A faithful braid group action on the stable category of tricomplexes

Bicomplexes of vector spaces frequently appear throughout algebra and geometry. In Section 2 we explain how to think about the arrows in the spectral sequence of a bicomplex via its indecomposable summands. Polycomplexes seem to be much more rare. In Section 3 of this paper we rethink a well-known faithful categorical braid group action via an action on the stable category of tricomplexes.


Introduction
"The impact of spectral sequences on algebraic topology was tremendous: Many major problems of topology, both solved and unsolved, became exercises for students..." -A. Fomenko and D. Fuchs [FF, Preface] Representation theory, which has been established for over a century, deals with linear actions of groups and algebras. Much more recent is the discovery of interesting categorical actions of group, primarily discrete groups. In these examples discrete groups act by symmetries of categories, which in many cases are triangulated, and the action preserves the triangular structure. One of the first nontrivial examples appeared in [KhS], see also [KH, ST]. There the n-strand braid group Br n acts on the homotopy category of complexes of modules over a particular finitedimensional algebra A n−1 . The action is by exact functors and on the Grothendieck group the action descends either to the Burau representation of the braid group (if one keep tracks of an additional grading on modules, in addition to the homological grading) or to the reduced permutation action of the symmetric group. Neither of these linear actions is faithful, but its categorical lifting was shown to be faithful in [KhS].
Algebra A n−1 (the zigzag algebra) is the quotient of the path algebra of the quiver with n − 1 vertices and edges connecting adjacent vertices in both directions (assuming n > 3, with minor changes necessary for n = 2, 3). (1.1) Generators of A n−1 corresponding to arrows in the quiver are denoted (i|i ± 1). The defining relations (i|i + 1|i + 2) = 0, (i|i − 1|i − 2) = 0, (i|i − 1|i) = (i|i + 1|i) (for i's for which both sides of a relation make sense) are quadratic, A n−1 is finite dimensional, with a basis made of idempotents (i), edges (i, i ± 1) and length two paths (i|i ± 1|i). For 1 < i < n indecomposable projective A n−1 module P i = A n−1 (i) is four-dimensional, with the basis {(i), (i − 1|i), (i + 1|i), (i|i − 1|i)} and can be visualized as a diamond.
These are exactly the relations on the two differentials in a bicomplex. A bicomplex is built out of vector spaces placed in the vertices of an integral lattice 2 , with the differentials going along the two coordinates, with the unit step each. One can introduce a grading on A n−1 by making, for instance, left-pointing arrows (edges) in the quiver to have degree one and right-pointing edges degree zero. The unit element of A n−1 decomposes as the sum of n − 1 idempotents, one for each vertex of the graph, 1 = (1) + (2) + · · · + (n − 1), inducing the decomposition of an A n−1 -module into a sum of vector spaces and the additional grading on M leads to the bigrading, with the left and right directed edges changing the bigrading by (1, 0) and (0, 1), respectively.
In this way, graded A n−1 -modules may be identified with bicomplexes with nonzero terms restricted to a suitable area of the lattice 2 . Changing the indexing of quiver vertices from {1, 2, . . . , n − 1} to by passing to the infinite in both directions quiver (see figure in equation (2.3)) results in a non-unital algebra A ∞ with a system of idempotents {(i)} i∈ such that graded A ∞ -modules naturally correspond to bicomplexes.
The braid group Br n acts on the homotopy category of (either graded or ungraded) A n−1modules by tensoring with a suitable complex of A n−1 -bimodules. This works as well in the limit of A ∞ -modules, with the braid group Br with strands (and generators σ i ) enumerated by integers.
Passing from modules over an algebra B to complexes of modules means working with suitably graded modules over the algebra B[∂]/(∂ 2 ). In our case, graded A n−1 or A ∞ modules can be identified with bicomplexes (more precisely, there is an equivalence of corresponding abelian categories). Consequently, complexes of A n−1 and A ∞ -modules may be identified with tricomplexes, with the homological grading in A n−1 [∂]/(∂ 2 ) corresponding to the additional, third, grading in tricomplexes.
Passing from complexes to the homotopy category of complexes (of modules over an algebra B) means modding out by null-homotopic morphisms. If one restricts to complexes of projective B-modules, which is a common and important subcategory of the category of complexes, this means killing morphisms which factor through a direct sum of objects of the form in various homological degrees. Specializing B to A ∞ , the above complex decomposes as a direct sum of terms of the form for various i ∈ . If keeping track of the additional grading, one can further shift these copies of A ∞ (i) and parametrize them by a pair of integers (i, j). Together with the homological grading k, one gets a 3-parameter family of possible indecomposable summands that each represent the zero complex in the homotopy category. If this picture is converted to the language of bicomplexes and tricomplexes, the module A ∞ (i) corresponds to a free rank one bicomplex in the bigrading associated to the idempotent (i) and grading j. The complex (1.2) will correspond to the free rank one tricomplex, placed in the suitable tridegree.
In the homotopy category of projective graded A ∞ -modules, a morphism is zero if it factors through the object which is a direct sum of complexes (1.2) over various i, j, and k, where i labels the idempotent, j is the additional grading parameter in A ∞ , and k is the homological grading. Converting this to tricomplexes, one unites the three integer grading parameters i, j, k of different origins into a single trigrading on tricomplexes. The complex (1.2) becomes a free tricomplex of rank one that can sit in any position relative to the trigrading. Killing morphisms that factor through sums of such free rank one tricomplexes is equivalent to the condition that one is working in the stable category of tricomplexes, that is in the category of tricomplexes modulo the ideal of morphisms that factor through a free tricomplex.
Tricomplexes can be described as trigraded modules over the algebra Λ 3 with generators ∂ 1 , ∂ 2 , ∂ 3 and relations ∂ 2 i = 0, i = 1, 2, 3, ∂ i ∂ j = ∂ j ∂ i , i = j. This 8-dimensional algebra is Frobenius, and it is even a Hopf algebra in the category of supervector spaces. Consequently, its stable category of trigraded modules is triangulated (and monoidal, due to the Hopf algebra structure).
The braid group action on the homotopy category of A n−1 and A ∞ -modules transfers to the stable category of tricomplexes. Note that these two categories are not equivalent, but rather admit equivalent subcategories, which, on the A n−1 side, is the homotopy category of complexes of projective modules. The braid group action respects these subcategories and the equivalence between them.
The braid group acts by exact functors on this triangulated category of tricomplexes. The actions does not respect the monoidal structure, though, and choosing the action requires singling out one differential out of three. Choosing different differentials gives three commuting braid group actions.
For now, we view this example as a curiosity. One natural question is whether our example fits into the more general framework of Hopfological algebra [Kh3,Qi], where stable categories of modules over Hopf algebras, such as Λ 3 , are used as base categories for new constructions of categorifications (see, e.g. [KQ]) or, perhaps, some other algebro-geometric structures. Another open problem is whether homotopy categories of complexes over other algebras of importance in categorification, such as arc algebras [Kh2], can be rethought through some generalization of the stable category of tricomplexes.
Tricomplexes seem to appear exceedingly rarely in mathematics. Currently, they have made appearances in the BRST theory [Sh], in the deformation theory of Hopf algebras [Ya], and in the algebraic K-theory [Ca]. A modified notion of a tricomplex, called quasi-tricomplex, occurs in the theory of variation [Ol].
The braid group action on the stable category of tricomplexes is constructed in Section 3 of this paper. In Section 2 we explain a way to think about arrows in the spectral sequence of a bicomplex of vector spaces via indecomposable modules over the rings A n−1 and A ∞ . This relation was independently discovered by Stelzig [St].
The first author learned algebraic topology for the first time from the Russian classic by Dmitry Fuchs and Anatolii Fomenko [FF] (in its first edition named Homotopic Topology), while the second author enjoyed teaching graduate courses out of this reprinted classic at his previous work institution. It is our great pleasure to dedicate this short note to Dmitry Fuchs on the occasion of his eightieth anniversary.
"The subject of spectral sequences is elementary, but the notion of the spectral sequence of a double complex involves so many objects and indices that it seems at first repulsive." -D. Eisenbud [Ei,Appendix 3.13] The standard textbook approach to spectral sequences makes them seem sophisticated and mysterious gadgets [CE, GM, Mc, Wl] and [Ei,Appendix 3.13]. Timothy Chow, in the introduction to his article on spectral sequences [Ch], quotes the opinions of experts who, essentially, say that the definition is so complicated that you just have to get used to it.
The goal of this section is to explain spectral sequences, restricted to bicomplexes of vector spaces, in a simple and straightforward way. Most of this section has appeared in lectures to graduate students by the first author, see for instance the informal lecture notes [Kh]. Similar results also appeared in Stelzig [St]. We warn the reader that this elementary approach works only for a bicomplex of vector spaces. Bicomplexes and filtered complexes that appear in spectral sequences in algebraic topology carry an enormous amount of extra structure, such as an action of the Steenrod algebra when working over /p, and cannot be easily understood in this elementary way. The complexity and beauty of these structures are captured in the Fomenko and Fuchs classic [FF] and other books, see McCleary [Mc].

Cohomology
Let k be a field and M · · · a complex of k-vector spaces. We allow unbounded complexes and infinite-dimensional vector spaces. It is easy to see that M decomposes into the direct sum of length zero complexes with a vector space H i in degree i, and length one complexes with two copies of a vector space W i in degrees i and i + 1. Thus, and hence W i are infinite-dimensional. The bulk of the complex Ω(X) is occupied by contractible "junk," while the "valuable part" (cohomology) has small size. If we equip X with a Riemannian metric g, the operator d * = ± * d * adjoint to d gives rise to the Laplace operator The Laplace operator provides a canonical embedding of each complex 0−→H i (X, Ê)−→0 into the complex (Ω(X), d), via the isomorphism H(X, Ê) ∼ = ker(∆).
A complex of k-vector spaces is the same as a graded module over the exterior k-algebra Λ 1 on one generator d of degree 1: The i-th homogeneous piece of a graded Λ 1 -module M is a vector space M i , and the action of d is exactly the differential d : The category M 1 of graded modules over Λ 1 is Krull-Schmidt, and any module (even infinitedimensional) decomposes into a direct sum of indecomposable modules S i and P i . Here S i is the one-dimensional k-vector space placed in degree i, and corresponds to the complex 0−→k−→0. The differential acts by 0 and the module S i is simple. The module P i = Λ 1 {i} is free and corresponds to the complex 0−→k and the cohomology of M only catches the first terms in the sum. Recall that an object M of an additive category is called indecomposable if M is not isomorphic to a direct sum N 1 ⊕ N 2 with both N 1 , N 2 nontrivial.

2.2 Bicomplexes
Let us now move on to bicomplexes. A bicomplex M over a field k is a family {M i,j } of vector spaces, for i, j ∈ , and maps Let Λ 2 be the exterior k-algebra on two generators ∂ 1 , ∂ 2 , so that the above equations are the defining relations for the generators. Λ 2 has a natural bigrading by A bicomplex M is the same as a bigraded left Λ 2 -module. We denote the category of bicomplexes by M 2 . We say that a bicomplex M is bounded if only finitely many M i,j are not 0.
Example 2.2. Let us describe some bounded indecomposable bicomplexes.
(1) The bicomplex S i,j is one-dimensional with a copy of k sitting in the (i, j)-th bidegree: In other words, S i,j is the simple Λ 2 -module sitting in bidegree (i, j).
(2) The indecomposable bicomplex P i,j ∼ = Λ 2 {i, j} is a free rank one module (looking like a square on a planar lattice), a copy of Λ 2 with bigrading shifted, so that the nonzero term in the southwest corner sits in (i, j)-th degree: The bicomplex Z i,j →,l has the top leftmost term in bidegree (i, j) and goes zigzag to the right and down. The number l ∈ AE denotes the number of nonzero arrows, l + 1 is the dimension of the vector space underlying this bicomplex.
(4) Likewise, the bicomplex Z i,j ↑,l starts from the bidegree (i, j) and goes zigzag down and to the right.
We will postpone the proof of the theorem until Section 2.4 Let Tot(M ) be the total complex of the bicomplex M, with the differential d = ∂ 1 + ∂ 2 and the terms given by direct sums over M i,j for i + j fixed, A common situation is that we want to compute the homology of Tot(M ) with respect to the differential d and already know the homology of M with respect to, say, differential ∂ 2 (the upward differential in our conventions). These homology groups H(M, d 2 ) are bigraded, and we would like to understand the relation between them and H(Tot(M ), d). If we write M as a (possibly infinite) direct sum of indecomposable bicomplexes M α , for α in some index set A, then both H(M, ∂ 2 ) and H(Tot(M ), d) decompose as direct sums of cohomology groups of M α : Hence, we will compare H(M, ∂ 2 ) and H(Tot(M ), d) for all types of indecomposable summands of M , case by case.
Case 1. S i,j contributes a copy of k to H i,j (M, ∂ 2 ) and a copy of k to H i+j (Tot(M ), d).
For the module Z i,j ↑,l , there are two sub-cases.
Case 3.a. Firstly, let l, the number of nonzero arrows in the zigzag, be odd in Z i,j ↑,l .
Cohomology of Z i,j ↑,l with respect to the vertical differential ∂ 2 is zero. The total complex of this zigzag has the form where d is an isomorphism and 2r = l + 1. Hence, cohomology of the total complex is zero as well.
Case 3.b. Suppose now that l in Z i,j ↑,l is even, l = 2r.
Cohomology with respect to ∂ 2 produce a single k in bidegree (i + r, j − r). The total complex has the form Cohomology of the total complex is k in degree i + j and zero elsewhere.
Case 4.a. For the module Z i,j →,l , there are two sub-cases as well. We start with even l = 2r.
Z i,j →,l (l = 2r) : Cohomology with respect to ∂ 2 give a single k in bidegree (i, j). The total complex is with a surjective d, and it has cohomology k in degree i + i and zero elsewhere.
Before we treat the last case, observe that in each of the above cases cohomology of the total complex is given by simply collapsing the bigrading of H(M, ∂ 2 ) into a single grading by adding i and j. Thus, if M does not contain any direct summands isomorphic to Z i,j →,l with odd l, Case 4.b. Lastly, consider Z i,j →,l with odd l = 2r + 1.
Z i,j →,l (l = 2r + 1) : Taking cohomology with respect to ∂ 2 produces two copies of k, in bigradings that differ by (r, 1 − r): Collapsing the bigrading in cohomology gives us two copies of k in adjacent degrees i + j and i + j + 1.
The total complex has the form with d an isomorphism, and the cohomology of the total complex is zero. Thus, for a general bounded bicomplex M , the cohomology H(M, ∂ 2 ), after the bigrading collapsed into a single grading, is isomorphic to the cohomology of the total complex of M , plus pairs of ground field in adjacent degrees (i + j, i + j + 1), for each direct summand of M isomorphic to Z i,j →,l with odd l. Since we want to know the cohomology of the total complex, the extraneous terms need to be eliminated. Ideally, we would locate all direct summands Z i,j →,2r+1 and kill off pairs of k, one for each summand, in the relative bigrading position (r, 1 − r). For a general r, we need to eliminate pairs in the relative positions (i, j) and (i + r, j − r + 1) by a map d i,j r : on the square lattice. This is exactly what the spectral sequence does. The E 1 -term of the spectral sequence of the bicomplex (M, ∂ 1 , ∂ 2 ) is the cohomology of M with respect to ∂ 2 : To pass to the E 2 -term, we remove contributions to H(M, ∂ 2 ) from the direct summands Z i,j →,1 , which are k 1 → k. Notice that the E 2 -term is simply the cohomology of H(M, ∂ 2 ) with respect to the differential ∂ 1 (more accurately, differential ∂ 1 on M descends to a differential on H(M, ∂ 2 ), which we also call ∂ 1 ): Going from E 2 to the E 3 -term, we remove pairs of one-dimensional vector spaces k which come from summands Z i,j →,3 and differ by (2, −1)-bigrading. In general, in the E r -term there are no contributions from summands Z i,j →,l for all odd l ≤ 2r − 1. The reader can find an accurate definition of spaces E i,j r and differentials d i,j r in almost any textbook on homological algebra, often done in a slightly different framework of a filtered complex rather than a bicomplex. However, we find the above approach via indecomposable bicomplexes more clarifying and intuitive than the standard textbook definition of pages E r and differentials d i,j r of a spectral sequence.

Bicomplexes and Hodge theory
The Hodge bicomplex [Wl, GH, DGMS]. Let X be a closed almost complex manifold. This means X is a smooth closed manifold equipped with an endomorphism J of its real tangent bundle poses into the direct sum of i and −i eigenspaces of J, This induces a direct sum decomposition of all exterior powers ∧ k T * of the complexified cotangent bundle T * (X) c : Let Ω k (X) be the space of smooth sections of ∧ k T * and (Ω (X), d) the complex with d the complexified deRham differential: Let Ω i,j (X) be the vector space of smooth sections of ∧ i,j T * . In general, d shows no respect for the direct sum decomposition However, Newlander and Nirenberg proved [NN] for all i, j if and only if the almost complex structure J of X comes from a complex structure on X. In this case d = ∂ +∂, where ∂ : is the composition of d with the projection onto the (i + 1, j)-component, and is the composition of d with the projection onto the (i, j + 1)-component. The relation d 2 = 0 splits into the relations ∂ 2 = 0,∂ 2 = 0, ∂∂ +∂∂ = 0.
Thus, to a complex manifold X there is assigned the Hodge bicomplex (Ω (X), ∂,∂). Its cohomology groups with respect to∂ are known as the Dolbeault cohomology, while the cohomology with respect to d = ∂ +∂ are the De Rham cohomology of X with coefficients in . The spectral sequence of this bicomplex, called the Hodge to De Rham spectral sequence, has the Dolbeault cohomology as the E 1 -term and converges to the De Rham cohomology of X. Assume now that X is a Kähler manifold. Then the ∂∂-lemma holds.
Lemma 2.4. If ω ∈ Ω (X) is a d-closed and either ∂-exact or∂-exact, then Since the lemma is true for Ω (X), it also holds for each indecomposable summand of X. A simple examination shows that the lemma fails for any zigzag Z i,j →,l and Z i,j ↑,l for l > 0 (when l = 0, the zigzag degenerates to the simple bicomplex S i,j ). We obtain immediately the following.
Proposition 2.5. For a compact Käher manifold X, every indecomposable summand of the bicomplex Ω (X) is isomorphic to either S i,j or P i,j for some i, j.
Equivalently, Ω (X) has no zigzags (including no zigzags of length 1, that is k 1 −→ k and its vertical conterpart).
Thus, the bicomplex Ω (X) decomposes into the direct sum where Ω s (X) is a finite-dimensional semisimple bicomplex (a direct sum of one-dimensional simple bicomplexes S i,j ), while Ω p (X) is an infinite-dimensional free bicomplex (a direct sum of free bicomplexes P i,j ). The first summand is finite-dimensional since Ω (X) has finite-dimensional cohomology groups, and The first three terms are bigraded vector spaces, and the second isomorphism says that, after collapsing the bigrading to a single grading, the groups become the usual De Rham cohomology groups of X. We see that the cohomology groups of a compact Kähler manifold X with respect to ∂,∂, and d are isomorphic; they are also isomorphic to the largest semisimple summand of Ω (X). The Hodge to De Rham spectral sequence for X degenerates at E 1 (E 1 = E ∞ ). Likewise, the ∂ counterpart of the Hodge to De Rham spectral sequence degenerates at E 1 = H(Ω (X), ∂).

Proof of Theorem 2.3
Let M be a graded module over Λ 2 . Suppose m ∈ M is a homogeneous vector of bidegree (i, j) such that ∂ 1 ∂ 2 (m) = 0. Then, it is clear that the submodule generated by m and spanned by vectors in the diagram below is isomorphic to Λ 2 , up to a grading shift, and thus is a projective submodule inside M . Since Λ 2 is graded Frobenius, the submodule is also graded injective, and therefore must be a direct summand of M . Iterating the process, we can decompose M ∼ = P ⊕ N , where P is a graded direct sum of projective-injectives of the form P i,j (case (2) of Example 2.2), and N is annihilated by the element ∂ 1 ∂ 2 = −∂ 2 ∂ 1 ∈ Λ 2 . Further, we may regard N as a module over the bigraded quotient algebra . Now assume N is a bounded bigraded Λ 2 -module which does not contain any projectiveinjective summands. By the above discussion, N is a bigraded module over Λ ′ 2 . Write for each term is the subspace annihilated by both ∂ 1 and ∂ 2 , and C i,j is an arbitrary complementary vector subspace to D i,j inside N i,j . Necessarily, since ∂ 1 ∂ 2 | N ≡ 0. Thus, there are two direct summands of N containing the subspaces C i,j and D i,j :  In particular, if we further assume that N as above is indecomposable, then there must be (i, j) ∈ 2 such that N is isomorphic to one of the above "zig-zag" modules, and either C i,j = N i,j or D i,j = N i,j . Flattening out the zig-zag, say, the first one, we may identify N with an indecomposable finite-dimensional representation of the A quiver with the alternating orientation By the classical result of Gabriel and Bernstein-Gelfand-Panomarev, such an indecomposable module must be of the form Such an indecomposable module translates back into either the simple module or a zig-zag module listed in Example 2.2 (cases (1), (3) and (4)). The theorem follows.
Remark 2.6 (Unbounded complexes). As the proof reveals above, one may extend Theorem 2.3 to the case of unbounded bicomplexes as well.
Case 5. Initially vertical and bounded from "below" or "above"; the bounded corner sitting in bidegree (i, j): Cohomology spaces of Z i,j ↑,± with respect to the vertical differential ∂ 2 are both zero. But the collapsed total complexes, which has the form have differental total cohomologies. It is readily seen that, for Z i,j ↑,+ , the total differential is both injective and surjective. However, for Z i,j ↑,− , the total differential is injective, but not surjective. The cokernel of d is given by k sitting in the bidegree (i, j).
Case 6. The module Z i,j →,± , which starts horizontally and is bounded from below or above, whose bounded corner lies in bidegree (i, j): Cohomology spaces with respect to ∂ 2 give a single k in bidegree (i, j). However, the total cohomology of the collapsed complexes behave differently. For Z i,j →,+ , the total differential is clearly injective, but not surjective. The cohomology classes represented by the vectors 1 sitting in bidegrees (i − r, j + r), r ∈ AE, are all cohomologous, and their images in the total complex represent the same cohomology class in degree i + j. On the other hand, the total differential of Z i,j →,− is an isomorphism, and thus there is no total cohomology. Case 7. The module Z i,j ± , which are unbounded in both directions. The underlined copy of k sits in bidegree (i, j). The modules are taken to be the same up to shifting (i, j) to (i + r, j − r), where r ∈ , and identifying Z i,j Again, the vertical cohomology with respect to ∂ 2 of Z i,j ± are both zero. The total cohomology for the collapsed complexes both have one-dimensional cohomology sitting in the cokernel of d.
Combining this discussion with those of Section 2.2, we see that, if a bicomplex M = ⊕ i,j∈ M i,j is bounded from below (resp. bounded from above) in the sense that, there exists a bigrading (k, l) such that M i,j = 0 whenever i ≥ k or j ≥ l (resp. M i,j = 0 whenever i ≤ k or j ≤ l), then M may contain additional summands of the form Z i,j ↑,+ and Z i,j →,+ , up to bigrading shifts, but taking ∂ 2 -cohomology first does not create additional classes that need to be killed of in the total cohomology. Corollary 2.7. If M is a bicomplex bounded from below, then there is a spectral sequence whose E 1 page equals (H ∂ 2 (M ), ∂ 1 ) converging to the total cohomology of M . Likewise, if M is a bicomplex bounded from above, then there is a spectral sequence starting at (H ∂ 1 (M ), ∂ 2 ) converging to the total cohomology of M .

Connection to zig-zag algebras
Let us point out the connection between the category M 2 of bicomplexes with the module category over (an infinite version of) the zig-zag algebra considered in [KhS].
Let Q ∞ be the following quiver whose vertices are labelled by r ∈ : Set kQ ∞ to be the path algebra associated to Q ∞ over the ground field. We use, for instance, notation (i|j|k), where i, j, k are vertices of the quiver Q ∞ , to denote the path which starts at vertex i, then goes through j (necessarily j = i ± 1) and ends at k. The composition of paths is given by (i i |i 2 | · · · |i r ) · (j 1 |j 2 | · · · |j s ) = (i i |i 2 | · · · |i r |j 2 | · · · |j s ) if i r = j 1 , 0 otherwise, where i 1 , . . . , i r and j 1 , . . . , j s are sequences of neighboring vertices in Q ∞ .
We make the zig-zag algebra graded by setting 1 deg(r) = deg(r|r + 1) = 0, deg(r|r − 1) = 1, (2.5) for all r ∈ . It is a non-unital algebra with a system of mutually orthogonal idempotents {(r)|r ∈ }. There is an obvious automorphism T on A, defined by T (r) := (r + 1), T (r|r + 1) := (r + 1|r + 2), T (r|r − 1) := (r + 1|r). (2.6) For a fixed pair of integers (r, i) ∈ 2 , there is a graded projective module P r i which is generated by the idempotent (r), whose degree is shifted up by i. More explicitly, P r i is the four-dimensional vector space with the basis ( 2.7) where σ i stands for the module generator sitting in degree i. We will consider the category of graded modules over A, which we denote by M(A), in what follows. The automorphism T of A induces an autoequivalence T of M(A), defined by T := (T −1 ) * . Clearly T (P r i ) = P r+1 i holds for all r, i ∈ .
Given a module M = ⊕ i,j∈ M i,j in M 2 , we place the homogeneous bigraded component of M i,j at (i, j) in the corresponding node of the two-dimensional lattice 2 . For each r ∈ , we collect together M i,j s on the line of slope one (depicted as the dashed line in the picture below): (2.8) Note that M r is singly graded, with its homogeneous degree j part M j r set to be M r+j,j . Since ∂ 1 and ∂ 2 have bidegrees (1, 0) and (0, 1), respectively, they induce maps These maps satisfy D 2 1 = 0, D 2 2 = 0 and D 1 D 2 = D 2 D 1 . We put the vector space M r at the rth vertex of A and declare the rightward (resp. leftward) going arrows to be the induced map D 1 (resp. D 2 ). We have thus obtained a graded A-module by summing over the r-degrees M ∞ := ⊕ r∈ M r .
Schematically, we depict the correspondence as follows.
Furthermore, a morphism f : M −→N in M 2 componentwise given by One has the associated morphism of bigraded Amodules, which is defined as Clearly D 1 f r = f r+1 D 1 and D 2 f r = f r−1 D 2 holds for all r ∈ , so that f ∞ is a morphism of bigraded A-modules. This defines a functor F ∞ : M 2 −→M(A).
As the above functorial assignment is clearly reversible, the functor F ∞ is invertible.
Proposition 2.9. The functor F ∞ : M 2 −→M(A) is an equivalence of abelian categories. Furthermore, the functor satisfies
Let Λ 3 be the exterior algebra over k with three generators ∂ 1 , ∂ 2 , ∂ 3 : We make Λ 3 a triply-graded k-algebra, by assigning degree e j to ∂ j . Let M 3 be the category of triply-graded left Λ 3 -modules with respect to tri-degree preserving maps. A module M consists of a collection of k-vector spaces together with linear maps ∂ j : M i −→M i+e j subject to the exterior algebra relations. It is useful to visualize M as a 3-dimensional object: the vector space M i sits in the i node of a 3-dimensional lattice and the maps ∂ j go along oriented edges of the lattice. Below is a portion of M depicted: The grading shift by i, denoted {i}, is an automorphism of M 3 . Any simple object of M 3 is isomorphic to S i := k{i}, for a unique i, Here k is a one-dimensional k-vector space, in tridegree 0, viewed as a Λ 3 -module with the trivial action of ∂ 1 , ∂ 2 , ∂ 3 .
Any indecomposable projective in M 3 is isomorphic to P i := Λ 3 {i}, for a unique i. Any projective in M 3 is isomorphic to the direct sum of P i 's, possibly with infinite multiplicities. Since Λ 3 is a trigraded Frobenius algebra, P i are also injective objects of M 3 . A module M contains P i as a direct summand (and not just as a submodule) if and only if ∂ 1 ∂ 2 ∂ 3 m = 0 for some m ∈ M i . Let Q = Λ 3 ω/Λ 3 ∂ 3 ω be the cyclic module with one generator ω in tri-degree 0 and relation ∂ 3 ω = 0. We depict Q as a square There is a graded isomorphism of modules Q ∼ = Λ 3 /∂ 3 Λ 3 . The algebra Λ 3 is a Hopf algebra in the category of trigraded (super) vector spaces, where the (super) /2 -grading is given by reducing i 1 + i 2 + i 3 modulo 2, and ∆(∂ r ) = ∂ r ⊗ 1 + 1 ⊗ ∂ r . Consequently, the tensor product M ⊗ N of trigraded Λ 3 -modules is a trigraded Λ 3 -module, with ∂ r acting by Similarly, there is a trigraded inner-hom on M 3 , defined by where the right hand side is the direct sum of homogeneous linear maps from M to N {i}. The inner hom space carries a natural Λ 3 action defined by, for any f ∈ Hom k (M, N {i 1 , i 2 , i 3 }) The spaces of Λ 3 -invariants under this action consist of morphisms in M 3 of all degrees: It is useful to regard Λ 2 and Λ 1 as certain graded Hopf subalgebras in Λ 3 . To do this, we break the apparent symmetry and define Λ 2 to be the subalgebra generated by ∂ 1 and ∂ 2 , while setting Λ ′ 1 to be the subalgebra generated by ∂ 3 . The natural algebra inclusions (3.6) which are respectively given by setting ∂ 3 or ∂ 1 , ∂ 2 to be zero. Using these subquotient algebras, we define a functor by taking "partial graded-hom" with respect to Λ ′ 1 , as follows. Fix i and j degrees. Given any M ∈ M 3 , set where in the last term, we only keep the Λ ′ 1 -module structure on ⊕ k M i,j,k . The functor extends naturally to morphisms in M 3 , and has the effect, on objects, of taking the direct sum of M i,j,k over k ∈ . It remembers the ∂ 3 -complex structure inherited from that of M , while making ∂ 1 , ∂ 2 act by 0.

A braid group action
In this section, we exhibit a braid group action on the stable category of trigraded Λ 3 -modules.
The tensor product Q ⊗ M i,j is an object of M 3 , with ∂ 1 , ∂ 2 acting only along Q (since their actions on M i,j are trivial) and ∂ 3 acting along M i,j .
Consider the functor Geometrically, we take the plane P r = {(i, j, k)|i − j = r} in 3 , with vector spaces M i sitting in the nodes, and form four copies of plane (the tensor product with Q) related by the differentials ∂ 1 and ∂ 2 . The differential ∂ 3 acts along edges (i, i + e 3 ) contained in the plane P r . We depict the summand Q ⊗ M i,j in the next diagram. For a fixed e 3 -degree k, Q ⊗ M i,j,k has four copies of M i,j,k sitting in degrees (i, j, k), (i + 1, j, k), (i, j + 1, k) and (i + 1, j + 1, k) respectively. They correspond to All maps except for act as identity maps, which is the negative identity map. Now, summing over k and keeping track of the differential ∂ 3 , we obtain the diagram Here the differential ∂ 3 points perpendicularly out of the plane.
Proposition 3.1. The following isomorphisms between endofunctors of M 3 hold: Proof. We start with the first equation. We compute the left hand side as Here, in the third equality, we have used that Q ⊗ M i,j has only two terms concentrated on the line k − l = r (see the above picture (3.9)). For the second isomorphism, we have (taking the r + 1 case) The last isomorphism is easy, and we leave it as an exercise to the reader.
Remark 3.2. Perhaps the cartoon below, in the scheme of equation (3.9), helps visualizing the equalities in the above proof. We show this for equation (3.12) as an example. Depict a copy of M i,j by a box in the lattices below. A black dot in a box indicates the term contributing to the functor on the outward arrow.
There exists a unique morphism in M 3 Q ⊗ M i,j −→M (3.14) which takes ω ⊗ m to m. This morphism takes ∂ 1 ω ⊗ m to ∂ 1 m, etc.
Summing over i, j such that i−j = r, morphisms (3.14) combine into a module homomorphism in r : U r (M )−→M (3.15) natural in M . Thus, in r : U r =⇒ Id is a natural transformation of functors on M 3 . Next, we construct a module homomorphism Denote by M ν the same underlying trigraded vector space as M , while only remembering the Λ ′ 1 -module structure. Consider the map where m ∈ M i,j,k is a homogeneous element. Proof. The map clearly commutes with ∂ 3 -actions on both sides, as ∂ 3 kills ω and anti-commutes with ∂ 1 and ∂ 2 . To verify that out also commutes with ∂ 1 and ∂ 2 requires a small computation. We check, for instance, that it commutes with ∂ 1 , and leave the ∂ 2 -computation to the reader.
On the one hand, if m ∈ M i,j,k , and using that ∂ 1 acts trivially on M ν , we have On the other hand, Comparing these expressions, the commutativity with the ∂ 1 -actions follows.
Since Q ⊗ M ν naturally decomposes into a direct sum of Λ 3 -modules for each r ∈ , we have a natural projection map of Λ 3 -modules We can thus define the composition map Componentwise, out r has the effect, for a homogeneous m ∈ M i,j,k , (3.21) We have thus obtained out r as a tri-grading preserving homomorphism of Λ 3 -modules, functorial in M . In other words, similarly as for in r , the map out r : Id ⇒ U r {−1, −1, 0} is a natural transformation of functors. Let SM 3 be the stable category of trigraded left Λ 3 -modules. It has the same objects as M 3 and the morphisms are those in M 3 modulo morphisms that factor through a projective object of M 3 . In particular, a projective trigraded Λ 3 -module is isomorphic to the zero object in SM 3 . The stable category is triangulated, with the shift functor [1] SM taking M to the cokernel of an inclusion M ⊂ P, where P is a projective module. For concreteness, we can choose P to be Λ 3 ⊗ M {−1, −1, −1}, with the inclusion taking m to ∂ 1 ∂ 2 ∂ 3 ⊗ m. The shift by {−1, −1, −1} makes the inclusion grading-preserving.
The cone of a morphism f : M −→N is defined as the cokernel of the inclusion which takes m to (f (m), ∂ 1 ∂ 2 ∂ 3 (m)). For more details, we refer the reader to Happel [Ha]. We will need the following result computing morphism spaces in SM 3 , bearing in mind the Λ 3 action defined in equation (3.3).
Lemma 3.4. Given two objects M, N ∈ SM 3 , there is an isomorphism .
We introduce another cone construction defined for morphisms in the abelian category M 3 . Given a morphism f : M −→N in M 3 , the ∂ 3 -cone C 3 (f ), as a trigraded vector space, is the object M {0, 0, −1} ⊕ N , on which the Λ 3 -generators act by ∂ 3 (m, n) = (−∂ 3 m, f (m) + ∂ 3 (n)), (3.22) and ∂ j (m, n) = (∂ j m, ∂ j n) for j = 1, 2. Alternatively, regard Λ ′ 1 = k[∂ 3 ]/(∂ 2 3 ) as a trigraded Λ 3 -module via the homomorphism ν (see equation (3.6)), the ∂ 3 -cone is defined as the push-out of f : M −→N and ∂ 3 ⊗ Id M : M −→Λ ′ 1 ⊗ M . This is the top square of the following diagram, whose columns are short exact in the abelian category because of the push-out property: Lemma 3.5. The functors R r , R ′ r descend to well-defined functors on the stable category SM 3 . Proof. It suffices to show that, if M is a projective Λ 3 -module, then R r (M ) and R ′ r (M ) are both projective. Let us do this for R r , and the R ′ r case is similar. By (3.23), R r (M ) fits into a short exact sequence of Λ 3 -modules Since Λ 3 is Frobenius, M is also injective and the above sequence splits. We are thus reduced to showing that U r (M ){0, 0, −1} is graded projective. Without loss of generality, we may assume that M ∼ = Λ 3 {i, j, k} is indecomposable. As Λ ′ 1 -modules, there is a direct sum decompostion , 1, 0}. Using this decomposition and the fact that The result follows.
Theorem 3.6. (i) The functors R r , R ′ r are invertible mutually-inverse endofunctors on the stable category SM 3 .
(ii) The following functor isomorphisms hold: (3.25b) Consequently, the collection of functors {R r |r ∈ } gives rise to an action of the infinite braid group of infinitely many strands Br ∞ on the triangulated category SM 3 .
The proof of the theorem will occupy the next subsection.
Remark 3.7. In this section, we have interpreted the three differentials of Λ 3 in two different ways: the Λ 2 ⊂ Λ 3 plays the role of the algebra A (c.f. Section 2.5), while ∂ 3 behaves more like a "homological differential". This apparent symmetry breaking allows one to construct three equivalent braid group actions on SM 3 as in Theorem 3.6, by the autormophism of Λ 3 permuting the indices {1, 2, 3}.

Proof of Theorem 3.6
Invertibility of R r . First we show R ′ r R r ∼ = Id. We check the effect of the left hand side on a trigraded Λ 3 -module M . (3.26) Here, in the diagram, the horizontal arrows are interpreted as the ∂ 3 -differential arising from the ∂ 3 -cone of in r , while the vertical arrows indicate that of out r . The differential action by ∂ 1 , ∂ 2 preserves the position of the node, while the ∂ 3 acts both internally at the nodes and transfer elements long the arrows (see equation (3.22)). By Proposition 3.1, we may decompose As in the proof of the Proposition, we further identify By the definition of the ∂ 3 -cone, the sum of terms on the lower horizontal line of (3.26) consitutes a Λ 3 -submodule of R ′ r (R r (M )). The morphism in r on the lower horizontal line of (3.26) maps the summand (3.27c) isomorphically onto As Q ⊗ Λ ′ 1 ∼ = Λ 3 is a tri-graded free Λ 3 -module, it is a not only a submodule in R ′ r (R r (M )) but also a direct summand, which is annihilated when passing to the stable category SM 3 . We thus may safely identify R ′ r (R r (M )) with the quotient of it by this submodule, which we denote by M 1 . Now M is clearly a Λ 3 -submodule in M 1 . We claim that M 1 /M is also a free Λ 3 -module, and hence is a direct summand in M 1 whose complement is isomorphic to M . It then follows that the natural inclusion map M ֒→ M 1 is an isomorphism in SM 3 .
To prove the claim, note that where the right hand side denotes a ∂ 3 -cone. If m ∈ M i,j is a homogeneous element, the map out r has, by equation (3.21), the effect The right hand side of the first equation contains elements in U r (M ){−1, −1, 0} (see equation (3.27c)), which has already been mod out in M 1 . The rest of the terms on the right hand side of the equations have their middle term ∂ 1 ∂ 2 ω. It follows that out r maps U r (M ){0, 0, −1} isomorphically onto U r (M ). The claim follows. (3.31) We will gradually strip off the projective-injective summands of this module, which, for brevity, we will call M 0 in what follows. By Proposition 3.1, we identify By the definition (3.14) of in r+1 , the external ∂ 3 -differential −in r+1 maps this term isomorphically onto the summand U r (M ){1, 1, −2} of In addition, the automorphism T of M(A) extends to an automorphism of C(A), denoted by the same letter, defined by termwise applying T on complexes: In what follows, we will also use the notation C(A-pmod) to stand for the full subcategory of C(A) consisting of complexes of graded projective A-modules up to homotopy.
We also re-interpret chain complexes of graded A-modules as differential graded modules over the graded dg algebra (A, d), where A sits in homological degree zero, and the natural grading of A is orthogonal to the homological grading. A chain complex of graded A-modules is equivalent to the data of a differential graded (A, d)-module Extending the (inverse) equivalence of Proposition 2.9, there is an auto-equivalence of abelian categories where, on the object G ∞ (M ) ∈ M 3 for a given M ∈ (A, d)-mod, the generator ∂ 3 acts by the differential (−1) k d : M k −→M k+1 . It follows from Proposition 2.9 that G ∞ commutes with the translation various shift functors as follows: As a result, one can deduce that G ∞ (P r j [k]) ∼ = Q{r + j, j, −k}. As each M µ k is a projective A-module, the result follows since, by Proposition 2.9, G(P r j ⊗ k[d]/(d 2 )[−k]) ∼ = Q{r, r + j, k} ⊗ Λ ′ 1 ∼ = Λ 3 {r, r + j, k} holds for any r, j, k ∈ . It remains to show that G is exact, i.e., it commutes with homological shifts and takes distinguished triangles into distinguished triangles.
Given a complex M of graded projective modules over A, there is a short exact sequence Applying G ∞ to the short exact sequence, we obtain a short exact sequence of M 3 : Applying G ∞ to this sequence, we obtain a short exact sequence of trigraded Λ 3 -modules. This sequence results in a distinguished triangle in SM 3 , being the image of the original triangle in C(A-pmod). The exactness of G now follows.
Denote by C b (A-pmod), C + (A-pmod) and C − (A-pmod) the full triangulated subcategories of C(A-pmod) consisting of, respectively, bounded, bounded-from-below and bounded-from-above complexes of graded projective modules over A. The localization functor from C(A-pmod) into D(A) restricts to equivalences of categories on these full-subcategories onto their respective images in the (dg) derived category D(A).
Proof. The proof is divided into three steps.
As the first step, we claim that G, when restricted to the full-subcategories C ± (A-pmod), is fully-faithful. To do this, we identify these categories with their images in D(A) under localization, and use the fact that the (dg) derived category of (A, d)-mod is compactly generated by the collection of objects {P r j [k]|r, j, k ∈ }. Then, in order to prove the claim, we just need to compare the morphism spaces between the generating objects P r j [k], r, j, k ∈ , and their images G(P r j [k]) = Q{r + j, j, −k} in SM 3 ( [Ke,Lemma 4.2]).