C ∗ n (cid:16) ¯ C ∗ n which is the identityin ∗ > ∗ = 0, so E m,n + m = H m C ∗ n = H m ¯ C ∗ n = 0 , m ≥ n ≥ . For n = 1 and char k (cid:54) = 2, we also have a surjective map C ∗ (cid:16) ¯ C ∗ for the samereason as above, and E m,m +13 = H m C ∗ = H m ¯ C ∗ = 0 , m ≥ . This completes the proof in case char k (cid:54) = 2. The problem in char k = 2 is that thedifferential C → C depends on the quadratic map (9.2) while ¯ C → ¯ C is justgiven by [ u, − ], so there is no obvious map C ∗ (cid:16) ¯ C ∗ .The last equation also holds for char k = 2, although it is more complicated tocheck. It suffices to construct a chain map C ∗ (cid:16) ¯ D ∗ +11 given by u · − for ∗ > ∗ = 0, sincethen E m,m +13 = H m C ∗ = H m +1 ¯ D ∗ = 0 , m ≥ . we define C → D as the following compositeZ , − (Λ( σ ) , Λ( σ )) (cid:16) HH , − (Λ( σ ) , Λ( σ )) β −→ HH , − (Λ( σ ) , Λ( σ ))where β ( x ) = u · x + { δ } · x . We must show that this β is surjective and completes the definition of a chain map C ∗ (cid:16) ¯ D ∗ +11 . Let us start with the second property. It suffices to check that u · α ( x ) = [ u, β ( x )]for any x ∈ HH , − (Λ( σ ) , Λ( σ )), where α is the morphism in (9.2) which definesthe differential d : E , → E , . This follows from Proposition 3.2, char k = 2, andthe laws of a Gerstenhaber algebra, since we know that [ u, u · x ] = u · [ u, x ], and[ u, { δ } · x ] = [ u, { δ } ] · x + { δ } · [ u, x ] · x + { δ } · x · [ u, x ]= u · x + { δ } · [ u, x ] · x + { δ } · [ u, x ] · x = u · x . We finish by showing the surjectivity of β . Actually, we will see that it is bijective.For this, we construct morphisms γ and λ fitting in the following commutativediagram, NHANCED FINITE TRIANGULATED CATEGORIES 35 HH , − (Λ( σ ) , Λ( σ ))HH , − (Λ( σ ) , Λ( σ ))HH , − (Λ( σ ) , Λ( σ ))HH , − (Λ( σ ) , Λ( σ )) β u ·−∼ = γu ·− ∼ = λ where the curved arrows are isomorphisms by Proposition 6.8. The morphisms γ and λ are defined as γ ( x ) = u · x + { δ } · x ,λ ( x ) = u · x + { δ } · x . Commutativity follows from γβ ( x ) = γ ( u · x + { δ } · x )= u · x + u · { δ } · x + { δ } · ( u · x + { δ } · x ) = u · x + u · { δ } · x + { δ } · u · x = u · x,λγ ( x ) = λ ( u · x + { δ } · x )= u · x + u · { δ } · x + { δ } · ( u · x + { δ } · x ) = u · x + u · { δ } · x + { δ } · u · x = u · x. Here we use Proposition 3.8, the commutativity of the cup product, and thatchar k = 2. (cid:3) The group Aut(Λ( σ )) also acts on the right of the graded algebra HH (cid:63), ∗ (Λ , Λ( σ ))by conjugation since Λ is the degree 0 part of Λ( σ ), so any automorphism of thelatter (co)restricts to an automorphism of the former. The following corollary is aconsequence of the previous theorem and Proposition 7.10. Corollary 9.3.
There is an
Aut(Λ( σ )) -equivariant bijection between ETS( T , Σ) and the set of edge units in HH , − (Λ , Λ( σ )) . Proposition 9.4.
The set
ETC( T , Σ) is non-empty if and only if Ω (Λ) ∼ = σ − Λ in Mod(Λ e ) . Moreover, in that case it is a singleton.Proof. By Theorem 8.9, ETC( T , Σ) is non-empty if and only if ETS( T , Σ) is.Therefore, the first part of the statement follows from the previous corollary andPropositions 5.7 and 6.5.By those previous results, in order to prove the second part, we must checkthat HH , − (Λ , Λ( σ )) × / Aut(Λ( σ )) is a singleton. Let Z (Λ) denote the center ofΛ. There is a group morphism g : HH , (Λ , Λ( σ )) × = HH , (Λ , Λ) × = Z (Λ) × → Aut(Λ( σ )) sending x ∈ Z (Λ) × to the automorphism g ( x ) : Λ( σ ) → Λ( σ ) defined on each degree n ∈ Z by right multiplication by x n . This morphism is obviously injec-tive, so we can regard HH , (Λ , Λ( σ )) × as a subgroup of Aut(Λ( σ )). Since any g ( x )is the identity in degree 0, the right action of HH , (Λ , Λ( σ )) × on HH , − (Λ , Λ( σ )) × is given by the comparison morphism HH , (Λ , Λ( σ )) × (cid:16) HH , (Λ , Λ( σ )) × , whichis surjective by Proposition 5.6, and left multiplication by the inverse. The quotientHH , − (Λ , Λ( σ )) × / HH , (Λ , Λ( σ )) × of the ‘left multiplication by the inverse’ action is a singleton since, given x, y ∈ HH , − (Λ , Λ( σ )) × , xy − ∈ HH , (Λ , Λ( σ )) × . Therefore, the quotient by the largergroup HH , − (Λ , Λ( σ )) × / Aut(Λ( σ )) is also a singleton. (cid:3) Definition 9.5.
We say that T has an enhanced triangulated structure if ( T , Σ)does for some automorphism Σ : T → T .Recall from [11, Proposition 3.8] that the Picard group
Pic(Λ) of invertible Λ-bimodules is isomorphic to the outer automorphism group Out(Λ) via Out(Λ) → Pic(Λ) : [ σ ] (cid:55)→ [ σ Λ ]. Moreover, Pic(Λ) is also the group of natural isomorphismclasses of self-equivalences of Mod(Λ), and [ σ ] corresponds to the restriction ofscalars ( σ − ) ∗ : Mod(Λ) → Mod(Λ) along σ − : Λ → Λ, which is naturally iso-morphic to − ⊗ Λ σ Λ . The advantage of the equivalence ( σ − ) ∗ is that it is anautomorphism. Corollary 9.6.
The finite category T (cid:39) proj(Λ) has an enhanced triangulatedstructure if and only if Ω (Λ) is isomorphic in Mod(Λ e ) to an invertible Λ -bimodule.Up to natural isomorphism, the possible suspension functors are the restrictionsof scalars σ ∗ : proj(Λ) → proj(Λ) , where σ runs over a set of representatives ofelements in Out(Λ) such that Ω (Λ) ∼ = σ − Λ in Mod(Λ e ) . Once the suspensionfunctor Σ is fixed, ETC( T , Σ) is a singleton. Corollary 9.7. If Λ is separable, then T (cid:39) proj(Λ) has an enhanced triangulatedstructure. Up to natural isomorphism, the possible suspension functors are therestrictions of scalars σ ∗ : proj(Λ) → proj(Λ) , where σ runs over a set of represen-tatives of elements in Out(Λ) . Once the suspension functor Σ is fixed, ETC( T , Σ) is a singleton. The case analyzed in this corollary is particularly simple, Λ is semisimple, T (cid:39) D c (Λ( σ )), and all exact triangles split here.Recall that the algebra Λ is connected if it cannot be decomposed as a productΛ ∼ = Λ × Λ , with Λ i a non-trivial algebra, i = 1 , Proposition 9.8.
Suppose Λ is connected and not separable. Given any Λ -bimodule M , denote by Ω( M ) the syzygy obtained ascovercover the kernel of a projectivecover, Ω( M ) (cid:44) → P (cid:16) M. Then, T = proj(Λ) has an enhanced triangulated structure if and only if Ω (Λ) is an invertible Λ -bimodule. In that case, the suspension functor is necessarily Σ = − ⊗ Λ Ω (Λ) − , where Ω (Λ) − is an inverse of Ω (Λ) in Pic(Λ) . Moreover,
ETC( T , Σ) is a singleton.Proof. The Λ-bimodule Λ is indecomposable, since Λ is connected as an algebra,hence Ω (Λ) too, and also any invertible Λ-bimodule σ Λ . Moreover, σ Λ cannotbe projective. Otherwise, it would be a direct summand of Λ ⊗ Λ, and then Λ
NHANCED FINITE TRIANGULATED CATEGORIES 37 would be a direct summand of σ − Λ ⊗ Λ ∼ = Λ ⊗ Λ, hence separable. The thirdsyzygy Ω (Λ) is not projective either since that would also imply that Λ would bea projective bimodule.Two Λ-bimodules M and N are isomorphic in Mod(Λ e ) if and only if thereare projective Λ-bimodules P and Q such that M ⊕ P ∼ = N ⊕ Q in Mod(Λ e ).Hence, M = Ω (Λ) is isomorphic to some N = σ Λ in Mod(Λ e ) if and only ifthey are isomorphic in Mod(Λ e ), because each of them is the only non-projectiveindecomposable factor on each side of an isomorphism M ⊕ P ∼ = N ⊕ Q as above.Now, this proposition follows from Corollary 9.6. (cid:3) Recall that the Nakayama algebra N nm is the quotient of the path algebra of theoriented cycle of length m ≥ m m − m − α m α α m − α α m − by the two-sided ideal generated by the paths of length n + 1, n ≥
1. This basic andself-injective algebra (hence Frobenius) has dimension m ( n + 1) over the groundfield k and it is connected but not separable, see [23, §
3] and [17, § Proposition 9.9. If k is algebraically closed, Λ = N nm , m divides n , and n > ,then Ω (Λ) is not stably isomorphic to an invertible Λ -bimodule, hence proj(Λ) cannot be endowed with a(n enhanced) triangulated category structure.Proof. In this proof we compute syzygies by using projective covers. By [23, § (Λ) = σ Λ for some automorphism σ of Λ. Since Λ is connected and not sepa-rable, if Ω (Λ) were stably isomorphic to an invertible Λ-bimodule then it wouldactually be isomorphic to it, by the same argument as in the proof of Proposition9.8 above. Recall from [11, Proposition 3.8] that any invertible Λ-bimodule is ofthe form µ Λ for some automorphism µ . If that happened, the minimal projectiveresolution of Λ as a bimodule would produce a short exact sequence of bimodules µ Λ (cid:44) → P (cid:16) σ Λ with projective middle term. If we twisted by σ − from the left we would obtainanother short exact sequence of bimodules with projective middle term σ − µ Λ (cid:44) → σ − P (cid:16) Λ , hence Ω(Λ) = σ − µ Λ . We know that dim k σ − P = dim k P = m ( n + 1) , see [23, § k σ − µ Λ = dim k Λ = m ( n + 1), so this can only happen if n = 1.The final conclusion follows from (1) in our main theorem in the enhanced caseand from [22, Theorem 1.2] in the non-enhanced case. (cid:3) For n = 1, Λ = N m satisfies Ω(Λ) = σ Λ for some automorphism σ of Λ, see theproof of [17, 4.2]. Hence Ω (Λ) = σ Λ , so in this case the Nakayama algebra Λdoes satisfy the assumptions of Corollary 9.6. References [1] Claire Amiot,
On the structure of triangulated categories with finitely many indecomposables ,Bull. Soc. Math. France (2007), no. 3, 435–474. MR 2430189[2] Maurice Auslander,
Representation Theory of Artin Algebras II , Communications in Algebra (1974), no. 4, 269–310 (en).[3] Maurice Auslander and Idun Reiten, On a theorem of E. Green on the dual of the transpose ,Representations of finite-dimensional algebras (Tsukuba, 1990), CMS Conf. Proc., vol. 11,Amer. Math. Soc., Providence, RI, 1991, pp. 53–65. MR 1143845[4] David Benson, Henning Krause, and Stefan Schwede,
Realizability of modules over Tatecohomology , Trans. Amer. Math. Soc. (2004), no. 9, 3621–3668. MR 2055748[5] Petter Andreas Bergh and David A. Jorgensen,
Tate-Hochschild homology and cohomologyof Frobenius algebras , J. Noncommut. Geom. (2013), no. 4, 907–937. MR 3148613[6] Jerzy Bia(cid:32)lkowski, Deformed preprojective algebras of Dynkin type E , Comm. Algebra (2019), no. 4, 1568–1577.[7] Jerzy Bia(cid:32)lkowski, Karin Erdmann, and Andrzej Skowro´nski, Deformed preprojective alge-bras of generalized Dynkin type , Trans. Amer. Math. Soc. (2007), no. 6, 2625–2650.MR 2286048[8] ,
Deformed preprojective algebras of generalized Dynkin type L n : classification andsymmetricity , J. Algebra (2011), 150–170. MR 2842059[9] Jerzy Bia(cid:32)lkowski and Andrzej Skowro´nski, Nonstandard additively finite triangulated cate-gories of Calabi-Yau dimension one in characteristic 3 , Algebra Discrete Math. (2007), no. 3,27–37. MR 2421775[10] Marta B(cid:32)laszkiewicz and Andrzej Skowro´nski,
On self-injective algebras of finite representa-tion type , Colloq. Math. (2012), no. 1, 111–126. MR 2945780[11] Michael L. Bolla,
Isomorphisms between endomorphism rings of progenerators , J. Algebra (1984), no. 1, 261–281. MR 736779[12] A. I. Bondal and M. M. Kapranov, Enhanced triangulated categories , Mat. USSR Sb. (1991), no. 1, 93–107.[13] A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations , LectureNotes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573[14] Alberto Canonaco and Paolo Stellari,
Uniqueness of dg enhancements for the derived cate-gory of a Grothendieck category , J. Eur. Math. Soc. (JEMS) (2018), no. 11, 2607–2641.MR 3861804[15] Justin Chen, Surjections of unit groups and semi-inverses , J. Commut. Algebra (2019), https://projecteuclid.org/euclid.jca/1545015624 .[16] W. G. Dwyer and D. M. Kan,
A classification theorem for diagrams of simplicial sets , Topol-ogy (1984), no. 2, 139–155.[17] Karin Erdmann and Thorsten Holm, Twisted bimodules and Hochschild cohomology for self-injective algebras of class A n , Forum Math. (1999), no. 2, 177–201. MR 1680594[18] Karin Erdmann and Andrzej Skowro´nski, Periodic algebras , Trends in representation theoryof algebras and related topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Z¨urich, 2008, pp. 201–251. MR 2484727[19] Ching-Hwa Eu and Travis Schedler,
Calabi-Yau Frobenius algebras , J. Algebra (2009),no. 3, 774–815. MR 2488552[20] Jens Franke,
Uniqueness theorems for certain triangulated categories possessing an Adamsspectral sequence , K-theory Preprint Archives (1996).[21] Peter Freyd,
Stable homotopy , Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965),Springer, New York, 1966, pp. 121–172. MR 0211399[22] Norihiro Hanihara,
Auslander Correspondence for Triangulated Categories , arXiv:1805.07585[math] (2018).[23] Thorsten Holm,
Hochschild cohomology of Brauer tree algebras , Comm. Algebra (1998),no. 11, 3625–3646. MR 1647106 NHANCED FINITE TRIANGULATED CATEGORIES 39 [24] T. V. Kadeishvili,
On the theory of homology of fiber spaces , Uspekhi Mat. Nauk (1980),no. 3(213), 183–188, International Topology Conference (Moscow State Univ., Moscow, 1979).[25] T. V. Kadeishvili, The algebraic structure in the homology of an A ( ∞ ) -algebra , Soob-shcheniya Akademii Nauk Gruzinsko˘ı SSR (1982), no. 2, 249–252 (1983), MR 720689[26] Hiroshige Kajiura, On A ∞ -enhancements for triangulated categories , J. Pure Appl. Algebra (2013), no. 8, 1476–1503. MR 3030547[27] Bernhard Keller, Introduction to A ∞ -infinity algebras and modules , Homology, HomotopyAppl. (2001), no. 1, 1–35.[28] , A remark on a theorem by Claire Amiot , C. R. Math. Acad. Sci. Paris (2018),no. 10, 984–986 (en).[29] Henning Krause,
Report on locally finite triangulated categories , J. K-Theory (2012), no. 3,421–458. MR 2955969[30] T. Y. Lam, Lectures on modules and rings , Graduate Texts in Mathematics, vol. 189,Springer-Verlag, New York, 1999. MR 1653294[31] Kenji Lef`evre-Hasegawa,
Sur les A ∞ -cat´egories , Ph.D. thesis, Universit´e Paris 7, 2003.[32] Valery A. Lunts and Dmitri O. Orlov, Uniqueness of enhancement for triangulated categories ,J. Amer. Math. Soc. (2010), no. 3, 853–908. MR 2629991[33] Barry Mitchell, Rings with several objects , Advances in Math. (1972), 1–161. MR 0294454[34] Fernando Muro, On the functoriality of cohomology of categories , J. Pure Appl. Algebra (2006), no. 3, 455–472. MR 2185612[35] ,
Moduli spaces of algebras over nonsymmetric operads , Algebr. Geom. Topol. (2014), no. 3, 1489–1539. MR 3190602[36] , Cylinders for non-symmetric DG-operads via homological perturbation theory , J.Pure Appl. Algebra (2016), no. 9, 3248–3281. MR 3486300[37] ,
Homotopy units in A-infinity algebras , Trans. Amer. Math. Soc. (2016), no. 3,2145–2184. MR 3449236[38] ,
The first obstructions to enhancing a triangulated category , Math. Z. (2019), https://doi.org/10.1007/s00209-019-02438-y .[39] ,
Enhanced A ∞ -obstruction theory , J. Homotopy Relat. Struct. (2020), no. 1,61–112 (en).[40] Fernando Muro and Constanze Roitzheim, Homotopy theory of bicomplexes , J. Pure Appl.Algebra (2019), no. 5, 1913–1939. MR 3906533[41] Irakli Patchkoria,
On exotic equivalences and a theorem of Franke , Bulletin of the LondonMathematical Society (2017), no. 6, 1085–1099 (en).[42] Piotr Pstr ą gowski, Chromatic homotopy is algebraic when p > n + n + 1, arXiv:1810.12250[math] (2018).[43] Dieter Puppe, On the formal structure of stable homotopy theory , Colloquium on AlgebraicTopology, Matematisk Institut, Aarhus Universitet, Aarhus, 1962, pp. 65–71.[44] Christine Riedtmann,
Algebren, Darstellungsk¨ocher, ¨uberlagerungen und zur¨uck , Comment.Math. Helv. (1980), no. 2, 199–224. MR 576602[45] , Representation-finite self-injective algebras of class A n , Representation theory, II(Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math.,vol. 832, Springer, Berlin, 1980, pp. 449–520. MR 607169[46] , Representation-finite self-injective algebras of class D n , Compositio Math. (1983),no. 2, 231–282. MR 704393[47] Alice Rizzardo and Michel Van den Bergh, A note on non-unique enhancements , Proc. Amer.Math. Soc. (2019), no. 2, 451–453. MR 3894882[48] Marco Schlichting,
A note on K-theory and triangulated categories , Invent. Math. (2002),no. 1, 111–116. MR 1930883[49] Stefan Schwede,
The stable homotopy category has a unique model at the prime 2 , Adv.Math. (2001), no. 1, 24–40. MR 1870511[50] ,
The stable homotopy category is rigid , Ann. of Math. (2) (2007), no. 3, 837–863.MR 2373374[51] Stefan Schwede and Brooke Shipley,
Algebras and modules in monoidal model categories ,Proc. London Math. Soc. (3) (2000), no. 2, 491–511. MR 1734325[52] , A uniqueness theorem for stable homotopy theory , Math. Z. (2002), no. 4, 803–828. MR 1902062 [53] Andrzej Skowro´nski,
Selfinjective algebras: finite and tame type , Trends in representationtheory of algebras and related topics, Contemp. Math., vol. 406, Amer. Math. Soc., Provi-dence, RI, 2006, pp. 169–238. MR 2258046[54] Andrzej Skowro´nski and Kunio Yamagata,
Frobenius algebras. I , EMS Textbooks in Mathe-matics, European Mathematical Society (EMS), Z¨urich, 2011. MR 2894798[55] Ross Street,
Homotopy classification of filtered complexes , Ph.D. thesis, University of Sydney,1969.[56] Mariano Suarez-Alvarez,
The Hilton-Heckmann argument for the anti-commutativity of cupproducts , Proc. Amer. Math. Soc. (2004), no. 8, 2241–2246. MR 2052399[57] Gon¸calo Tabuada,
Invariants additifs de DG-cat´egories , Int. Math. Res. Not. (2005), no. 53,3309–3339. MR 2196100[58] ,
Une structure de cat´egorie de mod`eles de Quillen sur la cat´egorie des dg-cat´egories ,C. R. Math. Acad. Sci. Paris (2005), no. 1, 15–19. MR 2112034[59] ,
Addendum `a Invariants Additifs de dg-Cat´egories , Int. Math. Res. Not. (2006), Art.ID 75853, 3. MR 2276352[60] ,
Corrections `a Invariants Additifs de DG-cat´egories , Int. Math. Res. Not. IMRN(2007), no. 24, Art. ID rnm149, 17. MR 2377018[61] Bertrand To¨en and Gabriele Vezzosi,
Homotopical algebraic geometry. II. Geometric stacksand applications , Mem. Amer. Math. Soc. (2008), no. 902, x+224. MR 2394633[62] Jie Xiao and Bin Zhu,
Locally finite triangulated categories , J. Algebra (2005), no. 2,473–490. MR 2153264[63] Yuji Yoshino,
Cohen-Macaulay modules over Cohen-Macaulay rings , London Mathemati-cal Society Lecture Note Series, vol. 146, Cambridge University Press, Cambridge, 1990.MR 1079937
Universidad de Sevilla, Facultad de Matem´aticas, Departamento de ´Algebra, Avda.Reina Mercedes s/n, 41012 Sevilla, Spain
E-mail address : [email protected] URL ::