The Singularity and Cosingularity Categories of C∗BG for Groups with Cyclic Sylow p-subgroups

We construct a differential graded algebra (DGA) modelling certain A∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A_{\infty }$\end{document} algebras associated with a finite group G with cyclic Sylow subgroups, namely H∗BG and H∗ΩBGp∧\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H_{*}{\Omega } BG{^{^{\wedge }}_p}$\end{document}. We use our construction to investigate the singularity and cosingularity categories of these algebras. We give a complete classification of the indecomposables in these categories, and describe the Auslander–Reiten quiver. The theory applies to Brauer tree algebras in arbitrary characteristic, and we end with an example in characteristic zero coming from the Hecke algebras of symmetric groups.


Introduction
Our purpose is to study the singularity categories of certain A ∞ algebras over a field k.We were led to these examples from the representation theory in characteristic p of finite groups with cyclic Sylow p-subgroups, but our earlier work [10], on which we build, showed these examples were members of a more general family of examples: the general case illuminates those we first considered, and the other examples also occur elsewhere.
In fact the examples occur in Koszul dual pairs A and B. The BGG correspondence shows that it is illuminating to consider both members of the pair together: the classical example occurs with A an exterior algebra and B the Koszul dual polynomial algebra.The singularity category of A is equivalent to the cosingularity category of B, which by a theorem of Serre is the bounded derived category of quasicoherent sheaves on Proj(B).Since A is finite dimensional, its cosingularity category is trivial; since B is regular, its singularity category is trivial.In this case both A and B are formal as k-algebras.
We consider here a family of the next simplest cases consisting of a non-formal A ∞ kalgebra A, usually with homology where τ has even degree 2b and ξ has odd degree 2a − 1.The family of examples we study is determined by a, b and two further parameters h, ℓ 2 related by ah−bℓ = 1.The parameter h is the length of the shortest non-trivial Massey product (when h = 2 the homology ring is a little different to that above, since ξ 2 = −τ ℓ ).We give a full description of A in Section 4. It is shown in [24] that the BGG correspondence extends to a more general A ∞ context, and it is again natural to consider the Koszul dual B. In fact B is of exactly the same form as A but with different degrees, and the parameters h and ℓ exchanged.In most cases it again has homology of form H * (B) = Λ(t) ⊗ k[x] where x is of even degree −2a and t is of odd degree −2b − 1.The parameter ℓ is the length of the shortest non-trivial Massey product (when ℓ = 2 the homology is a little different to that above, since t 2 = −x h ).We give a full description of B in Section 9.In this case both A and B have singularity categories that are non-trivial and we are able to give a complete description: they each have finitely many indecomposable objects and we describe their Auslander-Reiten quivers.This behaviour is rather special.In general, even when the singularity category of A has finitely many indecomposables, the singularity category of B can have infinitely many.The behaviour of A and B is also quite different to the behaviour of the formal algebras H * (A) and H * (B), whose singularity and cosingularity categories all have infinitely many indecomposable objects.
Our first task is to describe a small and explicit DG algebra Q in the same quasiisomorphism class as A, with some good properties that make it suitable for both theoretical and computational work.As a step towards Q, we first introduce an auxiliary DG algebra R in Section 2, which embodies the algebra of an odd element all of whose Massey powers vanish: it is generated by elements ξ 1 , ξ 2 , . .., and has homology H * (R) = Λ(ξ).The algebra Q can be viewed as a universal object for an algebra with an h-fold Massey power of an element of odd degree which is an ℓth power of an element of even degree.It is generated by elements τ , ξ 1 , . . ., ξ h−1 , with τ and ξ 1 representing elements τ , ξ in H * Q ∼ = A. Explicit formulas for the relations and the action of the differential d are given in Section 3. The element τ of Q is central, so Q may be regarded as an algebra over k[τ ].Our principal goal is to determine the structure of the singularity and cosingularity categories of A and B (see Section 6 for definitions).Our main theorems classify the indecomposable objects in these categories, see Theorems 9.3 and 15.3. ) ≃ D sg (B).The latter categories satisfy the Krull-Schmidt theorem, and have |b|(h − 1) isomorphism classes of indecomposable objects, in [h/2] orbits of the shift functor Σ.The Auslander-Reiten quiver is isomorphic to ZA h−1 /T |b| , where T is the translation functor Σ −2a .This is a cylinder of height h − 1 and circumference |b|.The functor Σ switches the two ends of the cylinder.
We give explicit descriptions of the indecomposable objects, both as elements of D csg (A) ≃ D b (A[τ −1 ]) and as elements of D sg (B).Reversing the roles of A and B, swapping h and ℓ, and replacing a by −b and b by −a gives us the structure of D sg (A) ≃ D csg (B), with |a|(ℓ −1) isomorphism classes of indecomposable objects, coming in [ℓ/2] orbits of Σ.We also give an explicit description of the Auslander-Reiten quivers of these categories, and explain their position in the Amiot's classification of finite triangulated categories.This leads to some explicit models such as D sg (B) ≃ D b (A h−1 )/T |b| in Section 16.
Of course the simplicity of the A ∞ structure is essential for explicit calculations, but key structural features making a complete analysis possible are the τ -periodicity and the Tate duality of Theorem 10.8.The key ingredient for Tate duality is the fact that finitely generated modules are automatically dualizable, which we proved in this case using a Hochschild cohomology calculation.
We are especially interested in the following occurrence of the A ∞ algebras A and B. Let p be an odd prime, let G be a finite group with cyclic Sylow p-subgroups P of order p n and inertial index q > 1, and let k be a field of characteristic p. Omitting notation for coefficients in the field k, and writing ΩBG ∧ p for the loop space on the Bousfield-Kan mod p completion of BG, we showed in [10] that the A ∞ algebra structures on A = H * ΩBG ∧ p and B = H * BG were of the above form with a = q, b = q − 1, h = p n − (p n − 1)/q, ℓ = p n .Then the DG algebra Q describes a model for the DG algebra C * ΩBG ∧ p up to quasi-isomorphism, and by reversing the roles of A and B we obtain a model for C * BG.This is a finite Krull-Schmidt triangulated category with (q−1)(h−1) indecomposable objects in [h/2] orbits of the shift functor.It also induces equivalences This is a finite Krull-Schmidt triangulated category with q(p n − 1) indecomposable objects in [p n /2] orbits of the shift functor.
Other examples of the A ∞ algebras A and B occur every time an algebra is described by a Brauer tree of finite representation type.These occur throughout representation theory, both in characteristic zero and in prime characteristic.For the sake of describing an example in characteristic zero, we discuss the Hecke algebras of symmetric groups.Let H = H(n, q) be the Hecke algebra of the symmetric group of degree n over a field k of characteristic zero, where q is a primitive ℓth root of unity with n = ℓ > 2. Then letting A be the principal block of H and B be Ext * H (k, k), we obtain an example with a = n − 1, b = n − 2, h = n − 1 and ℓ = n.We spell out the consequences of our main theorem in this case, in Theorem 18.3.
Acknowledgements.The authors are grateful to EPSRC: the second author is supported by EP/P031080/1, which also enabled the first author to visit Warwick.The authors are also grateful to Bernhard Keller and Greg Stevenson for conversations related to this work.
The authors would also like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for providing an opportunity to work on this project during the simultaneous programmes 'K-theory, algebraic cycles and motivic homotopy theory' and 'Groups, representations and applications: new perspectives' (one author was supported by each programme).

The DG Hopf algebra R
We begin by looking at the DG Hopf algebra R over a field k.As a graded algebra over k, R is free with odd degree generators ξ 1 , ξ 2 , ξ 3 , . . ., and the differential is given by , and so on.Remark 2.1.To motivate this, we factor out the differential ideal of R generated by ξ i for i h + 1 and take the DG-subalgebra R h generated by ξ i for i h − 1.The element µ h = d(ξ h ) lies in R h and represents the h-fold Massey power of the homology class of ξ 1 up to sign.Thus a DGA map from θ : R h → C in which θ(ξ 1 ) = c shows that the h-fold Massey power of [c] is defined and gives an element [θ(µ The antipode S on R is the anti-automorphism of algebras given on generators by S(ξ i ) = −ξ i .The comultiplication ∆ : R → R ⊗ R is defined on generators by Note that if ξ 1 has degree 2a − 1 then ξ i has degree 2ia − 1.So R is either connected or coconnected, according to whether 2a − 1 is positive or negative.We shall assume that a = 0, so that |ξ 1 | = −1, which implies that each graded piece is finite dimensional.Lemma 2.2.In R we have d 2 = 0.
Proof.To show that d 2 = 0, we note that dd(ξ i ) has two terms for each way of writing i as a sum of three positive integers.They have opposite signs, so they cancel.
Lemma 2.3.The map ∆ : R → R ⊗ R is a map of DG algebras.
Proof.As an algebra, R is free, so specifying the map on generators gives a well defined map of algebras.We must check that it commutes with the differential.Since the ξ j have odd degree and are primitive, ξ 2 j and ξ j ξ k + ξ k ξ j are also primitive.So d(ξ i ) = j+k=i ξ j ξ k is also primitive, and hence d∆(ξ i ) = ∆d(ξ i ).
Proposition 2.4.The definitions above make R into a cocommutative DG Hopf algebra.
Proof.Lemmas 2.2 and 2.3 show that R is a DG bialgebra.It is easy to check that the antipode satisfies the identity S(x (1) )x (2) = x (1) S(x (2) ) = 0 in Sweedler notation, for elements of non-zero degree; this only needs checking on the generators, where it is clear.Cocommutativity also only needs checking on generators.
Proof.Define a linear map δ : R → R sending a monomial of the form ξ 1 ξ i f to ξ i+1 f , and sending all other monomials to zero.Then we have Thus δd + dδ is the identity on all monomials apart from 1 and ξ 1 , on which it vanishes.So δ defines a homotopy from the identity map of R to the projection onto the linear span of 1 and ξ 1 .It follows that H * R = Λ(ξ 1 ).
Definition 2.6.The weight of a monomial in R is the sum of the subscripts (and zero for constants).The height of a monomial in R is the number of generators ξ i that have to be multiplied to give the monomial (we might call it degree, if that didn't already have a different meaning).Multiplication in R adds weights, and adds heights.If f (ξ 1 , . . ., ξ n ) is an element of R, we write f i,j (ξ 1 , . . ., ξ n ) for the sum of the terms of f with weight i and height j.Thus f = i,j f i,j .
The differential d preserves weight, and increases height by one, so that d(f i,j ) = (df ) i,j+1 .Thus if df = 0 then each d(f i,j ) = 0.

The DG Hopf algebra Q
In this section, we let h and ℓ be integers 2 and describe the DG Hopf algebra Q = Q h,ℓ .In terms of the previous section, the idea is that Q h,ℓ is obtained from R h by adjoining an ℓth root to the element −µ h .Thus a DGA map θ : Q h,ℓ → C shows that the h-fold Massey power of c = θ(ξ 1 ) is defined and contains an ℓth power of the adjoined variable.
The generators for Q are ξ 1 , . . ., ξ h−1 in odd degree and τ in even degree.The relations and differential are as follows: The antipode is the algebra anti-automorphism given by S(ξ i ) = −ξ i , S(τ ) = −τ , and the comultiplication is given by We In particular, a and b are coprime, as are h and ℓ.
As well as this homological grading, we give Q a second, internal grading by setting It is easy to check that the relations, differential, and Hopf structure above respect this second grading.
Example 3.1.If h = 2, the algebra Q is generated by ξ 1 and τ with relations , the algebra Q is generated by ξ 1 , ξ 2 , ξ 3 and τ with relations In the algebra Q, every element has a unique expression of the form Proof.The algebra relations (ignoring the differential) can be rewritten in the form with 1 i h − 1 (note that φ h−1 = 0).Thus all occurrences of ξ h−1 may be moved to the beginning, and ξ 2 h−1 = 0.There are no relations among ξ 1 , . . ., ξ h−2 .Definition 3.5.We shall refer to a monomial in ξ 1 , . . ., ξ h−2 , or ξ h−1 times such a monomial, as a standard monomial in the variables ξ 1 , . . ., ξ h−1 .The lemma shows that the standard monomials form a basis for Q as a free module over k[τ ].Lemma 3.6.In the algebra Q, we have d 2 = 0.
Proof.The differential is given by On the free subalgebra generated by ξ 1 , . . ., ξ h−2 , the differential is the same as in the algebra R above, and so by Lemma 2.2 we have d 2 = 0 on this subalgebra.Thus we have Proof.As in Lemma 2.3, for each i > 0, j+k=i ξ j ξ k is primitive, so both the relations and the elements d(ξ i ) are primitive.Hence ∆ takes relations to relations, and d∆ = ∆d.Proposition 3.8.The definitions above make Q into a cocommutative DG Hopf algebra.
Proof.The proof of this is similar to the proof of Proposition 2.4, but using Lemmas 3.6 and 3.7 in place of Lemmas 2.2 and 2.3.

The A ∞ algebras A and B
In this section we introduce an A ∞ algebra A which is quasi-isomorphic to the DG algebra Q of the last section.We then describe the Koszul dual B of A. These are the A ∞ algebras discussed in our previous paper [10].
We extend the definition to h = 2 by letting A be the formal and with all m i apart from m 2 equal to zero.
We next show that the DG algebra Q (forgetting the Hopf structure) is quasi-isomorphic to A as an A ∞ -algebra.In other words, A ∼ = H * (Q), with the A ∞ structure given by Kadeishvili's theorem [29].
Theorem 4.2.There is a quasi-isomorphism from the DG algebra Q to the A ∞ algebra A, sending τ to τ and ξ 1 to ξ.
Proof.First, we show that H * Q is isomorphic to A as an algebra over k[τ ].The proof is similar to the proof of Lemma 2.5, but working over k[τ ] instead of k.Namely, we define a linear map δ : Q → Q sending a monomial of the form ξ 1 ξ i f to ξ i+1 f for 1 i h − 2, and all other standard monomials (see Definition 3.5) to zero.Thus δ(f + ξ h−1 g) = δ(f ).The same computation as in Lemma 2.5 shows that δd + dδ is the identity on all monomials except those in k[τ ] + ξ 1 k[τ ], where it is zero.Thus δ defines a homotopy from the identity map of Q to the projection onto k[τ ] + ξ 1 k[τ ].It follows that H * Q is isomorphic to A as a ring, with τ and ξ 1 corresponding to τ and ξ.The maps m i on H * Q are easy to calculate using the elements ξ i , and give zero for i > 2 except in the case of m h , where it gives m h (ξ, . . ., ξ) = (−1) h(h−1)/2 τ ℓ .Using Kadeishvili's theorem [29] completes the proof.
, and all m i with i > 2 are zero on all other monomials.In the exceptional case where ℓ = 2, we define B to be the formal A ∞ algebra k[x, t]/(t 2 + x h ).It was shown in [10] that A and B are Koszul dual.Thus A is quasi-isomorphic to End D b (B) (k) and B is quasi-isomorphic to End D b (A) (k).Here, End D b (B) (k) denotes the A ∞ endomorphism ring whose homology is and so on.

Hochschild cohomology
We will recall the definition of the Hochschild cohomology of an A ∞ -algebra and then calculate it for the A ∞ algebras A and B described in the previous section.The point of this is that nilpotent elements in (for example) HH * (A) control certain uniform processes of construction in the category of A-modules: we will make essential use of this in our proof of Theorem 9.7.
The bar resolution B(a) = n 0 a ⊗(n+2) of an A ∞ algebra a is described in Section 3 of Getzler and Jones [21], see also Definition 12.6 of Stasheff [42].The action of the differential on a ⊗(n+2) in bar notation is where the signs are determined by the usual sign conventions.Taking a-a-bimodule homomorphisms to a bimodule M, we obtain the differential on Hochschild cochains Hom a,a (a ⊗(n+2) , M) ∼ = Hom k (a ⊗n , M) as follows: see also Section 1 of Roitzheim and Whitehouse [41].The cohomology of this complex is HH * (a, M).If M = a, we write HH * (a) for HH * (a, a).
We filter B(a) by number of bars, n+2) .This gives a filtration on Hochschild cochains, for which F i is the cochains which vanish on F i B(a).With this filtration, F 0 is the whole complex and i F i = 0.This leads to a spectral sequence in which the differentials d n are given by the terms involving ±m n+1 .Thus the E 1 page is the Hochschild complex of H * a with coefficients in H * M, and the E 2 page is HH * (H * a, H * M).So the spectral sequence takes the form (5.1) We are numbering everything homologically, so the Hochschild degrees in HH * H * a are negative, and the spectral sequence lives in the second and third quadrants.

Applying lim
← with respect to i to the exact sequences 0 So the spectral sequence is conditionally convergent, and is strongly convergent if and only if lim 0 in the spectral sequence, see for example Theorem 7.1 of Boardman [11].In particular, if each graded piece of E r is finite dimensional for some r, then the spectral sequence (5.1) is strongly convergent.
Theorem 5.2.Given a map of exact couples (D, E) → (D ′ , E ′ ), where both spectral sequences are conditionally convergent and live in the (homologically indexed) left half plane, if the map E r → E ′ r is an isomorphism for some r then D → D ′ is an isomorphism of filtered graded groups.
Proof.This follows from Theorem 7.2 of Boardman [11], since the hypotheses imply that and May [23].
We need to know that A, B and Q have the same Hochschild cohomology.Since our equivalences are slightly indirect we need some machinery to see the isomorphism preserves structure.Keller's article [30] recalls the definition of B ∞ -algebras.
Proposition 5.3.There are isomorphisms in the homotopy category of B ∞ algebras between the Hochschild complexes of Q, A and B.
Proof.For the algebras Q and A, we apply the main theorem in Section 3.2 of [30] to the quasi-isomorphism Q → A of Theorem 4.2 to obtain an equivalence of Hochschild complexes in the homotopy category of B ∞ algebras.
Let C be the cobar construction on Q.This is an augmented DG coalgebra, which is finite dimensional in each degree.As in Section 2 of Keller [36], this is the Koszul-Moore dual of Q.Its graded k-linear dual C * is "the" Koszul dual of Q, and is quasi-isomorphic to B. Using the fact that C is finite dimensional in each degree, the Hochschild complex for C is isomorphic to that for C * as a B ∞ algebra.Applying Theorem 3.3 of [36], the Hochschild complexes of Q and C are equivalent in the homotopy category of B ∞ algebras.
Certainly an isomorphism in the homotopy category of B ∞ algebras induces an isomorphism of cohomology rings.
The only non-zero differential is d ℓ−1 , and this is given by There are no ungrading problems in the spectral sequence.
Proof.The element t on the E 2 page corresponds to the cochain t : [ ] → t in the Hochschild complex.Applying the formula for the differential, we have Using this, and the rather simple form of the A ∞ structure on B, it is not hard to see that d t = ±ℓx h τ ℓ−1 since the two cochains take the same value on all elements of the bar resolution.
The element ξ on the E 2 page corresponds to the cochain ξ : Then applying the formula for the differential, we have Using this, again we find that d ξ = ±hx h−1 τ ℓ since both cochains take the same value on all elements of the bar resolution.
Examining the locations of these terms in the filtration of the bar complex giving rise to the spectral sequence, we deduce that these correspond to the differential d ℓ−1 taking t to ±ℓx h τ ℓ−1 and ξ to ±hx h−1 τ ℓ .
Next, we show that the possible values of n for which d n is non-zero are very restricted.The possible tridegrees (u, v, w) at the E 2 -term lie in three parallel planes.Take N = (ℓ − 2, ℓ, −2a) as normal direction and consider the dot products N Since n 2, we first deduce that all differentials are zero on elements with N • (u, v, w) = 0, and hence on the polynomial generators x and τ .Next, we deduce that the smallest value of n for which We computed above the value of d ℓ−1 on the exterior generators t and ξ.Finally, since h and ℓ are coprime, d ℓ−1 is injective on elements with N • (u, v, w) = 4 − 2ℓ, and so there is no room for further differentials.
For the ungrading problem, we note that moving down one place in the filtration replaces (u, v, w) by (u−1, v+1, w) and so the dot product with N increases by N •(−1, 1, 0) = 2, while w is unchanged.The relations hx h−1 τ ℓ = 0 and ℓx h τ ℓ−1 = 0 therefore have no ungrading problems, and hold in HH * B. The relations ξ 2 = 0 and t 2 = 0 have no ungrading problems, because there are no candidates with the correct value of w and with larger dot product with N.
Theorem 5.6.There are three cases for HH * Q ∼ = HH * A ∼ = HH * B, according to the characteristic of the field k.
Proof.If h > 2 and ℓ > 2 then this follows from Theorem 5.5, after checking that there are no ungrading problems.The element u in case (iii) represents ), and we can use the method of Buchweitz and Roberts [15] to compute HH * B. If ℓ = 2 then we can use the same method on HH * A.
Corollary 5.7.We have Proof.This follows from Theorem 5.6: in all three cases we have x h τ ℓ = 0. Note that if h > 2 and ℓ > 2 then we see directly that the differential d ℓ−1 in Theorem 5.5 takes ±aξ ± bt (with appropriate signs) to x h τ ℓ .

The derived category
Suppose first that a is a DG algebra.We write D(a) for the derived category of a.This is the triangulated category having as objects the left DG a-modules, and as arrows the homotopy classes of morphisms of DG modules, with the quasi-isomorphisms inverted.The shift functor is the suspension Σ defined by (ΣM) n = M n−1 , so the triangles take the form X → Y → Z → ΣX.
In the case where H * a is a Noetherian graded ring, we write D b (a) for the thick subcategory of D(a) whose objects are the a-modules X such that H * X is finitely generated as an H * amodule.We regard this as the analogue of the bounded derived category in this context; an extended discussion motivating this can be found in Greenlees and Stevenson [24].Definition 6.1.An a-module is homotopically projective if the functor Hom a (X, −) preserves quasi-isomorphisms (see Section 9.1 of Avramov, Foxby and Halperin [7]).Semifree modules are homotopically projective (Lemma 9.3.5 of [7]) and every module has a semifree resolution (Theorem 8.3.2 of [7]).It follows that given any module X there exists a homotopically projective module X ′ and a surjective quasi-isomorphism X ′ → X.We call this a homotopically projective resolution of X.
Homomorphisms in D(a) may be described as follows.Given DG a-modules X and Y , choose a homotopically projective module X ′ and a quasi-isomorphism X ′ → X.Then It can be seen that X ′ → X is a fibrant replacement with respect to the projective model structure (see Hovey [28]) on a-modules, and D(a) is the corresponding homotopy category.
Next, we describe the derived category of an A ∞ algebra.Suppose that a is an A ∞ algebra.In this case, the modules do not form an abelian category, because of the definition of morphism of A ∞ modules.This time, the derived category D(a) is the triangulated category having as objects the left A ∞ modules over a, and as arrows the homotopy classes of A ∞ morphisms.Unlike in the DG context, A ∞ quasi-isomorphisms automatically have A ∞ inverses.This is again a triangulated category, with triangles of the form X → Y → Z → ΣX.For details, see Keller [31,32].As before, in the case where H * a is Noetherian, we write D b (a) for the thick subcategory whose objects are the modules with finitely generated homology.
In the case where a is a DG algebra regarded as an A ∞ algebra with m i = 0 for i > 2, the two definitions agree up to canonical equivalences of triangulated categories.If X and Y are DG a-modules, the homotopy classes of morphisms of A ∞ modules from X to Y are canonically isomorphic to the homotopy classes of morphisms of DG modules from X ′ to Y , where X ′ is a homotopically projective resolution of X.A suitable set of details can be found in Théorème 2.2.2.2 and Sections 2.4 and 4.1 of the thesis of Lefèvre-Hasegawa [37].See also Theorem 4.5 of Keller [34].
In the case of the DG algebra Q of Section 3 and the A ∞ algebra A of Section 4, we have the following.Proof.A quasi-isomorphim of A ∞ algebras induces an equivalence of derived categories, see for example [37], Section 4.1.3.So it follows from Theorem 4.2 that D b (Q) is equivalent to D b (A).Remark 6.3.Although the A ∞ algebras Q, A and B carry an internal grading to make them bigraded, we do not require that the objects in the derived category carry an internal grading respected by the morphisms.Nonetheless, we shall make use of internal gradings in identifying Auslander-Reiten triangles in D sg (B) in Section 14.As we shall see, the reason this works is that the duality established in Section 10 respects grading for objects that admit one.

A spectral sequence
In this section, we give a brief reminder of the construction and convergence properties of the spectral sequence for computing Homs in the derived category D(a) of an A ∞ algebra a: We shall make use of this in Section 12 to compute some endomorphism rings, as a preliminary to applying Auslander-Reiten theory.The construction is taken from Adams [1], and a discussion of convergence may be found in Boardman [11].
Let a be an A ∞ algebra.If X is an A-module, then taking homology gives isomorphisms Choosing a set of generators of H * X, we obtain a morphism F 0 → X, where F 0 is a direct sum of shifts of a, with the property that , and the map i : X 0 → X 1 is zero in homology.Repeating this construction, we obtain a sequence of triangles where the maps marked j involve a degree shift.This has the property that the resulting sequence Proof.Any map Σ j a → lim → i X i factors through some X i , and then the composite If Y is another a-module, then taking Homs in D(a) from the above resolution of X to Y , we obtain a diagram of long exact sequences The direct sum of all these long exact sequences is an exact couple The spectral sequence of this exact couple has as its E 1 term The differential is the composite and so we have , and the abutment of the spectral sequence is Hom D(a) (X, Y ).Thus we have a filtration and In this filtration, F 0 is the whole thing, and by Lemma 7.2 we have j As in the spectral sequence for Hochschild cohomology described in Section 5, the spectral sequence is conditionally convergent, and strongly convergent if and only if lim In particular, if each graded piece of E i is finite dimensional, the the spectral sequence is strongly convergent.

Inverting τ
The advantage of the explicit model Q is that the element τ is represented by a central cycle, so it is elementary to invert it.Definition 8.1.We write K for the graded field k[τ, τ −1 ], and we define Proof.This follows from the fact that τ is central in Q with dτ = 0, together with the fact that H * (X) is finitely generated over H * (Q).

Similarly, we write
A as an A ∞ -algebra, and we have a quasi-isomorphism Proposition 8.3.We have an equivalence of bounded derived categories If X and Y are objects in D b (A) then Proof.This follows from Proposition 6.2 and Lemma 8.2.

Koszul duality and singularity categories
We recapitulate the development of [24] in our more concrete setting.For the A ∞ algebras Q and A introduced in Sections 3 and 4 and the A ∞ algebras Q[τ −1 ] and A[τ −1 ] described in Section 8 we have the following calculation.
Theorem 9.3.We have equivalences of triangulated categories Proof.By Propositions 6.2 and 8.3, we have compatible equivalences Using Lemma 9.2, it kills exactly Thick(k), and therefore induces an equivalence ) kills exactly Thick(k), and therefore induces an equivalence Proof.This follows from Greenlees and Stevenson [24], Theorem 9.1 and Example (10.6).
Corollary 9.5.We have an equivalence of categories Proof.This follows from Theorems 9.3 and 9.4.
Thus we can regard the central element τ of A as a periodicity operator on D sg (B) of degree 2b, namely a natural isomorphism from the identity to Σ 2b .In Section 13, we shall see explicitly how to interpret this element in terms of resolutions.
Proof.This follows from Proposition 8.3 using the equivalence of categories given in Corollary 9.5.
It is also in Thick(F ), and therefore in Thick(A[τ −1 ]).Hence so is N.
Proof.This follows from Theorem 9.7, using the fact that the equivalence Proof.These hold for X = Q[τ −1 ], and hence for every object in Thick(Q[τ −1 ]), which by Corollary 9.8 is every object in Let Q[τ −1 ] be the algebra described in Section 8.In this section, we prove a form of Tate duality for the bounded derived category of Q[τ −1 ]-modules, see Theorem 10.6.This combine the dualities Hom K (−, K) (Brown-Comenetz duality) and Hom Q ) (Spanier-Whitehead duality).The proof makes essential use of Corollary 9.9.
Definition 10.1.We write Q[τ −1 ] * for the dual of Q with respect to K, Hom k[τ ] (Q, K).Left and right multiplication make Q[τ −1 ] * into a Q-bimodule, and simultaneously a K-module with compatible actions of τ , and hence a Q[τ −1 ]-bimodule.Note that Similarly, if X is any K-vector space (i.e., graded K-module), we write X * for Hom K (X, K).
For brevity, we have recorded the shift as |ξ 1 | = 2a−1, however in the presence of the τ -periodicity, this is only well defined modulo 2b.It is more helpful to say that the shift in Tate duality is one less than the Gorenstein shift (2a − 2b in this case) of A as in [25,Proposition 4.1].
We should also keep track of the internal degrees, and write Proof.The standard monomials of Definition 3.5 form a K-basis for Q It takes all standard monomials to zero except 1 and ξ 1 .It takes 1 to the element of Q[τ −1 ] * taking value 1 on ξ 1 and value zero on all other standard monomials, and it takes ξ 1 to the element of Q[τ −1 ] * taking value 1 on 1 and value zero on all other standard monomials.It is easy to check that this is a map of Q[τ −1 ]-bimodules, and using Corollary 4.3 that it a quasi-isomorphism.
Proof.This is standard.
Corollary 10.5.If X is homotopically projective then we have a quasi-isomorphism Proof.Using Propositions 10.2 and 10.4, and the fact that X is homotopically projective, we have Theorem 10.6.Let X and Y be objects in Then there is a functorial duality of graded K-modules Proof.We may replace X and Y by homotopically projective resolutions and work with Q[τ −1 ]-module homomorphisms.Combining Corollary 9.9 (ii) with Corollary 10.5, we have Hence using Proposition 10.4 and Corollary 9.9 (i), we have Corollary 10.7.Let X and Y be objects in Then there is a functorial duality of graded K-modules If X and Y carry an internal grading, the shift is Σ 1−2a,−ℓ .
Proof.This follows from Theorem 10.6, using the equivalence of categories described in Proposition 8.3.
Theorem 10.8.Let X and Y be objects in D sg (B).Then there is a functorial duality of graded K-modules Proof.This follows from Corollary 10.7 using the equivalence of categories given in Corollary 9.5.

A tour of the two worlds
Our aim is to understand the singularity category of B, and in particular to construct indecomposable objects.In this interlude we give a topological account of the strategy before returning to give an elementary implementation in algebra.The section can be entirely ignored by those with a strong algebraic compass.
Bearing in mind that B itself is trivial in the singularity category, it is natural to start with X 1 = k and then construct objects from that.Since Hom Dsg(B) (k, k) = A[τ −1 ] it is natural to focus on ξ, which supplies a map Σ 2a−1 k → k, with mapping cone X 2 with cells in degree 0 and 2a.We then attempt to iterate this construction by forming In fact X s exists if and only if the (s−1)-fold Massey product ξ, ξ, • • • , ξ exists and contains zero.In our case the indeterminacy is always zero.We may thus construct X 1 , . . ., X h , but not X h+1 since the h-fold Massey product is nonzero.One may also check inductively that these complexes are unique up to equivalence.Our main result will show that up to suspension this does give all indecomposables.The counterparts Y s = Hom B (k, X s ) in A-modules will have a cell structure Under the derived equivalence between finitely generated B-modules and A-modules, small B-modules (i.e., modules in Thick(B)) correspond to A-modules with finite dimensional homology, and vice versa.Thus a B-module N is small if and only if [k, N] B * is finite dimensional.Similarly, N corresponds to a small A-module if and only if H * (N) is finite dimensional.
Exchanging the roles of A and B, we may construct A-modules for s = 1, 2, . . ., ℓ but not V ℓ+1 since the ℓ-fold Massey product is nonzero.These correspond to the small B-modules . Our actual method will take full advantage of the explicit and easily described models, and give concrete representatives for the objects W s .
Remark 11.1.For the graded rings H * (A) with h > 2 and H * (B) with ℓ > 2 (i.e., the formal case where the algebras are polynomial tensor exterior and all products m i are zero for i > 2) the singularity and cosingularity categories are well understood, for example through the theory of maximal Cohen-Macaulay modules.See for example Proposition 2.6 of Ene and Popescu [18], which discusses modules over the ungraded completion, but the modules over the graded ring are very similar.In that case the objects X s , Y s , V s , W s exist for all s 0 and the terms in each sequence are inequivalent.By contrast with our case, the singularity and cosingularity categories for H * (A) and H * (B) each contain infinitely many indecomposable objects.
12. The B-modules W i in Thick(B) Following the pattern set out in Section 11 we will construct explicit small B-modules (i.e., modules in Thick(B)) W 1 , W 2 , . . ., W ℓ , starting with W 1 = B, and we will do so in a bigraded fashion.Background on modules over A ∞ -algebras may be found in [34].
Consider the map t : Then using the long exact sequence in homology, as a module over where k The element v 2 defines a map Σ −4b−1 B → W 2 , which we complete to a triangle We have

Continuing this way, we construct objects
The second to last stage of this process is a module W ℓ−1 with Then something different happens.The map Σ ), and the process cannot be continued any further.The composite is equivalent to the quotient of B by the ideal (x h , t).What we have seen is the following.

Indecomposables objects in D sg (B)
In this section, we discuss the indecomposable objects which we eventually wish to show form a complete list in D sg (B).They come with an internal grading, which we keep track of.The ideal in B generated by x h and t is a bigraded A ∞ ideal, and the quotient In other words, the only non-zero operation m i on the quotient is the multiplication m 2 .We begin with a discussion of B-modules.The indecomposable modules are the shifts of the quotients Mi = k[x]/(x i ) for 1 i h of B. For i < j we have short exact sequences of B-modules We also have almost split sequences of B-modules Pulling back the k[x]/(x h )-module Mi (1 i h) to B, we obtain a B-module M i and a map B → M i .In particular, M 1 is the field object, with H * M 1 ∼ = k.We shall write k for the B-module M 1 .By Theorem 12.1, M h is in the subcategory Thick(B) of D b (B), and therefore vanishes in D sg (B).We write X i for the fibre of B → M i .Applying H * to the triangle X i → B → M i , we see that H * X i is the ideal generated by t and x i in H * B = k[x] ⊗ Λ(t).In D sg (B), we have ΣX i ≃ M i and X h ≃ 0.
The exact sequence (13.1) of B-modules gives rise to triangles in D b (B) In particular, taking j = h, we obtain the following, critical in linking odd and even suspensions.
Proposition 13.2.There is an equivalence Similarly, the almost split sequences of B-modules give rise to triangles in D b (B) Bearing in mind that X h is zero in D sg (B), the last of these becomes We shall eventually see that these are almost split triangles in D sg (B).
Next, we compute the spectral sequence (7.1) We first assume that ℓ > 2, and later describe the modifications necessary for the case ℓ = 2.
Resolving H * X i as a module over H * B, we obtain the sequence Thus H * X i is a maximal Cohen-Macaulay H * B-module corresponding to the following matrix factorisation of the relation t 2 = 0.
If X i and X j are B-modules of this form, we take homomorphisms from the resolution of H * X i to H * X j to obtain The differential sends The case j i: The kernel is generated by ( 0 t ) and .
The image contains t times these and x i times these.We write α for the element of the kernel represented by the first of these elements ( 0 t ), and β for the element represented by the second tx j−i x j . Thus as a module over H * B, the degree zero homology of the above complex is Hom H * B (H * X i , H * X j ), and is generated by α and β with relation x j α = tβ.Recalling that H * (A) = Hom D b (B) (X 1 , X 1 ), the periodicity element of degree (−1, 2b + 1, h) in the above resolution represents the element τ ∈ H 2b A, so we shall call it τ by abuse of notation.As a module over H * B[τ ] we have Ext * , * H * B (H * X i , H * X j ) = α, β /(tα, x j α − tβ, tτ β, x i τ β) with α in degree (0, 2ja − 2b − 1, jℓ − h) and β in degree (0, 0, 0).
(ii) The case i j: The kernel is generated by ( 0 t ) and ( t 0 ) + x i−j 0 x j = t x i .The image contains t times these, and x j times these.
(iii) The case i = j.The cases i = j in (i) and (ii) coincide, but there is one extra piece of structure to determine, namely the ring structure.This is determined by the statements that β is the identity endomorphism of X i , and α is the endomorphism sending x i to t and t to zero.So the extra relations are α 2 = 0 and β = 1.Thus x i α = t, so t is a redundant generator, and the presentation becomes , α]/(α 2 , x i τ ).For the computations above, we assumed that ℓ > 2. We now explain the modifications necessary for the case ℓ = 2.In this case, the matrix in the resolution of H * X i is This corresponds to the following matrix factorisation of the relation t 2 + x h = 0.
Taking homomorphisms into H * X j , the matrix for the differential is the transpose, t x h−i There are more cases this time, but we content ourselves with computing Ext * , * H * B (X i , X j ) in the case i = j h/2, which is all we need.In this case, the kernel is generated by the element α representing −x h−i t and β representing t x i .Again β is the identity endomorphism of X i and α is the endomorphism sending x i to t and t to −x h−i , so α 2 is multiplication by −x h−2i .This time the presentation becomes , and λ some scalar in k.
Proof.First assume that ℓ > 2. We have to show that there are no non-zero differentials in the spectral sequence Section 7) whose E 2 page is given by (13.5), and we then have to address the ungrading problem for the E ∞ term.This is easier if we take into account the internal degrees, which have to be preserved by the differentials.So elements in the spectral sequence are triply graded, with |x| As in the proof of Theorem 5.5, the possible tridegrees (u, v, w) at the E 2 -term are very restricted.This time they lie in two parallel planes.We use the same normal vector N = (ℓ − 2, ℓ, −2a), and consider the dot products N 2, we deduce that either d n = 0 or (2n − ℓ) + (2 − ℓ) = 0, so n = ℓ − 1.In the latter case, d ℓ−1 sends x and τ to zero, and α to an element of degree (−ℓ + 1, 2ia − 2b + ℓ − 3, iℓ − h).There is only one such monomial in x and τ , namely x h−i τ ℓ−1 .The assumption that 2i h implies that this is zero, since x i τ = 0.
For the ungrading problem, since α 2 has even degree it cannot involve αx i 1 τ i 2 .If α 2 has the same bidegree as x i 1 τ i 2 then equating bidegree and solving, we find that i The element x i τ has degree (−1, 2b − 1 − 2ia, h − iℓ).There are no non-zero monomials in E ∞ with this internal degree, so x i τ ungrades to zero.
Finally, in the case ℓ = 2, B and X i are formal, and the E 2 page (13.6) of the spectral sequence computes End D b (B) (X i ) with no non-zero differentials or ungrading problems.
Remark 13.8.If ℓ > 2 and i h/3 then the element x h−2i τ ℓ−2 is zero, so the value of λ only matters when h/3 < i h/2.Using the models we produce in Section 16, it turns out that λ = 1 always holds, as we saw above in the case ℓ = 2.We shall not need this information in what follows.
Theorem 13.9.We have ).This has finite length over K, and in particular, it is an Artinian local graded ring.The (homological) degree zero part is k[x |b| τ |a| ]; note that x |b| τ |a| is nilpotent, and often zero.As a module over End Dsg(B) (X i ), Hom Dsg(B) (X i , X j ) has finite length.The Krull-Schmidt theorem holds for finite direct sums of shifts of the objects X i .

Proof. By Corollary 9.6 we have End
h it follows from Theorem 13.7 that this is isomorphic to K[x, α]/(α 2 + λx 2h−i τ ℓ−2 , x i ).The structure for 2i h then follows from the isomorphism Next we compute the homological degree zero part.The degree of α is 2ia − 2b − 1, which is odd, while the degrees of x and τ are even.If x j τ m has degree zero then ja = mb.Since a and b are coprime, this implies that j is divisible by b, m is divisible by a, and the element is a power of x |b| τ |a| (note that ah − bℓ = 1, and h and ℓ are positive, a and b have the same sign).
The E 2 page of the spectral sequence (13.4) is finitely generated over k[τ ], and therefore The final statement about the Krull-Schmidt theorem follows from the fact that the endomorphism rings of the X i are local rings.
Corollary 13.10.The socle of the homological degree 2a − 1 part of End Dsg(B) (X i ), as a module over the degree zero part, is spanned by the element αx i−1 τ in degree (2a − 1, ℓ).None of the other monomials of homological degree 2a − 1 has internal degree ℓ.
Proof.Using Theorem 13.9, the monomials of homological degree 2a Corollary 13.11.The socles of the homological degree 2a − 1 parts of End Dsg(B) (X i ) for 1 i h are represented by the triangles (13.3) coming from the almost split sequences of B-modules.
Proof.The connecting homomorphism for the triangle (13.3) for X i is a non-zero element of degree (2a − 1, ℓ) in End Dsg(B) (X i ).By Corollary 13.10, it is therefore a non-zero multiple of αx i−1 τ .
Remark 13.12.The images in D b (A) of the objects X i of D b (B) are, up to shifts, the analogues of the modules W i described in Section 12.
Write Y i = Hom B (k, X i ) for the image of X i in D b (A).Then Y i is a free k[τ ]-module on two generators, u i of degree 2a − 1 and v i in degree −2(i − 1)a.The action of ξ is given by and the remaining m j vanish.

Auslander-Reiten triangles
For background on Auslander-Reiten triangles in triangulated categories, see Happel [26,27].We shall construct them in the category D sg (B) and use them to classify the indecomposables.Although we make use of the internal grading to identify these triangles, the grading does not interfere with their existence and use for classification.
Suppose that X is an indecomposable, internally gradable object in D sg (B) with local endomorphism ring End Dsg(B) (X).A map from another, not necessarily internally gradable object Y to X has a right inverse (i.e., induces an isomorphism from a direct summand of Y to X) if and only if the induced map Hom Dsg(B) (X, Y ) → End Dsg(B) (X) is surjective.
Using Theorem 10.8, the dual Hom Dsg(B) (X, Σ 1−2a X) has a simple socle as a module over End Dsg(B) (X), and the socle is a map from X to Σ 1−2a,−ℓ X.For the objects X i , this socle is identified in Corollary 13.10.Choose a non-zero morphism α X : X → Σ 1−2a,−ℓ X in this socle.This has the property that a map Y → X has a right inverse if and only if the composite ).This then has the following lifting property.If a map Y → X does not have a right inverse, then it lifts to a map Y → Z.
Y These are the defining properties of an Auslander-Reiten triangle, sometimes also called an almost split triangle.
Proof.This follows from Corollary 13.11.
If Z ′ is a direct summand of Z then the composite Z ′ → Z → X is called an irreducible morphism.It has the property that it is not an isomorphism, but for any factorisation Z ′ → U → X, either Z ′ → U has a left inverse or U → X has a right inverse.This property is symmetric, so that if Z ′ is a direct summand of Z then the composite Σ −2a X → Z → Z ′ is also an irreducible morphism.
The Auslander-Reiten quiver is the quiver (directed graph) whose vertices correspond to the isomorphism classes of indecomposable objects and whose directed edges correspond to the irreducible morphisms.This comes with a translation Σ −2a with the property that there is an arrow from Z to X if and only if there is an arrow from Σ −2a X to Z.
For the objects Σ j X i , the Auslander-Reiten quiver has the following form. . . .This wraps round to form a cylinder, since Σ 2b X i is isomorphic to X i via τ .Since a and b are coprime, the circumference of the cylinder is b.The height is h − 1, and X h−1 ∼ = Σ 2a−1 X 1 is in the bottom row.This gives a total of b(h − 1) isomorphism classes of objects.In the usual language of translation quivers, we therefore have the following.
Theorem 14.2.The objects Σ j X i in D sg (B) form a connected component of the Auslander-Reiten quiver, isomorphic to ZA h−1 /T |b| , where T is the translation functor.
is even there is just one of them with length |b|, namely the middle one, and the rest have length 2|b|.

Classification of indecomposables
The goal of this section is to show that every indecomposable object in D sg (B) is isomorphic to some Σ j X i , so that the Auslander-Reiten quiver is that given in Theorem 14.2.
The following proposition plays a role analogous to that of the Harada-Sai lemma in the representation theory of finite dimensional algebras, see for example Lemma 4.14.1 of [8].
Proposition 15.1.The composite of any h composable irreducible morphisms between the objects Σ j X i is equal to zero.
Proof.For each i and j, the sum of the composites Σ j−2a X i → Σ j X i+1 → Σ j X i and Σ j−2a X i → Σ j−2a X i−1 → Σ j X i is the composite of two adjacent maps in an Auslander-Reiten triangle, and therefore equal to zero.Using these relations, any composite of h composable irreducible morphisms can be rewritten as a composite involving Σ j−2a X 1 → Σ j X 2 → Σ j X 1 , which is zero.In other words, we can deform any path of length h so that it hits the top edge of the cylinder, without moving the ends of the path.Proposition 15.2.If X is any non-zero object in D sg (B) then for some n ∈ Z there is a non-zero morphism Σ n X 1 → X.
Proof.By Corollary 9.6, the image of if there is no non-zero morphism from any Σ n X 1 to X then H * Y = 0, so Y is quasi-isomorphic to zero in D b (A[τ −1 ]) and hence X is quasi-isomorphic to zero in D sg (B).
Theorem 15.3.Every indecomposable object in D sg (B) is isomorphic to some Σ j X i with 1 i < h, 0 j < |b|.
Proof.First we note that the set of Σ j X i with 1 i < h and 0 j < b is the set of vertices in the Auslander-Reiten quiver described in Section 14.
Let X be an indecomposable object.Then by Proposition 15.2 there is a non-zero morphism Σ n X 1 → X for some n ∈ Z. Since X is indecomposable, if this has a left inverse then it is an isomorphism, and we are done.Otherwise, it factors as Σ n X 1 → Σ n+2a X 2 → X.If Σ n+2a X 2 → X has a left inverse, again we are done.Otherwise, we obtain a factorisation not an isomorphism, then at the next stage, we obtain a statement that a sum of two composites is non-zero, so at least one of them has to be non-zero.Continuing this way, we obtain either an isomorphism Σ j X i → X for some i, j, or a factorisation through a composite of at least h irreducible morphisms between the objects X i .In the latter case, by Proposition 15.1, it follows that Σ n X 1 → X is the zero map, contradicting the way it was chosen.Proof.This follows from the last statement of Theorem 13.9 together with Theorem 15.3.
Remark 15.6.In Remark 11.1 we noted that for the formal graded rings H * (B) the objects X s exist for all s 1 and are inequivalent.The singularity category retains the τ -periodicity, since D sg (H * (B)) ≃ D csg (H * (A)) ≃ D b (H * (A)[1/τ ]), but now (in the absence of Proposition 13.2) the Auslander-Reiten quiver consists of two semi-infinite cylinders of circumference |b|, one containing the even suspensions of the X i and one containing the odd suspensions of the X i .16. Models for D sg (B) In this section we exhibit some more familiar looking categories that are equivalent as triangulated categories to D sg (B).
Theorem 15.3 gives us a presentation for the category D sg (B) in terms of the mesh category of the quiver.The definition of the mesh category k(Γ) of a translation quiver Γ comes from Riedtmann [40] (see also Bongartz and Gabriel [13]), and can be found in Section I.5.6 on pages 54-55 of Happel [27].
In the case of the Auslander-Reiten quiver ZA h−1 /T |b| of Corollary 15.4, the generators for the morphisms in the mesh category are the irreducible morphisms between the Σ j X i .The mesh relations come from the Auslander-Reiten triangles, and say that for each i and j, the sum of the composites Σ j−2a X i → Σ j X i+1 → Σ j X i and Σ j−2a X i → Σ j−2a X i−1 → Σ j X i is equal to zero.At the boundaries, i = 1 and i = h − 1, only one of these composites makes sense, and the corresponding relation says that this composite is equal to zero.
The classification of Krull-Schmidt triangulated categories with finitely many isomorphism classes of indecomposables is described in Section 6 of Amiot [4], see also Chapter 2 of Amiot [5] and the paper of Xiao and Zhu [43].Let Γ be the translation quiver ZA h−1 /T |b| .Then it is shown in Theorem 6.5 of [4] that given a triangulated category T with Auslander-Reiten quiver Γ, we have an equivalence of k-linear categories between the full subcategory ind (T) of indecomposables in T and the mesh category k(Γ).This induces an equivalence between T and the additive closure of k(Γ).Applying this to D sg (B), we obtain a k-linear equivalence k(ZA h−1 /T |b| ) → ind D sg (B), the full subcategory of indecomposables, which then extends to an equivalence from the additive closure of k(ZA h−1 /T |b| ) to D sg (B).
There is another triangulated category with the same Auslander-Reiten quiver.Let D b (kA h−1 ) be the bounded derived category of modules for the quiver A h−1 over k.The Auslander-Reiten quiver of D b (kA h−1 ) is the quiver ZA h−1 .The translation T of this quiver lifts to the translation T of D b (kA h−1 ).It is shown in Keller [33]  is a triangle functor.The Auslander-Reiten quiver of the orbit category D b (kA h−1 )/T |b| is isomorphic to ZA h−1 /T |b| .So Theorem 6.5 of [4] shows that there is a k-linear equivalence D sg (B) ≃ D b (kA h−1 )/T |b| inducing the isomorphism of Auslander-Reiten quivers.We would like to know that this is an equivalence of triangulated categories.This proves to be more delicate, but another theorem of Amiot comes to our rescue.Proof.We would like to apply Amiot [4,Theorem 7.2].This states that given a finite triangulated category T which is connected, algebraic, and standard, there exists a Dynkin diagram Γ of type A, D or E, and a self-equivalence Φ of D b (kΓ), such that T is triangle equivalent to Keller's orbit category D b (kΓ)/Φ.To apply the theorem, we need to check the conditions.
To say that T is connected means that the Auslander-Reiten quiver is connected, so we have already established that D sg (B) is connected.To say that T is standard means that it is equivalent to a mesh category as a k-linear category, so we have also already established that D sg (B) is standard.
To say that T is algebraic means that there is a Frobenius category E such that T is triangle equivalent to the stable category E, see Keller [35,Section 3.6].By Proposition 8.3 and Corollary 9.5 we have D We have therefore checked the conditions for applying the theorem of Amiot.Since the Auslander-Reiten quiver of D sg (B) is ZA h−1 /T |b| , we have Γ = A h−1 and Φ = T |b| , and the theorem now follows.
There is another category that looks very similar, and we apply similar techniques to make the comparison.We write B for the formal A ∞ algebra k[x]/(x h ) where |x| = −2a.Thus there is an obvious map B → B sending x to x and t to zero.We consider the bounded derived category D b ( B) and its quotient, the singularity category D sg ( B) formed by quotienting out all objects finitely built from the ring.We have objects Mi = k[x]/(x i ) in this category for 1 i h, and Mh is zero.The analogue of Corollary 10.7 in this situation says that Hom Dsg( B) (X, Y ) is the graded vector space dual of Hom Dsg( B) (Y, Σ 1−2a X), and hence we have Auslander-Reiten triangles However, in contrast to the situation for D sg (B), the periodicity is given by Mi Mi , and 2ha − 2 = 2ℓb.The Auslander-Reiten quiver again consists of the Σ j Mi .It is in the form of a cylinder, and again the height of the cylinder is h − 1, but the circumference is ℓ|b| instead of |b|.The generators and relations for this category are given in the same way as that of D sg (B) in terms of the irreducible morphisms and the Auslander-Reiten triangles.
We put all these equivalences together in the following theorem.
Theorem 16.4.We have equivalences of triangulated categories Each of these is a finite Krull-Schmidt category with |b|(h − In this section, we apply our main results to the A ∞ algebras H * BG and H * ΩBG ∧ p for G a finite group with cyclic Sylow p-subgroups, discussed in [10]. We are interested in the following occurrences of the A ∞ algebra A of Section 4. Let p be an odd prime and k a field of characteristic p.Let G be a finite group with cyclic Sylow subgroup P of order p n and inertial index q = |N G (P ):C G (P )| > 1.Let C * BG be the cochains on the classifying space BG with coefficients in k, and C * ΩBG ∧ p be the chains on the p-completed loop space of BG, again with coefficients in k.
The In the case of A = H * ΩBG ∧ p , we have a = q, b = q − 1, h = p n − (p n − 1)/q and ℓ = p n , and we must assume that q > 1.In the case of A = H * BG (homologically graded in negative degrees), the roles are reversed and we have a = −(q − 1), b = −q, h = p n , and ℓ = p n − (p n − 1)/q.

Brauer trees and Hecke algebras
In this section, we describe what our main theorem tells us about Brauer tree algebras.In general, a Brauer tree algebra is described by a planar embedding of a tree with e edges corresponding to the simple modules.The vertices are assigned multiplicities, which are all equal to one with the possible exception of a single vertex of multiplicity λ > 1; otherwise we set λ = 1.This parameter λ is called the exceptional multiplicity, even when it equals one.These data are sufficient to describe the algebra up to Morita equivalence, and an algorithm for computing projective resolutions was described by Green [22].Brauer tree algebras were first introduced in order to describe blocks of defect one in the representation theory of finite groups by Brauer [14], and the analysis was extended to all blocks of cyclic defect by Dade [17].A nice treatment in this context may be found in the book of Alperin [3].They also appear in many other contexts in representation theory, and we shall give an example of this in characteristic zero below.
We say that a simple module is a leaf module if it corresponds to an edge one end of which has valency one and multiplicity one.The leaf modules are all syzygies of each other, so they have isomorphic Ext algebras.The singularity category of Ext * A (M, M) has finite representation type, with λ(e − 1) 2 isomorphism classes of indecomposable objects, in [(λ(e − 1) + 1)/2] orbits of the shift functor Σ.The Auslander-Reiten quiver is isomorphic to ZA λ(e−1) /T e−1 , where T is the translation functor Σ −2e .
Proof.It was shown by Gabriel and Riedtmann [19] that all Brauer tree algebras with e edges and exceptional multiplicity λ are stably equivalent (indeed, even more is true: Rickard [39], see also Aihara [2], showed that these are all derived equivalent, but we don't need to go that far).In particular, they are all stably equivalent to the Brauer star algebra which has e vertices of valency one and multiplicity one, surrounding the one remaining vertex in the middle, which has multiplicity λ.The Brauer star algebras have exactly the same presentation as the algebras in Section 2 of [10], except for the change of characteristic.In more detail, the projective indecomposables are uniserial, and the radical filtration is isomorphic to its associated graded.Some readers may find it helpful to refer to the paper Bogdanic [12] which recalls details of Brauer tree algebras and shows that the stable grading comes from a grading of the Brauer tree algebras themselves.
Under such a stable equivalence, the leaf modules correspond to simple modules or first syzygies of simple modules, for the corresponding Brauer star algebra.It follows that we may compute the A ∞ structure on these Ext algebras using exactly the same computation as in [10], and the result is as stated in the theorem.
Theorem 1.2 now describes the singularity and cosingularity categories.
Remark 18.2.In the remaining case e = 1, a Brauer tree algebra is Morita equivalent to a truncated polynomial algebra, so the A ∞ structure on the Ext ring of the simple module is just like that of a cyclic p-group.
As an example, let H = H(n, q) be the Hecke algebra of the symmetric group of degree n over a field k of characteristic zero, where q is a primitive ℓth root of unity with ℓ 2. This has generators T 1 , . . ., T n−1 satisfying braid relations together with the relations (T i + 1)(T i − q) = 0.The cohomology H * (H, k) = Ext * H (k, k) was computed by Benson, Erdmann and Mikaelian [9].
In the case where n = ℓ > 2, H * (H, k) has the form k[x] ⊗ Λ(t) where |x| = −2n + 2 and |t| = −2n + 3.This corresponds to the fact that in this case the principal block of H is a Brauer tree algebra for a tree which is a straight line with n vertices, n − 1 edges, and λ = 1, with the trivial module k at one end as a leaf module.In particular, we can apply Theorem 18.1 in this context.So the Massey product of n copies of t is equal to −x n−1 , and writing a = n − 1, b = n − 2, ℓ = n, h = n − 1, we have Ext * H (k, k) = B as an A ∞ algebra.Theorem 18.3.Let H = H(n, q) be the Hecke algebra of the symmetric group of degree n over a field k of characteristic zero, where q is a primitive ℓth root of unity.In the case n = ℓ > 2, the singularity category of the A ∞ algebra Ext Other examples of Hecke algebras described by Brauer trees may be found in Geck [20], Ariki [6].

Corollary 5 . 4 .Theorem 5 . 5 .
We have HH * Q ∼ = HH * A ∼ = HH * B. Suppose that h > 2 and ℓ > 2. In the spectral sequence HH * H * B ⇒ HH * B the E 2 page is given by and N • |τ | = 0.So the only possible values of N • (u, v, w) on the E 2 page are 0, 2 − ℓ, and 4 − 2ℓ.Furthermore, the elements with N • (u, v, w) = 4 − 2ℓ are multiples of tξ.The differential d n decreases u by n, increases v by n − 1, and leaves w unchanged.It therefore increases N • (u, v, w) by 2n − ℓ.

Proposition 6 . 2 .
The bounded derived category D b (Q) is equivalent to D b (A).

Definition 9 . 1 .Lemma 9 . 2 .
Let a be an A ∞ algebra with Noetherian homology.(i) The singularity category D sg (a) is the quotient of D b (a) by Thick(a).(ii) The cosingularity category D csg (a) is the quotient of D b (a) by Thick(k).Suppose that a is an A ∞ algebra such that H * a is local with residue field k.If M is an a-module such that H * M has finite length, then M is in Thick(k).Proof.A copy of k of lowest degree in H * M lifts to a map k → M. Completing to a triangle k → M → N, the length of H * N is one less than the length of H * M.So by induction on length of H * M, it follows that M is in Thick(k).

Theorem 9 . 4 .
The functor Hom A (k, −) induces a triangulated equivalence of derived categories D b (A) ∼ − → D b (B), that sends A to k and k to B. It induces triangulated equivalences

Corollary 9 . 6 .
If X and Y are objects in D b (B) then

Proof.
Consider the element x ∈ B. For any B-module M with H * M finitely generated, the fibre F of x : M → Σ −2a M has the property that H * F has finite length.So by Lemma 9.2, F is in Thick(k).Now under the equivalence D b (B) ≃ D b (A), the image of k is A. So regarding x as an element of HH * B ∼ = HH * A, we have an action of x on D b (A), and for any A-module N in D b (A), the fibre of x : N → Σ −2a N is in Thick(A).So we have a triangle F → N x − → Σ −2a N with F in Thick(A).Inverting τ , we have such a triangle in D b (A[τ −1 ]) with F in Thick(A[τ −1 ]).Now by Corollary 5.7, we have x h τ ℓ = 0 in HH and these are the only possible values of N • (u, v, w) for degrees of elements at E 2 .The differential d n decreases u by n, increases v by n − 1, and leaves w unchanged.It therefore increases N • (u, v, w) by 2n − ℓ.Since n
that the orbit category D b (kA h−1 )/T |b| , whose Hom sets are by definition n∈Z Hom D b (kA h−1 ) (X, T n|b| (Y )), is a triangulated category in such a way that the canonical functor D b (kA h−1 ) → D b (kA h−1 )/T |b|

Theorem 16 . 2 .
There is a triangulated equivalence D sg ( B) ≃ D b (kA h−1 )/T ℓ|b| .Proof.This is proved in the same way as Theorem 16.1, using Amiot [4, Theorem 7.2].The functor D sg ( B) → D sg (B) sends Mi to ΣX i and irreducible morphisms to irreducible morphisms.It wraps the Auslander-Reiten quiver of D sg ( B) around that of D sg (B) exactly ℓ times.Thus it corresponds to the functor on orbit categories D b (kA h−1 )/T ℓ|b| → D b (kA h−1 )/T |b| .There is another way to achieve this wrapping around.Namely, instead of considering differential Z-graded modules for B, we consider differential Z/2|b|-graded modules.Let us write D b ( B, Z/2|b|) for this bounded derived category D sg ( B, Z/2b) for the corresponding singularity category.Theorem 16.3.There is a triangulated equivalence D sg ( B, Z/2|b|) ≃ D b (kA h−1 )/T |b| .There is an equivalence of categories D sg ( B, Z/2|b|) → D b (B) making the following diagram commute.D sg ( B) / / D sg (B) A ∞ algebra structures on H * BG and on H * ΩBG ∧ p coming from the DG algebras C * BG and C * ΩBG ∧ p are described in [10].They are Koszul dual, and we can apply the results of this paper either with A = H * BG, B = H * ΩBG ∧ p , or with A = H * ΩBG ∧ p and B = H * BG.

Proof of Theorem 1 . 2 .
By Proposition 6.2, the bounded derived categories of the DGA algebras C * BG and C * ΩBG ∧ p are equivalent to those of the A ∞ algebras H * BG and H * ΩBG ∧ p .The equivalences of categories follow by applying Theorem 9.4 to the A ∞ algebras H * BG and H * ΩBG ∧ p .The classification of the indecomposable objects in these categories follow from Theorem 16.4.

Theorem 18 . 1 .
Let A be a Brauer tree algebra with e > 1 edges and exceptional multiplicity λ, and let M be a simple leaf module for A. Then the A ∞ algebra Ext * A (M, M) is the algebra B described in Section 4, with parameters a = e, b = e − 1, ℓ = λe + 1 and h = ℓ − λ = λ(e − 1) + 1.
* H (k, k) has finite representation type, with (n − 2) 2 isomorphism classes of indecomposables, in [(n − 1)/2] orbits of the shift functor.The Auslander-Reiten quiver is a cylinder of height n − 2 and circumference n − 2.The cosingularity category of the A ∞ algebra Ext * H (k, k) also has finite representation type, with (n − 1) 2 isomorphism classes of indecomposables, in [n/2] orbits of the shift functor.The Auslander-Reiten quiver is a cylinder of height n − 1 and circumference n − 1. 9