Enhanced finite triangulated categories

ENHANCED FINITE TRIANGULATED CATEGORIES FERNANDO MURO Abstract. We give a necessary and sufficient condition for the existence of an enhancement of a finite triangulated category. Moreover, we show that enhancements are unique when they exist, up to Morita equivalence. Contents Introduction 1 Standing assumptions and notation 4 Acknowledgements 5 1. Hochschild cohomology 5 2. The Gerstenhaber square in the kernel 8 3. The Euler derivation 8 4. Hochschild cohomology of weakly stable graded categories 11 5. Hochschild–Tate cohomology 13 6. Edge units 18 7. Lifting edge units 20 8. Enhancements 24 9. Enhanced triangulated structures and edge units 29 References 38 Introduction Let us fix a perfect ground field k. Recall that a field kis perfect if char(k) = 0 or if char(k) = pand any element in khas a pth root, e.g. finite and algebraically closed fields, but not function fields in positive characteristic. A (linear) additive category Tis 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. Such a category is essentially small and satisfies the Krull–Remak–Schmidt property, i.e. any object decomposes as a finite direct sum of indecomposables with local endomorphism algebra in an essentially unique way. If X∈Tis a basic additive generator, consisting of a direct sum of one indecomposable for each isomorphism class, then T(X, −): T→proj(Λ) is an equivalence onto the category of finitely generated projective right Λ-modules for Λ = T(X, X), which is a basic algebra. 2010 Mathematics Subject Classification. 18E30, 16E40, 18G40, 55S35. 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 grant MTM2016-76453-C2-1-P (AEI/FEDER, UE) and by the Andalusian Ministry of Economy and Knowledge and the Operational Program FEDER 2014–2020 under the grant US-1263032. 1 arXiv:1810.10068v4 [math.KT] 11 May 2020 2 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 = P1⊕ · · · ⊕ Pnwere each Pi is indecomposable and PiPjfor i6=j. Any finite-dimensional algebra is Morita equivalent to a basic one, which is unique up to isomorphism. Finite triangulated categories arise commonly in representation theory, and have been thoroughly studied from that viewpoint [62, 1, 29]. If Tadmits a triangulated structure, then Λ is a Frobenius algebra. Indeed, Freyd [21] showed that projective objects 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 it is Frobenius since it is basic [54, Proposition 3.9]. In this paper, we care about enhancements. An enhanced triangulated category is just a DG-category A. Its underlying triangulated category is Dc(A), the derived category of compact objects. An enhanced triangulated structure on Tconsists of a DG-category Aand an equivalence Dc(A)≃T. We may want to incorporate the suspension functor Σ: T→Tto the picture. The pair (T,Σ) has an enhanced triangulated structure if there is an equivalence Dc(A)≃Twhich commutes with the suspension functors up to natural isomorphism. Recall that two DG- categories Aand Bare Morita equivalent if there is a DG-functor A→Bsuch that the induced functor Dc(A)→Dc(B) is an equivalence, or a zig-zag of such DG-functors connecting both. Our main result is the following theorem, where Λedenotes the enveloping algebra of Λ. Theorem. Let T≃proj(Λ) be a finite category over a perfect field kwith Λa basic Frobenius algebra: (1) Thas an enhanced triangulated structure if and only if the third syzygy Ω3 Λe(Λ) is stably isomorphic to an invertible Λ-bimodule. (2) The possible suspension functors Σ: T→Tsuch that (T,Σ) admits an enhanced triangulated structure are Σ∼ =− ⊗ΛI, where Iis an invertible Λ-bimodule whose inverse is stably isomorphic to Ω3 Λe(Λ). (3) If Σis as above, any two enhancements of (T,Σ) are Morita equivalent. Most triangulated categories appearing in the literature are born with an enhancement. This theorem is still interesting for finite triangulated categories for which an enhancement is known, since it shows uniqueness up to Morita equivalence. The stable category mod(A) of finite-dimensional modules over a selfinjective finite-dimensional algebra Aof finite representation type fits in this framework, e.g. A=k[x]/(xn). Over an algebraically closed field, these algebras where classified by Riedtmann [44, 45, 46]. Over an arbitrary field, there are even more examples and the general picture is yet unknown, see e.g. [10]. Also the stable category MCM(A) of maximal Cohen–Macaulay modules over a commutative complete local algebra Aof finite Cohen–Macaulay representation type fits here [29, §2], e.g. A=k[[x, y]]/(y2+xn) when nis odd and kis algebraically closed [63, Proposition 5.11]. This theorem also applies to the non-standard finite 1-Calabi-Yau categories defined in [1] from deformed preprojective algebras of generalized Dynkin type over an algebraically closed field of characteristic 2 [7, Theorem 1.3], [6, Corollary]. So far, these categories were only known to be triangulated in the ordinary sense, no enhancements were known (except for those of type Ln[8, Theorems 2 and 3]) but (1) shows that an enhancement indeed exists. The simplest of these examples, of ENHANCED FINITE TRIANGULATED CATEGORIES 3 Dynkin type D4, is the algebra Λ obtained as the quotient of the path algebra of the quiver 0 2 3 1 a0 a2 ¯a0 ¯a1 ¯a2 a1 over an algebraically closed field of characteristic 2 by the two-sided ideal generated by the following five elements ¯a0a0,¯a1a1, a2¯a2, a0¯a0+a1¯a1+ ¯a2a2+a1¯a1a0¯a0, a0¯a0a1¯a1+a1¯a1a0¯a0. There are infinitely many known examples of this kind associated to the Dynkin quivers Dn,n≥4, E7, and E8. There may actually be many more, even in characteristic 6= 2, since deformed preprojective algebras associated to these quivers are not 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 Thas an enhanced triangulated structure, then there is an essentially unique suspension functor in (2) so, by (3), the enhancement of Tis unique up to Morita equivalence (Proposition 9.8). Deformed preprojective algebras of generalized Dynkin type fit in this family, e.g. the example above. If kis algebraically closed, Λ is connected, Tis standard, i.e. equivalent to the mesh category of its Auslander–Reiten quiver, and we know that an enhancement exists, then (3) was established in [28], and was already implicit in [1], where all possible such Λ are classified. The simplest of these examples is the algebra of dual numbers Λ = k[x]/(x2), and more generally, the algebra whose representations are periodic 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 Bn,n≥3, also in characteristic 2, see [18, Example 9.1]. We do not know any examples in characteristic 6= 2. The claimed examples in characteristic 3 in [9] are flawed, since the deformations do not satisfy the admissibility condition for the quivers En,n= 6,7,8, described in [18]. Nevertheless, there are non-standard Frobenius algebras in arbitrary characteristic [53, §4.12], so there may be many non-standard examples yet to be discovered. Disconnected examples are also interesting, because we can play with different suspension functors. For instance, Λ = knsatisfies (1) for all n≥1 because it is separable. In this case Tis the n-fold power of the category mod(k) of finitedimensional vector spaces. Moreover, the enveloping algebra of Λ is k2n, which is semi-simple, hence Σ can be the tensor product with an arbitrary invertible Λ-bimodule, by (2). In this case this is the same as saying that Σ can be any permutation of nelements, regarded as an automorphism of T= mod(k)nin the obvious way, see [11, Proposition 3.8]. By (3), the pair (T,Σ) has a unique k-linear enhancement 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. This is actually the triangulated category considered in [47, Example] when n > 0 is even and k=`(x1, . . . , xn+1) is a function field on n+ 1 variables (which is perfect in characteristic 0). In that paper the authors show that this particular (T,Σ) has 4 FERNANDO MURO non-Morita equivalent enhancements over `. We deduce that only one of them can be defined over kas per (3). Hanihara [22, Theorem 1.2] proved recently that, under the assumptions of the previous theorem, Tadmits an ordinary triangulated structure if and only if Ω3 Λe(Λ) is stably isomorphic to an invertible Λ-bimodule, see also [11, Proposition 3.8]. He actually did not use the octahedral axiom at all. Hence, we deduce that Tadmits an ordinary triangulated structure in the sense of Puppe [43] if and only if it admits an enhanced triangulated structure. Beware that the underlying ordinary triangulated structure of the latter need not coincide with the former, therefore there may be finite triangulated categories over a perfect field without enhancements (there may even be finite Puppe triangulated categories not satisfying the octahedral axiom!). It would be interesting to know whether this can really happen. There are many Frobenius algebras which do not satisfy (1) in the previous theorem. Nakayama algebras yield a whole family of examples (Proposition 9.9). Among these, the simplest are Λ = k[x]/(xn), n > 2. Theorems on uniqueness of enhancements exist in the literature for triangulated categories coming from algebraic geometry [32, 14] or algebraic topology [49, 52, 50]. Nevertheless, enhancements are in general not unique when they exist, see [48, 26, 47] for some algebraic examples and [20], [49, §2.1], [41], [42] for topologically flavored examples. Despite our main result looks very algebraic, some crucial proofs in this paper use homotopy theory. Sets of enhancements appear as connected components of homotopical moduli spaces of enhancements. Some of our papers on the homotopy theory of operads from the last years were developed with this paper’s main result in mind. Here we use them together with some Hochschild cohomology computations. The main theorem is proved at the end, as Corollary 9.6. The claim in the connected but not separable case is Proposition 9.8. Standing assumptions and notation. Throughout this paper, we will work over a fixed ground field k, which acts everywhere. In particular, all categories are assumed to be k-linear, except for few clear exceptions. The tensor product over kwill simply be denoted by ⊗, unless otherwise indicated. In our main results, we need kto 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 a finite category T, Λ will denote the endomorphism algebra of a basic additive generator, as above. We will often work with graded objects, such as graded vector spaces, graded algebras, graded categories, etc. This means Z-graded, and the Koszul 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. We will 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 denote the Grothendieck abelian category of right C-modules by Mod(C), and mod(C) will be the full subcategory of finitely presented objects. If Cis graded, then we will actually consider the graded categories of graded modules. For Cungraded, modules will also be ungraded, except when explicitly stated otherwise. For a Frobenius algebra Λ, we will also consider the stable module categories Mod(Λ) and mod(Λ), obtained by quotienting out maps factoring through a projective-injective object. Left modules are the same as right modules over the opposite category ENHANCED FINITE TRIANGULATED CATEGORIES 5 Cop, and bimodules are right modules over the enveloping category Ce=C⊗Cop. Further notation and conventions will be introduced the first time we use them. Acknowledgements. This paper has benefited from valuable conversations, email exchange, and MathOverflow interaction with many colleagues, including Jerzy Bia lkowski, Marco Farinati, Ram´on Flores, Norihiro Hanihara, Dolors Herbera, Bernhard Keller, Henning Krause, Georges Maltsiniotis, Victor Ostrik, Jeremy Rickard, Antonio Rojas, Manuel Saor´ın, Stefan Schwede, and Mariano Su´arez- ´ Alvarez (in surname alphabetical order). The author wishes to express his gratitude to all of them and to an anonymous referee. 1. Hochschild cohomology The Hochschild cohomology of a graded category Cwith coefficients in a C- bimodule Mis HH?,∗(C, M) = Ext?,∗ Ce(C, M). Here ?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|the degree of an element xin a singly graded object, or the total degree of an element in a bigraded object; horizontal and vertical degrees will be denoted by |x|hand |x|v, respectively. Hochschild cohomology can be computed as the cohomology of the Hochschild cochain complex C?,∗(C, M), given by Cn,∗(C, M) = Y X0,...,Xn∈C Hom∗ k(C(X1, X0)⊗ · · · ⊗ C(Xn, Xn−1), M(Xn, X0)), with bidegree (1,0) differential d(ϕ)(f1, . . . , fn+1) = (−1)|ϕ|v|f1|f1·ϕ(f2, . . . , fn+1) + n X i=1 (−1)iϕ(...,fifi+1, . . . )+(−1)n+1ϕ(f1, . . . , fn)·fn+1. This complex arises from the bar projective resolution B?(C) of Cas a C-bimodule, called standard complex in [33, §17]. It is given by Bn(C) = M X0,...,Xn∈C C(X0,−)⊗C(X1, X0)⊗ · · · ⊗ C(Xn, Xn−1)⊗C(−, Xn), with differential d(f0⊗ · · · ⊗ fn+1) = n X i=0 (−1)i· · · ⊗ fifi+1 ⊗ · · · and augmentation ε:B?(C)−→ C, ε(f0⊗f1) = f0f1. We will now describe the algebraic structure on Hochschild cohomology, following the sign conventions in [39, §1]. Therein, we define the Hochschild complex of 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 we use the language of operads, although it is implicit). Therefore, some formulas here 6 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 Mand N, we have the cup-product operation, ^: Cp,q(C, M)⊗Cs,t(C, N)−→ Cp+s,q+t(C, M ⊗CN), which is a map of bigraded complexes defined by the following formula, (ϕ ^ ψ)(f1, . . . , fp+s)=(−1)tPp i=1 |fi|ϕ(f1, . . . , fp)⊗ψ(fp+1, . . . , fp+s). This operation comes from the differential graded comonoid structure of B?(C) in the category of C-bimodules defined by the comultiplication ∆: B?(C)−→ B?(C)⊗CB?(C), ∆(f0⊗ · · · ⊗ fn+1) = n X i=0 (f0⊗ · · · ⊗ fi⊗1Xi)⊗(1Xi⊗fi+1 ⊗ · · · ⊗ fn+1). The counit is the augmentation ε. The induced cup-product on Hochschild cohomology will be denoted in the same way. If Mis a monoid in the category of C-bimodules then the cup-product induces a differential bigraded algebra structure on C?,∗(C, M) and a bigraded algebra structure on HH?,∗(C, M). The main example is M=Citself, the tensor unit in C-bimodules. The cup-product is bigraded commutative in HH?,∗(C,C), satisfying the Koszul sign rule with respect to both degrees separately. If we tweak the differential and the cup product in the following way d0(ϕ)=(−1)|ϕ|vd(ϕ), ϕ·ψ= (−1)tpϕ ^ ψ, then the product becomes a map of complexes at the cochain level, and graded commutative in HH?,∗(C,C), all with respect to the total degree. Actually, this dot product and d0correspond to the cup product and the differential used in [39]. Therefore, the bigraded commutativity of the cup product in HH?,∗(C,C), as defined here, follows formally from the graded commutativity of the dot product with respect to the total degree, checked in [39], by purely formal reasons, compare [40, Remark 2.8]. The Gerstenhaber bracket endows C?,∗(C,C) with a DG-Lie algebra structure of degree −1 for the differential d0, [−,−]: Cp,q(C,C)⊗Cs,t(C,C)−→ Cp+s−1,q+t(C,C). It is defined as the commutator [ϕ, ψ] = ϕ•ψ−(−1)(p+q−1)(s+t−1)ψ•ϕ of the pre-Lie product (1.1) •: Cp,q(C,C)⊗Cs,t(C,C)−→ Cp+s−1,q+t(C,C) given by (ϕ•ψ)(f1, . . . , fp+s−1) = p X i=1 (−1)(s−1)(p−i)+t(p−1+Pi−1 j=1 |fj|)ϕ(f1, . . . , fi−1, ψ(fi, . . . , fi+s−1), fi+s, . . . , fp+s−1). ENHANCED FINITE TRIANGULATED CATEGORIES 7 If m2∈C2,0(C,C) denotes the composition in C, then d0= [m2,−]. This pre-Lie product corresponds to the brace operation ϕ{ψ}in [39]. The square of the pre-Lie product Sq(ϕ) = ϕ•ϕ induces in cohomology an operation, called Gerstenhaber square, Sq: HHp,q(C,C)−→ HH2p−1,2q(C,C) whenever p+qis even or char k= 2. The relations satisfied by the product, the Lie bracket, and the Gerstenhaber square in HH?,∗(C,C) are: (x·y)·z=x·(y·z), x·y= (−1)|x||y|y·x, [x, y] = −(−1)(|x|−1)(|y|−1)[y, x], [x, x]=0,|x|odd, [x, [y, z]] = [[x, y], z]+(−1)(|x|−1)(|y|−1)[y, [x, z]], [x, [x, x]] = 0,|x|even, [x, y ·z]=[x, y]·z+ (−1)(|x|−1)|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)·y2+x·[x, y]·y+x2·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 that 2·Sq(x)=[x, x] whenever the Gerstenhaber square is defined. The Gerstenhaber square and the relations where it appears are redundant if char k6= 2. The bivariant functoriality of Hochschild cohomology can be described as in [34]. A graded functor F:D→Cand a C-bimodule morphism τ:N→Minduce morphisms on Hochschild cochain complexes that we denote by F∗: C?,∗(C, M)−→ C?,∗(D, M(F, F )), F∗(ϕ)(g1, . . . , gn) = ϕ(F(g1), . . . , F(gn)); τ∗: C?,∗(C, N)−→ C?,∗+|τ|(C, M), τ∗(ϕ)(f1, . . . , fn) = τ(ϕ(f1, . . . , fn)). The induced morphisms on Hochschild cohomology, denoted in the same way, F∗: HH?,∗(C, M)−→ HH?,∗(D, M(F, F )), τ∗: HH?,∗(C, N)−→ HH?,∗+|τ|(C, M), satisfy F∗τ∗=τ(F, F)∗F∗. These induced morphisms are compatible with the cupproduct in the obvious way, actually at the cochain level. Hochschild cohomology is Morita invariant, i.e. F∗is an isomorphism for any coefficient bimodule provided Finduces an equivalence of categories Mod(C)≃Mod(D). Ungraded objects are graded objects concentrated in degree 0. An ungraded category Chas graded and ungraded modules and bimodules. If Cis an ungraded 8 FERNANDO MURO category and Mis a graded C-bimodule, then graded and ungraded Hochschild cohomology are related by the following obvious formula, HHp,q(C, M) = HHp(C, Mq). 2. The Gerstenhaber square in the kernel In this short section we prove that the Gerstenhaber square operation restricts to the kernel of any morphism induced by a graded functor. Proposition 2.1. Let F:D→Cbe a graded functor. If x∈HH?,∗(C,C)is in the kernel of F∗: HH?,∗(C,C)→HH?,∗(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 •: Cp,q(C,C)⊗Cs,t(D,C(F, F)) −→ Cp+s−1,q+t(D,C(F, F)) by essentially the same formula as the pre-Lie product in (1.1), (ϕ•ψ)(g1, . . . , gp+s−1) = p X i=1 (−1)(s−1)(p−i)+t(p−1+Pi−1 j=1 |gj|)ϕ(F(g1), . . . , F(gi−1), ψ(gi, . . . , gi+s−1), F(gi+s), . . . , F(gp+s−1)). We recover (1.1) for Fthe identity. This new operation and the pre-Lie product are clearly related by the formula F∗(ϕ•ϕ0) = ϕ•F∗(ϕ0). The proof of the previous proposition is based on the following formula, which is a slight generalization of [39, (1.4)]. Lemma 2.2. Given ϕ∈C?,∗(C,C)and ψ∈C?,∗(D,C(F, F)), the following formula holds, F∗(ϕ)·ψ−(−1)|ϕ||ψ|ψ·F∗(ϕ)=(−1)|ϕ|(d0(ϕ•ψ)−d0(ϕ)•ψ+ (−1)|ϕ|ϕ•d0(ψ)). The proof of this lemma is straightforward but rather tedious, hence we omit it. Proof of Proposition 2.1. Let ϕ∈C?,∗(C,C) be a representative of x, so d0(ϕ) = 0. Since F∗(x) = 0, there exists ψ∈C?,∗(D,C(F, F)) with F∗(ϕ) = d0(ψ). If xhas even total degree, so does ϕ. Therefore |ψ|is odd and the Leibniz rule for d0 together with the formula in Lemma 2.2 yield, d0(ψ·ψ) = F∗(ϕ)·ψ−ψ·F∗(ϕ) =d0(ϕ•ψ) + ϕ•F∗(ϕ). Hence, the cohomology class of ϕ•F∗(ϕ) = F∗(ϕ•ϕ) is trivial. The same formula proves the result in char k= 2, where signs do not matter.  3. The Euler derivation In this section we recall the Euler class in Hochschild cohomology and we study its interaction with the Gerstenhaber algebra structure. Definition 3.1. Given a graded category C, the Euler derivation δ∈C1,0(C,C) is the cochain defined as δ(f) = |f| · f, f ∈C(X1, X0). ENHANCED FINITE TRIANGULATED CATEGORIES 9 If Cis ungraded then δ= 0, so everything in this section pertains to the graded setting. Proposition 3.2. Given a Hochschild cochain ϕ∈Cp,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)(f1, . . . , fp) = δ(ϕ(f1, . . . , fp)) = q+ p X i=1 |fi|!·ϕ(f1, . . . , fp) ϕ•δ= p X i=1 ϕ(f1, . . . , fi−1, δ(fi), fi+1, . . . , fp) = p X i=1 |fi|ϕ(f1, . . . , fp).  Corollary 3.3. The Euler derivation δis a Hochschild cocycle. Proof. Simply note that [m2, δ] = −[δ, m2] = 0·m2= 0 since m2∈C2,0(C,C).  The Euler class is the Hochschild cohomology class of the Euler derivation {δ} ∈ HH1,0(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∈HHn,−1(C,C)with nodd, and x∈HHp,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].  Remark 3.7.The formula in the previous proposition shows that {δ} · − is a chain null-homotopy for y· −, if we think of [y, −] as a differential. If Sq(y) = 0, then [y, −] is a differential and multiplication by yis 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}2= 0 ∈HH2,0(C,C). 16 FERNANDO MURO where fis defined as f(a⊗b⊗(Pici⊗di)) = Pi(abci⊗1−a⊗bci)⊗1⊗di. This proves the claim.  We can similarly choose an adjoint inverse cosyzygy functor by taking a short exact sequence in Modp(Λe) Λj0 ,→P0p0 Ω−1(Λ) with P0projective-injective and setting Ω−1(M) = Ω−1(Λ) ⊗M. The counit is determined by an isomorphism in Modp(Λe) ξ: Ω(Λ) ⊗Ω−1(Λ) ∼ =Λ given by any choice of map of extensions in Modp(Λe) as follows Ω(Λ) ⊗Ω−1(Λ) P⊗Ω−1(Λ) Λ ⊗Ω−1(Λ) ∼ =Ω−1(Λ) ΛP0Ω−1(Λ) j⊗1p⊗1 j0p0 Once ξis chosen, there is only one possible isomorphism in Modp(Λe) ξ0: Λ ∼ =Ω−1(Λ) ⊗Ω(Λ) defining the unit. It is easy to check, using the coherence of ζand Lemma 5.4, that ξ0=−ζΩ−1(Λ)ξ−1. There are also coherent isomorphisms in Modp(Λe) (5.5) ζ0 M:M⊗Ω−1(Λ) ∼ =Ω−1(Λ) ⊗M, like those in (5.3). The isomorphisms ζMand ζ0 Mare compatible in the sense that the two isomorphisms Ω(Λ) ⊗M⊗Ω−1(Λ) ∼ =Mwhich can be constructed by using these isomorphisms and ξcoincide. Now, if we define Ωn(Λ), n∈Z, as a tensor power of Ω(Λ) or Ω−1(Λ), according to the sign of n, the isomorphisms ξand ξ0define associative isomorphisms Ωp(Λ)⊗ Ωq(Λ) ∼ =Ωp+q(Λ), p, q ∈Z. Given two objects Mand Nin Modp(Λe), there is a natural associative cupproduct operation HHp(Λ, M)⊗kHHq(Λ, N)−→ HHp+q(Λ, M ⊗N) defined by (Ωp(Λ) f →M)^(Ωq(Λ) g →N) = (Ωp+q(Λ) ∼ =Ωp(Λ) ⊗Ωq(Λ) f⊗g −→ M⊗N). In particular, HH?(Λ, M) is a graded algebra if Mis a monoid in Modp(Λe), even a bigraded algebra if Mis graded. The comparison maps from Hochschild to stable Hochschild cohomology preserve the cup-product. Let σ: Λ →Λ be an algebra automorphism. We will need some knowledge on units in the bigraded algebra HH?,∗(Λ,Λ(σ)). Proposition 5.6. The comparison map HH0(Λ,Λ) HH0(Λ,Λ) (co)restricts to a surjection between groups of units HH0(Λ,Λ)×HH0(Λ,Λ)×. This follows from [15, Corollary 2.3] since HH0(Λ,Λ) is the center of Λ, and hence a finite-dimensional commutative algebra. ENHANCED FINITE TRIANGULATED CATEGORIES 17 Proposition 5.7. An element f∈HHp,q(Λ,Λ(σ)) = HHp(Λ,σqΛ1)is a unit in HH?,∗(Λ,Λ(σ)) if and only if f: Ωp(Λ) →σqΛ1is an isomorphism in Modp(Λe). Proof. Given g: Ωp0(Λ) →σq0Λ1, if we use the isomorphisms Ωp(Λ) ⊗Ωp0(Λ) ∼ = Ωp+p0(Λ) as identifications, the cup-product f ^ g above coincides with (f⊗ σq0Λ1)(Ωp(Λ) ⊗g)=(σqΛ1⊗g)(f⊗Ωp0(Λ)). Assume fis invertible. Then 1Λ=f ^ f−1= (f⊗σ−qΛ1)(Ωp(Λ) ⊗f−1) = (σqΛ1⊗f−1)(f⊗Ω−p(Λ)). The functor Ωp= Ωp(Λ)⊗− is an equivalence, and so is −⊗Ω−p(Λ), which is naturally isomorphic to Ω−pvia ζor ζ0. Moreover, σqΛ1⊗− is also an equivalence of categories since σqΛ1is an invertible Λ-bimodule with inverse σ−qΛ1, and similarly − ⊗ σ−qΛ1. Therefore, fis both a split epimorphism and a split monomorphism in Modp(Λe), and hence an isomorphism. Conversely, if fis an isomorphism in Modp(Λe), then so is f⊗σ−qΛ1, and the inverse isomorphism Ω−p(f⊗σ−qΛ1)−1is a cup-product inverse for f. Remark 5.8.For any p > 0 and f∈HHp,q(Λ,Λ(σ)) = Extp Λ(Λ,σqΛ1), we can take a representing extension σqΛ1,→Pp−1→ · · · → P0Λ with Pp−2, . . . , P0projective-injective. Using Proposition 5.7, we see that fis a unit in HH?,∗(Λ,Λ(σ)) if and only if Pp−1is also projective-injective. We can also use the previous choice of Ω(Λ) to fix a syzygy functor in the stable category of right modules Mod(Λ), namely Ω(M) = M⊗Ω(Λ), and similarly for the cosyzygy functor Ω−1. The extended Yoneda product in Mod(Λ), Extp Λ(M, N)⊗kExtq Λ(L, M)−→ Extp+q Λ(L, N), can be computed as follows, HomΛ(Ωp(M), N)⊗kHomΛ(Ωq(L), M)−→ HomΛ(Ωp+q(L), N), (M⊗Ωp(Λ) f →N)⊗(L⊗Ωq(Λ) g →M)7→ (L⊗Ωp+q(Λ) ∼ =L⊗Ωq(Λ) ⊗Ωp(Λ) g⊗1 −→ M⊗Ωp(Λ) f →N). Remark 5.9.As above, f∈Extp Λ(M, N) is a Yoneda unit if and only if f: Ωp(M)→ Nis an isomorphism in Mod(Λ). For p > 0, Extp Λ(M, N) = Extp Λ(M, N) so we can take an extension N→Pp−1→ · · · → P0M representing fwith Pp−2, . . . , P0projective-injective, and fis a Yoneda unit if and only if Pp−1is also projective-injective. For any Λ-bimodule M, there is a natural map HHn(Λ, M)−→ HH0(mod(Λ),Extn Λ(−,− ⊗ M)) defined as (Ωn(Λ) f →M)7→ L7→ Ωn(L) = L⊗Ωn(Λ) L⊗f −→ L⊗M. Let σ: Λ →Λ be an automorphism. These maps assemble to a map (5.10) ε: HH?,∗(Λ,Λ(σ)) −→ HH0,∗(mod(Λ),Ext?,∗ Λ(−,− ⊗ Λ(σ))). The coefficient mod(Λ)-bimodule on the right is a monoid for the Yoneda composition. Indeed, since each σqΛ1is left and right projective, the functor − ⊗ σqΛ1 18 FERNANDO MURO is exact (actually, an exact equivalence since σqΛ1is invertible) and the product of g∈Extp,q Λ(L, M ⊗Λ(σ)) = Extp Λ(L, M ⊗σqΛ1) and f∈Exts,t Λ(M, N ⊗Λ(σ)) = Exts Λ(M, N ⊗σtΛ1) is the Yoneda product of gand f⊗σqΛ1. In particular, an element in the target is a unit if and only if it is pointwise a Yoneda unit. Remark 5.11.We can regard mod(Λ) as the degree 0 part of mod(Λ(σ)), compare [38, §3]. Therefore, mod(Λ(σ)) is a graded Frobenius abelian category and graded Tate Extn,∗ 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 along Λ⊂Λ(σ). Therefore, Extn,∗ Λ(−,− ⊗ Λ(σ)) coincides with Extn,∗ Λ(σ)as a monoid in mod(Λ)-bimodules (also in the non-Tate case), and we can exchange them in the target of (5.10). Proposition 5.12. If kis perfect, then the morphism εin (5.10) preserves and reflects units for ? > 0. Proof. Recall from Remark 5.8 that any x∈HHn,s(Λ,Λ(σ)) with n > 0 can be represented by an extension of finite-dimensional Λ-bimodules σsΛ1,→Pn−1→ · · · → P0Λ with P0, . . . , Pn−2projective. Moreover, xis a unit if and only if Pn−1is also projective. All bimodules in this sequence are projective as left or right Λ-modules. For each finitely presented right Λ-module M,ε(x)(M)∈Extn Λ(M, M ⊗σsΛ1) is represented by the extension M⊗σsΛ1,→M⊗Pn−1→ · · · → M⊗P0M⊗Λ∼ =M obtained by applying M⊗− to the bimodule extension. All right Λ-modules M⊗Pi 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⊗Pn−1is also a projective right Λ-module for all M, see Remark 5.9. This clearly holds if Pn−1is projective. The converse is also true when kis perfect by [3, Theorem 3.1].  6. Edge units Throughout this section, we will assume that the ground field kis a perfect field, since we will derive consequences of Proposition 5.12. Moreover, Λ is a Frobenius algebra 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) Ep,q 2= HHp,∗(mod(Λ(σ)),Extq,∗ Λ(σ)) =⇒HHp+q,∗(Λ(σ),Λ(σ)). It has edge morphisms HH?,∗(Λ(σ),Λ(σ)) −→ HH0,∗(mod(Λ(σ)),Ext?,∗ Λ(σ)). Definition 6.2. An element in HH?,∗(Λ(σ),Λ(σ)) is an edge unit if its image along HH?,∗(Λ(σ),Λ(σ)) edge −→ HH0,∗(mod(Λ(σ)),Ext?,∗ Λ(σ))comp. −→ HH0,∗(mod(Λ(σ)),Ext?,∗ Λ(σ)) is a unit. ENHANCED FINITE TRIANGULATED CATEGORIES 19 Note that it is really crucial to get to Tate Ext coefficients. Otherwise, there would only be edge units in horizontal degree ?= 0. There is another spectral sequence (6.3) Ep,q 2= HHp,∗(mod(Λ),Extq,∗ Λ(σ)) =⇒HHp+q,∗(Λ,Λ(σ)) with ungraded first variables, see [38, Remark 5.4]. It has edge morphisms HH?,∗(Λ,Λ(σ)) −→ HH0,∗(mod(Λ),Ext?,∗ Λ(σ)). Definition 6.4. An element in HH?,∗(Λ,Λ(σ)) is an edge unit if its image along HH?,∗(Λ,Λ(σ)) edge −→ HH0,∗(mod(Λ),Ext?,∗ Λ(σ))comp. −→ HH0,∗(mod(Λ),Ext?,∗ Λ(σ)) is a unit. These edge units have the following smarter characterization in positive Hochschild degree. Proposition 6.5. For n > 0, an element in HHn,∗(Λ,Λ(σ)) is an edge unit if and only if its image along the comparison morphism HHn,∗(Λ,Λ(σ)) −→ HHn,∗(Λ,Λ(σ)) is a unit in HH?,∗(Λ,Λ(σ)). Proof. By [38, Remark 5.7], the following square commutes, HHn,∗(Λ,Λ(σ)) HH0,∗(mod(Λ),Extn,∗ Λ(σ)) HHn,∗(Λ,Λ(σ)) HH0,∗(mod(Λ),Extn,∗ Λ(σ)) edge comparison comparison ε Here, the bottom horizontal arrow is the horizontal degree npart of (5.10) (see Remark 5.11), which preserves and reflects units for n > 0 by Proposition 5.12. Hence, we are done.  Proposition 6.6. The morphism induced by the inclusion i: Λ ⊂Λ(σ), i∗: HHn,∗(Λ(σ),Λ(σ)) −→ HHn,∗(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 a commutative diagram HH?,∗(Λ(σ),Λ(σ)) HH0,∗(mod(Λ(σ)),Ext?,∗ Λ(σ)) HH0,∗(mod(Λ(σ)),Ext?,∗ Λ(σ)) HH?,∗(Λ,Λ(σ)) HH0,∗(mod(Λ),Ext?,∗ Λ(σ)) HH0,∗(mod(Λ),Ext?,∗ Λ(σ)) edge comp. edge comp. whose vertical arrows are induced by Λ ⊂Λ(σ). The two last vertical arrows are injective 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.  20 FERNANDO MURO Corollary 6.7. For n > 0, an element in HHn,∗(Λ(σ),Λ(σ)) is an edge unit if and only if its image along HHn,∗(Λ(σ),Λ(σ)) i∗ −→ HHn,∗(Λ,Λ(σ)) comp. −→ HHn,∗(Λ,Λ(σ)) is a unit in HH?,∗(Λ,Λ(σ)). In order to conclude this section, we record a property of edge units which is crucial for later computations. Lemma 6.8. If x∈HH3,−1(Λ(σ),Λ(σ)) is an edge unit, then the map x· −: HHp,q(Λ(σ),Λ(σ)) −→ HHp+3,q−1(Λ(σ),Λ(σ)) given by left multiplication by xis an isomorphism for all p≥2and q∈Zand an epimorphism for p= 1 and all q∈Z. Proof. By the previous corollary i∗(x)∈HH3,−1(Λ,Λ(σ)) = HH3,−1(Λ,Λ(σ)) is a unit in HH?,∗(Λ,Λ(σ)). Since the Hochschild–Tate cohomology coincides with Hochschild’s in positive Hochschild degrees, and the latter surjects onto the former in Hochschild degree 0, the map i∗(x)· −: HHp,q(Λ,Λ(σ)) −→ HHp+3,q−1(Λ,Λ(σ)) is an isomorphism for all p≥1 and q∈Zand an epimorphism for p= 0 and all q∈Z. By Proposition 4.3, we have a map of long exact sequences . . .. . . HHp,q(Λ(σ),Λ(σ)) HHp+3,q−1(Λ(σ),Λ(σ)) HHp,q(Λ,Λ(σ)) HHp+3,q−1(Λ,Λ(σ)) HHp,q(Λ,Λ(σ)) HHp+3,q−1(Λ,Λ(σ)) HHp+1,q(Λ(σ),Λ(σ)) HHp+4,q−1(Λ(σ),Λ(σ)) . . .. . . i∗ x·− i∗ id −σ−1 ∗σ∗ i∗(x)·− id −σ−1 ∗σ∗ ∂ i∗(x)·− ∂ x·− Hence, the proposition follows from the previous paragraph and the five lemma.  7. Lifting edge units The goal of this section is to lift edge units along the morphism in Proposition 6.6, mainly in bidegree (3,−1). We will use the characterization of edge units in Proposition 6.5, as well as other consequences of Proposition 5.12. Therefore, in this section the ground field kwill be a perfect field. We start with some consequences of 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, ENHANCED FINITE TRIANGULATED CATEGORIES 21 we will work in the monoidal triangulated category Modp(Λe) introduced in Section 5, and ⊗will exceptionally stand for ⊗Λ. Recall the natural isomorphism ζMfrom (5.3). The following result is an immediate consequence of Lemma 5.4. Corollary 7.1. For n≥1, the automorphism ζΩn(Λ) of Ω(Λ)⊗n+1 is (−1)n. Corollary 7.2. If f: Ωn(Λ) →Mis a map, n≥1, then the following diagram commutes, Ω(Λ) ⊗Ω(Λ)⊗nΩ(Λ)⊗n⊗Ω(Λ) Ω(Λ) ⊗M M ⊗Ω(Λ) (−1)n 1⊗f f⊗1 ζM This follows by naturality. Corollary 7.3. If f: Ωn(Λ) →Mis a map, n≥1, then the following diagram commutes in Modp(Λe), Ω(Λ)⊗n⊗Ω(Λ)⊗nΩ(Λ)⊗n⊗Ω(Λ)⊗n Ω(Λ)⊗n⊗M M ⊗Ω(Λ)⊗n (−1)n2 1⊗f f⊗1 Qn i=1 1⊗i−1⊗ζM⊗1⊗n−i We use again naturality here. Corollary 7.4. If f: Ωn(Λ) →Mis an isomorphism in Modp(Λe),n≥1, then the following diagram commutes in this category, Ω(Λ)⊗n⊗M M ⊗Ω(Λ)⊗n M⊗M M ⊗M Qn i=1 1⊗i−1⊗ζM⊗1⊗n−i f⊗1 1⊗f (−1)n2 Proof. It suffices to consider the following commutative diagram of isomorphisms, Ω(Λ)⊗n⊗Ω(Λ)⊗nΩ(Λ)⊗n⊗Ω(Λ)⊗n Ω(Λ)⊗n⊗M M ⊗Ω(Λ)⊗n M⊗M M ⊗M (−1)n2 1⊗f f⊗f f⊗1 f⊗f Qn i=1 1⊗i−1⊗ζM⊗1⊗n−i f⊗1 1⊗f  The functor σΛ1⊗−⊗1Λσ: Modp(Λe)−→ Modp(Λe) is an exact equivalence, so it gives rise to a triangulated equivalence σΛ1⊗−⊗1Λσ: Modp(Λe)−→ Modp(Λe). 22 FERNANDO MURO Part of this triangulated equivalence is a natural isomorphism in Modp(Λe), which shows that this functor commutes with the syzygy functor, ωM: Ω(σΛ1⊗M⊗1Λσ)∼ =σΛ1⊗Ω(M)⊗1Λσ. It is simply given by ωM=ζσΛ1⊗1M⊗1Λσ: Ω(Λ) ⊗σΛ1⊗M⊗1Λσ∼ =σΛ1⊗Ω(Λ) ⊗M⊗1Λσ. We also have an exact equivalence Modp(Λe)−→ Modp(Λe): M7→ σMσ, which is exactly the same thing as the restriction of scalars along σ: Λ →Λ. It is naturally isomorphic to the previous one via the map ρM:σΛ1⊗M⊗1Λσ∼ =σMσ, ρM(a⊗x⊗b) = axb. In addition, we obtain an induced triangulated equivalence Modp(Λe)−→ Modp(Λe): M7→ σMσ, also naturally isomorphic to the previous one in the same way, as triangulated functors. Again, part of this triangulated equivalence is a natural isomorphism in Modp(Λe), which shows that this functor commutes with the syzygy functor, ¯ωM: Ω(σMσ)∼ =σΩ(M)σ. The natural isomorphism of triangulated equivalences is compatible with the natural isomorphisms ωand ¯ω, in the sense that the following square of natural isomorphisms commutes, Ω(σΛ1⊗M⊗1Λσ)σΛ1⊗Ω(M)⊗1Λσ Ω(σMσ)σΩ(M)σ ωM Ω(ρM)ρΩ(M) ¯ωM Let us denote ω(n) M= n Y i=1 Ωi−1(ωΩn−i(M)),¯ω(n) M= n Y i=1 Ωi−1(¯ωΩn−i(M)). These are isomorphisms fitting in a similar diagram, (7.5) Ωn(σΛ1⊗M⊗1Λσ)σΛ1⊗Ωn(M)⊗1Λσ Ωn(σMσ)σΩn(M)σ ω(n) M Ωn(ρM)ρΩn(M) ¯ω(n) M Proposition 7.6. If f: Ω(Λ)⊗n= Ωn(Λ) →σΛ1is an isomorphism in Modp(Λe), n≥1, then the following diagram commutes in this category, Ωn(Λ) Ωn(σΛσ)σΩn(Λ)σ σΛ1σ2Λσ Ωn(σ)¯ωn Λ σfσ f (−1)n2σ Proof. By the previous corollaries and various naturality properties, we have the following commutative diagram of isomorphisms, ENHANCED FINITE TRIANGULATED CATEGORIES 23 It therefore suffices to check that the bottom composite is (−1)n2σ. This is very easy to check. Actually, if we recall that ρis defined by multiplication, we readily see that, starting with a∈σΛ1in the bottom left corner, 1⊗σ(a)⊗1 (−1)n21⊗σ(a)⊗1 a1⊗a1⊗σ(a) (−1)n2σ(a)  Corollary 7.7. For n > 0, if nis odd or char k= 2, then i∗: HHn,1(Λ(σ),Λ(σ)) −→ HHn,1(Λ,Λ(σ)) induces a surjection on edge units. Proof. This morphism induces a map on edge units by Proposition 6.6. Consider the long exact sequence in Proposition 4.3 for TΣ= Λ(σ) and ∗= 1. Proposition 7.6 shows that any edge unit in the target is in the kernel of id −σ−1 ∗σ∗. Here we use that n2is odd and that the graded σis defined on the degree 1 part of Λ(σ), which is σΛ1, as the ungraded −σ, see Remark 4.2. Therefore, any edge unit in the target has a preimage, which must also be an edge unit by Proposition 6.6.  Since Λ(σ) is the same as Λ(σ−1) reversing the degrees, we also deduce the following result. Corollary 7.8. For n > 0, if nis odd or char k= 2, then i∗: HHn,−1(Λ(σ),Λ(σ)) −→ HHn,−1(Λ,Λ(σ)) induces a surjection on edge units. Proposition 7.9. Given an edge unit u∈HH3,−1(Λ(σ),Λ(σ)), there exists y∈ HH2,−1(Λ(σ),Λ(σ)) such that Sq(u+{δ} · y) = 0. Proof. For any ywe have Sq(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· −: HH2,−1(Λ(σ),Λ(σ)) →HH5,−2(Λ(σ),Λ(σ)) is an isomorphism by Lemma 24 FERNANDO MURO 6.8, there exists a unique ysuch that Sq(u) + u·y= 0. Let us fix this y. If we prove that [u, y] = 0, then we will be done. Again, u· −: HH4,−2(Λ(σ),Λ(σ)) → HH7,−3(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· −: HH7,−3(Λ(σ),Λ(σ)) → HH10,−4(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)) =−2·Sq(u)2 = 0. Here we use that |Sq(u)|= 5 −2 = 3 is odd, so its square is 2-torsion by graded commutativity.  Proposition 7.10. The map i∗: HH3,−1(Λ(σ),Λ(σ)) −→ HH3,−1(Λ,Λ(σ)) induces a bijection between the set of edge units uin 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 check surjectivity. Any edge unit in HH3,−1(Λ,Λ(σ)) has a preimage uby Corollary 7.8. The element u+{δ} · yin Proposition 7.9 is another preimage since Λ is ungraded so i∗({δ}) = 0 and therefore i∗(u+{δ} · y) = i∗(u). Moreover, u+{δ} · yis also an edge unit by Proposition 6.6. We now check injectivity. Let u, u0∈HH3,−1(Λ(σ),Λ(σ)) be edge units with Sq(u) = Sq(u0) = 0 and i∗(u) = i∗(u0). The element z=u0−uis in the kernel of i∗, hence Sq(z) too by Proposition 2.1. Using the Gerstenhaber algebra laws and Proposition 3.6 we obtain 0 = Sq(u0)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 Proposition 4.3, and i∗({δ}) = 0. By Lemma 6.8, uz = 0 implies that z= 0, so u=u0. 8. Enhancements Recall that a DG-functor between DG-categories A→Bis a Morita equivalence if it induces an equivalence between their derived categories D(A)→D(B), or equivalently between their full subcategories of compact objects. Definition 8.1. Let Tbe a finite category and Σ: T→Tan automorphism. The set of enhanced triangulated categories with underlying suspended category (T,Σ), denoted by ETC(T,Σ), ENHANCED FINITE TRIANGULATED CATEGORIES 25 is the set of Morita equivalence classes of DG-categories Asuch that the derived category of compact objects Dc(A) is equivalent to (T,Σ) as a suspended category. A suspended category is a category equipped with a self-equivalence, and two suspended categories (T,Σ), (T0,Σ0) are equivalent if there is an equivalence of categories F:T→T0such that FΣ∼ =Σ0F. This is equivalent to the existence of a graded category equivalence TΣ≃T0 Σ0. The aim of this paper is to show that, under mild conditions, ETC(T,Σ) is either empty or a singleton. In this section, we give alternative descriptions of this set which will help in proving that. Let DGCatMor be the category of DG-categories endowed with Tabuada’s Morita model structure [57, 59, 60]. Let DGCatMor(T,Σ) be the full subcategory of DG- categories Awith Dc(A) equivalent to (T,Σ). The set ETC(T,Σ) has been implicitly defined as the set of connected components of the classification space |DGCatMor(T,Σ)|in the sense of [16]. This space is the nerve of the category of Morita weak equivalences between DG-categories Awith Dc(A) equivalent to (T,Σ). This space contains much more information than its mere set of connected components. In a sense, it knows about the homotopy symmetries of each DG- category in the Morita model structure [61, Corollary A.0.4]. Let DGCatEq be the category of DG-categories endowed with the model structure in [58], whose weak equivalences are quasi-equivalences. Let us denote by DGCatEq(T,Σ) be the full subcategory of pre-triangulated DG-categories A, in the sense of [12], such that H0(A) is equivalent to (T,Σ) as a suspended category. A DG-category Ais pre-triangulated if the canonical inclusion functor H0(A),→D(A) identifies H0(A) with a triangulated subcategory of the target. A DG-category Ais fibrant in DGCatMor precisely if it is pre-triangulated and H0(A) is idempotent complete. This is equivalent to saying that the previous canonical inclusion induces an equivalence H0(A)≃Dc(A) with the full subcategory of compact objects in D(A). In particular, objects in DGCatEq(T,Σ) are Morita fibrant. Since DGCatMor is a left Bousfield localization of DGCatEq, Morita equivalences between Morita fibrant DG-categories are the same as quasiequivalences. Proposition 8.2. There is a homotopy equivalence |DGCatEq(T,Σ)| ≃ | DGCatMor(T,Σ)|. Proof. We have an inclusion functor j: DGCatEq(T,Σ) ⊂DGCatMor(T,Σ). A fibrant replacement functor Ron DGCatMor (co)restricts to R: DGCatMor(T,Σ) → DGCatEq(T,Σ). Both functors jand Rpreserve weak equivalences. Moreover, both composites jR and Rj are equipped with natural weak equivalences to the corresponding identity functor. Hence, we are done.  Let Λ be the endomorphism algebra of a basic additive generator of T, that we assume to be Frobenius as a necessary condition for ETC(T,Σ) to be nonempty, σ: Λ →Λ an automorphism induced by Σ, and DGCatEq(Λ(σ)) the full subcategory of DGCatEq spanned by the DG-categories Bsatisfying the following properties: (1) The category H0(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). 32 FERNANDO MURO Excluding E00 2, where d2is not defined, the second page of the truncated spectral sequence splits into two families of cochain complexes C∗ nand D∗ n,n∈Z, C∗ −1 C∗ 0 C∗ 1 C∗ 2 C∗ 3 C∗ −3 C∗ −2 D∗ 0 D∗ 1 D∗ −1 D∗ −2 D∗ 2 D∗ 3 with the following descriptions, Cm n=E2m,n+m 2, m ≥1,or m= 0 and n≥1; 0,elsewhere; Dm n=E2m+1,n+m 2, m ≥0; 0,elsewhere. The differential is of course d2in all cases. The Ep,q 3terms are defined for q≥p≥0, and for p≥3 and q∈Z. All of them are given by the homology of d2, except for E00 3and E11 3, i.e. HmC∗ n=E2m,n+m 3, m ≥2,or m≥0 and n≥1; HmD∗ n=E2m+1,n+m 3, m ≥1,or m≥0 and n≥2; In order to compute these groups, we consider new related families of cochain complexes, ¯ C∗ nand ¯ D∗ n,n∈Z, which agree with the former almost everywhere, and can be depicted as follows ENHANCED FINITE TRIANGULATED CATEGORIES 33 ¯ C∗ 0 ¯ C∗ 1 ¯ C∗ 2 ¯ C∗ 3 ¯ C∗ −3 ¯ C∗ −2 ¯ C∗ −1 ¯ D∗ 0 ¯ D∗ 1 ¯ D∗ −2 ¯ D∗ −1 ¯ D∗ 2 ¯ D∗ 3 Here, on any coordinate (p, q), p≥0, q∈Z, we place the Hochschild cohomology group HHp+2,−q(Λ(σ),Λ(σ)). Therefore, for m≥0 and n∈Z, ¯ Cm n= HH2m+2,−n−m(Λ(σ),Λ(σ)),¯ Dm n= HH2m+3,−n−m(Λ(σ),Λ(σ)), and ¯ Cm n=¯ Dm n= 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−1, for n∈Z, depicted below in red and blue, respectively, ¯ C∗ 0 ¯ C∗ 1 ¯ C∗ 2 ¯ C∗ 3 ¯ C∗ −3 ¯ C∗ −2 ¯ C∗ −1 ¯ D∗ 0 ¯ D∗ 1 ¯ D∗ −2 ¯ D∗ −1 ¯ D∗ 2 ¯ D∗ 3 By Lemma 6.8, fand gare injective, the cokernel of fis concentrated in degree ∗=−1, and the cokernel of gis concentrated in degrees ∗=−2,−1. Therefore, the associated long exact sequences induce the following isomorphisms in cohomology for m≥1 and n∈Z, f∗:Hm¯ C∗ n∼ =Hm+1 ¯ D∗ n, g∗:Hm¯ D∗ n∼ =Hm+2 ¯ C∗ n. 34 FERNANDO MURO By Proposition 3.6 and Remark 3.7, fand gare null-homotopic, so, for m≥1 and n∈Z, Hm¯ C∗ n=0=Hm¯ D∗ n. Clearly, D∗ n=¯ D∗ n, hence E2m+1,n+m 3=HmD∗ n= 0, m ≥1, n ∈Z. For n≤0, C∗ nis the naive truncation of ¯ C∗ nat ∗ ≥ 1, therefore E2m,n+m 3=HmC∗ n=Hm¯ C∗ n= 0, m ≥2 and n≤0. For n≥2, there is an obvious surjective cochain map C∗ n¯ C∗ nwhich is the identity in ∗>0 and the natural projection of Hochschild cocycles onto Hochschild cochains in ∗= 0, so E2m,n+m 3=HmC∗ n=Hm¯ C∗ n= 0, m ≥1 and n≥2. For n= 1 and char k6= 2, we also have a surjective map C∗ 1¯ C∗ 1for the same reason as above, and E2m,m+1 3=HmC∗ 1=Hm¯ C∗ 1= 0, m ≥1. This completes the proof in case char k6= 2. The problem in char k= 2 is that the differential C0 1→C1 1depends on the quadratic map (9.2) while ¯ C0 1→¯ C1 1is just given by [u, −], so there is no obvious map C∗ 1¯ C∗ 1. The last equation also holds for char k= 2, although it is more complicated to check. It suffices to construct a chain map C∗ 1¯ D∗+1 1given by u· − for ∗>0, which is an isomorphism by Lemma 6.8, and which is surjective for ∗= 0, since then E2m,m+1 3=HmC∗ 1=Hm+1 ¯ D∗ 1= 0, m ≥1. we define C0 1→D1 1as the following composite Z2,−1(Λ(σ),Λ(σ)) HH2,−1(Λ(σ),Λ(σ)) β −→ HH5,−2(Λ(σ),Λ(σ)) where β(x) = u·x+{δ} · x2. We must show that this βis surjective and completes the definition of a chain map C∗ 1¯ D∗+1 1. Let us start with the second property. It suffices to check that u·α(x)=[u, β(x)] for any x∈HH2,−1(Λ(σ),Λ(σ)), where αis the morphism in (9.2) which defines the differential d2:E0,1 2→E2,2 2. This follows from Proposition 3.2, char k= 2, and the laws of a Gerstenhaber algebra, since we know that [u, u ·x] = u·[u, x], and [u, {δ} · x2]=[u, {δ}]·x2+{δ} · [u, x]·x+{δ} · x·[u, x] =u·x2+{δ} · [u, x]·x+{δ} · [u, x]·x =u·x2. We finish by showing the surjectivity of β. Actually, we will see that it is bijective. For this, we construct morphisms γand λfitting in the following commutative diagram, ENHANCED FINITE TRIANGULATED CATEGORIES 35 HH2,−1(Λ(σ),Λ(σ)) HH5,−2(Λ(σ),Λ(σ)) HH11,−4(Λ(σ),Λ(σ)) HH23,−8(Λ(σ),Λ(σ)) β u3·− ∼ = γ u6·− ∼ = λ where the curved arrows are isomorphisms by Proposition 6.8. The morphisms γ and λare defined as γ(x) = u2·x+{δ} · x2, λ(x) = u4·x+{δ} · x2. Commutativity follows from γβ(x) = γ(u·x+{δ} · x2) =u3·x+u2· {δ} · x2+{δ} · (u·x+{δ} · x2)2 =u3·x+u2· {δ} · x2+{δ} · u2·x2 =u3·x, λγ(x) = λ(u2·x+{δ} · x2) =u6·x+u4· {δ} · x2+{δ} · (u2·x+{δ} · x2)2 =u6·x+u4· {δ} · x2+{δ} · u4·x2 =u6·x. Here we use Proposition 3.8, the commutativity of the cup product, and that char k= 2.  The group Aut(Λ(σ)) also acts on the right of the graded algebra HH?,∗(Λ,Λ(σ)) by conjugation since Λ is the degree 0 part of Λ(σ), so any automorphism of the latter (co)restricts to an automorphism of the former. The following corollary is a consequence 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 HH3,−1(Λ,Λ(σ)). Proposition 9.4. The set ETC(T,Σ) is non-empty if and only if Ω3(Λ) ∼ =σ−1Λ1 in Mod(Λe). 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 and Propositions 5.7 and 6.5. By those previous results, in order to prove the second part, we must check that HH3,−1(Λ,Λ(σ))×/Aut(Λ(σ)) is a singleton. Let Z(Λ) denote the center of Λ. There is a group morphism g: HH0,0(Λ,Λ(σ))×= HH0,0(Λ,Λ)×=Z(Λ)×→ Aut(Λ(σ)) sending x∈Z(Λ)×to the automorphism g(x): Λ(σ)→Λ(σ) defined on 36 FERNANDO MURO each degree n∈Zby right multiplication by xn. This morphism is obviously injective, so we can regard HH0,0(Λ,Λ(σ))×as a subgroup of Aut(Λ(σ)). Since any g(x) is the identity in degree 0, the right action of HH0,0(Λ,Λ(σ))×on HH3,−1(Λ,Λ(σ))× is given by the comparison morphism HH0,0(Λ,Λ(σ))×HH0,0(Λ,Λ(σ))×, which is surjective by Proposition 5.6, and left multiplication by the inverse. The quotient HH3,−1(Λ,Λ(σ))×/HH0,0(Λ,Λ(σ))× of the ‘left multiplication by the inverse’ action is a singleton since, given x, y ∈ HH3,−1(Λ,Λ(σ))×,xy−1∈HH0,0(Λ,Λ(σ))×. Therefore, the quotient by the larger group HH3,−1(Λ,Λ(σ))×/Aut(Λ(σ)) is also a singleton.  Definition 9.5. We say that Thas 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]7→ [σΛ1]. Moreover, Pic(Λ) is also the group of natural isomorphism classes of self-equivalences of Mod(Λ), and [σ] corresponds to the restriction of scalars (σ−1)∗: Mod(Λ) →Mod(Λ) along σ−1: Λ →Λ, which is naturally isomorphic to − ⊗ΛσΛ1. The advantage of the equivalence (σ−1)∗is that it is an automorphism. Corollary 9.6. The finite category T≃proj(Λ) has an enhanced triangulated structure if and only if Ω3(Λ) is isomorphic in Mod(Λe)to an invertible Λ-bimodule. Up to natural isomorphism, the possible suspension functors are the restrictions of scalars σ∗: proj(Λ) →proj(Λ), where σruns over a set of representatives of elements in Out(Λ) such that Ω3(Λ) ∼ =σ−1Λ1in Mod(Λe). Once the suspension functor Σis fixed, ETC(T,Σ) is a singleton. Corollary 9.7. If Λis separable, then T≃proj(Λ) has an enhanced triangulated structure. Up to natural isomorphism, the possible suspension functors are the restrictions of scalars σ∗: proj(Λ) →proj(Λ), where σruns over a set of representatives of elements in Out(Λ). Once the suspension functor Σis fixed, ETC(T,Σ) is a singleton. The case analyzed in this corollary is particularly simple, Λ is semisimple, T≃ Dc(Λ(σ)), and all exact triangles split here. Recall that the algebra Λ is connected if it cannot be decomposed as a product Λ∼ =Λ1×Λ2, with Λia non-trivial algebra, i= 1,2. Proposition 9.8. Suppose Λis connected and not separable. Given any Λ-bimodule M, denote by Ω(M)the syzygy obtained ascovercover the kernel of a projective cover, Ω(M),→PM. Then, T= proj(Λ) has an enhanced triangulated structure if and only if Ω3(Λ) is an invertible Λ-bimodule. In that case, the suspension functor is necessarily Σ = − ⊗ΛΩ3(Λ)−1, where Ω3(Λ)−1is an inverse of Ω3(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 Ω3(Λ) too, and also any invertible Λ-bimodule σΛ1. Moreover, σΛ1cannot be projective. Otherwise, it would be a direct summand of Λ ⊗Λ, and then Λ ENHANCED FINITE TRIANGULATED CATEGORIES 37 would be a direct summand of σ−1Λ⊗Λ∼ =Λ⊗Λ, hence separable. The third syzygy Ω3(Λ) is not projective either since that would also imply that Λ would be a projective bimodule. Two Λ-bimodules Mand Nare isomorphic in Mod(Λe) if and only if there are projective Λ-bimodules Pand Qsuch that M⊕P∼ =N⊕Qin Mod(Λe). Hence, M= Ω3(Λ) is isomorphic to some N=σΛ1in Mod(Λe) if and only if they are isomorphic in Mod(Λe), because each of them is the only non-projective indecomposable factor on each side of an isomorphism M⊕P∼ =N⊕Qas above. Now, this proposition follows from Corollary 9.6.  Recall that the Nakayama algebra Nn mis the quotient of the path algebra of the oriented cycle of length m≥1 m1 m−1 2 m−2 3 αm α1 αm−1 α2 αm−2 by the two-sided ideal generated by the paths of length n+1, n≥1. This basic and self-injective algebra (hence Frobenius) has dimension m(n+ 1) over the ground field kand it is connected but not separable, see [23, §3] and [17, §3]. Proposition 9.9. If kis algebraically closed, Λ = Nn m,mdivides n, and n > 1, then Ω3(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, §3], Ω2(Λ) = σΛ1for some automorphism σof Λ. Since Λ is connected and not separable, if Ω3(Λ) were stably isomorphic to an invertible Λ-bimodule then it would actually be isomorphic to it, by the same argument as in the proof of Proposition 9.8 above. Recall from [11, Proposition 3.8] that any invertible Λ-bimodule is of the form µΛ1for some automorphism µ. If that happened, the minimal projective resolution of Λ as a bimodule would produce a short exact sequence of bimodules µΛ1,→PσΛ1 with projective middle term. If we twisted by σ−1from the left we would obtain another short exact sequence of bimodules with projective middle term σ−1µΛ1,→σ−1P1Λ, hence Ω(Λ) = σ−1µΛ1. We know that dimkσ−1P1= dimkP=m(n+ 1)2, see [23, §3], and dimkσ−1µΛ1= dimkΛ = m(n+ 1), so this can only happen if n= 1. The final conclusion follows from (1) in our main theorem in the enhanced case and from [22, Theorem 1.2] in the non-enhanced case.  38 FERNANDO MURO For n= 1, Λ = N1 msatisfies Ω(Λ) = σΛ1for some automorphism σof Λ, see the proof of [17, 4.2]. 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