A categorification of cyclotomic rings

For any natural number $n \geq 2$, we construct a triangulated monoidal category whose Grothendieck ring is isomorphic to the ring of cyclotomic integers $\mathbb{O}_n$.


INTRODUCTION
1.1. Backround. The seminal paper of Louis Crane and Igor B. Frenkel [CF94] proposes that one should lift three-dimensional topological quantum field theories defined at a primitive nth root of unity to four-dimensional theories. The lifting process is usually referred to, in mathematics, as categorification. One aim is to replace algebras appearing in the construction of the three-dimensional topological quantum field theories by categories, such that the original algebras can be recovered by passing to the Grothendieck group. However, a foundational obstacle to the program is the lack of a monoidal category that categorifies the cyclotomic ring of integers O n at a primitive nth root of unity.
As an initial breakthrough, Khovanov [Kho16] observed that, when n " p is a prime number, the graded Hopf algebra H p " krds{pd p q (degpdq " 1) over a field k of characteristic p may be utilized to categorify O p . The basic idea is as follows. Inside the category of graded H p -modules, the projective modules, which coincide with the injectives since H p is graded Frobenius ( [LS69]), have their graded Euler characteristic equal to a multiple of that of the rank-one free module (1.1) rH p s " 1`v`¨¨¨`v p´1 " Φ p pvq.
Here Φ n pqq stands for the nth cyclotomic polynomial. Systematically killing projective-injective objects from H p -modules results in a triangulated monoidal category H p -gmod whose Grothendieck ring is isomorphic to The tensor triangulated category H p -gmod bears significant similarities with the usual homotopy category of abelian groups, and is thus also referred to as the homotopy category of pcomplexes.
The study of H p -gmod and algebra objects in these categories has been further developed in [Qi14]. The theory has since been applied to categorify various root-of-unity forms of quantum groups. We refer the reader to [QS17] for a brief summary and special phenomena at a pth root of unity.
1.2. Outline of the construction. In this paper, we construct a triangulated monoidal category O n whose Grothendieck group is isomorphic to the ring of cyclotomic integers O n . We work in any characteristic, including characteristic zero, as long as the ground field contains a primitive N th root of unity, where N " n 2 {m and m is the radical of n, the product of the distinct prime factors of n. The construction is motivated from pioneering works of Kapranov [Kap96] and Sarkaria [Sar95] on n-complexes. When n " p a is a prime power, our work is equivalent to a graded version of p-complexes. With this approach, we remove the restriction on n being a prime number.
Let us outline the construction. When n " p a is a prime power, one sees that p-complexes, up to homotopy, categorify O p a when the p-differential has degree p a´1 . The characteristic zero lift of the Hopf algebra krds{pd p q, which controls Kapranov-Sarkaria's p-complexes, is a Hopf algebra object in the braided monoidal category of q-graded vector spaces, where q is a primitive p 2a´1 th root of unity. Now suppose n ě 2 is a general integer. Let n " p a 1 1¨¨¨p at t be the prime decomposition of n and fix q, a primitive N th root of unity. By the one-factor case, it is natural to consider the Hopf algebra object H n , in q-graded vector spaces, generated by commuting differentials d 1 , . . . , d t subject to d p i i " 0, i " 1, . . . , t. The algebra H n is graded Frobenius, and thus has associated with it a well-behaved tensor triangulated stable category H n -gmod. However, the Grothendieck ring of H n -gmod is defined by setting the character of the free module H n equal to zero. The last relation is usually larger than the cyclotomic relation Φ n pvq " 0 which we would like to impose.
To obtain the correct relations in the Grothendieck ring, we use the elementary fact that Φ n pvq occurs as the greatest common divisor of some prime power cyclotomic relations (Lemma 5.13). On the categorical level, the ideal generated by the greatest common divisor is categorified by a "categorical ideal", i.e, a thick triangulated subcategory I inside H n -gmod which is closed under tensor product actions by H n -gmod. Verdier localization at the categorical ideal I yields the desired category O n , whose Grothendieck ring is defined by the desired relations and is isomorphic to O n .
Before moving on, let us also point out some connection to previous work. Furthering Kapranov and Sakaria, there is a significant amount of study on n-complexes in the literature. See, for instance, the lectures notes of Dubois-Violette [DV02] and the references therein. In [Bic03], Bichon considers a Hopf algebra Apqq " krxs{px n q ⋊ kZ for which there is a monoidal equivalence between the category of n-complexes and Apqq-comod. One may similarly define a Z{nZgraded version of Bichon's Hopf algebra, which resembles the classical Taft algebra. Recently, Mirmohades [Mir15] has introduced a tensor triangulated category arising from a suitable quotient of a tensor product of two Taft algebras. This category categorifies a primitive root of unity whose order is the product of two distinct odd primes. Our current work and [Bic03,Mir15] are both unified under the frame of hopfological algebra [Qi14].
1.3. Summary of contents. We now briefly describe the structure of the paper and summarize the contents of each section.
In Section 2, we review the construction of stable module categories in the particular case of finite-dimensional Hopf algebras. The Frobenius structure and the existence of an inner hom space allows an explicit identification of morphism spaces in the stable module categories.
In Section 3, we introduce the main object of study, a finite-dimensional braided Hopf algebra H n , depending on a given natural number n. The category H n -gmod has a tensor product depending on a certain root of unity q. The braided Hopf algebra H n is primitively generated by certain commuting p k -differentials d k , for k " 1, . . . , t. Using the Radford-Majid biproduct (or bosonization, see [Rad85,Maj95]) of the differentials by the group algebra of a finite cyclic group, one obtains a related Hopf algebra, for which graded H n -modules correspond to rational graded modules. We also point out that H n -gmod has the structure of a spherical monoidal category in the sense of Barrett-Westbury [BW99].
Next, we proceed, in Section 4, to define a tensor triangulated ideal I (Definition 4.12) in the (stable) module category of H n . Upon factoring out the ideal by localization, we show, in Section 5, that the quotient category has the desired Grothendieck ring O n (Theorem 5.15). The reason to introduce this ideal is as follows. On the Grothendieck ring level, we would like the objects of the ideal to have characters satisfying cyclotomic relations dividing q n´1 that are of lower order than the primitive condition Φ n pqq " 0. Systematically killing these objects by taking a Verdier quotient in the stable category H n -gmod gives the lower order relations in the Grothendieck ring. An example of such an object in the ideal is an n-complex which is freely generated by all but one of the differentials while the remaining differential acts by zero. This objects has self-extensions and it is thus natural to require a filtration condition on the modules on the abelian level giving a triangulated tensor ideal I k . The bulk of the work in Section 4 is devoted to showing that after passing to the stable category of H n -modules, the ideals I k are orthogonal so that their sum I is indeed closed under tensor product, extensions and direct summands. Now, the standard machinery of Verdier localization (quotient) can be used to obtain a triangulated quotient category O n with a tensor product structure. Finally, in Theorem 5.15, we prove that the Grothendieck ring of O n is isomorphic to O n .

Comparison and further directions.
To conclude this introduction, let us make some comparison between our construction and the works [Mir15,Bic03], as well as indicate some further directions.
We employ multiple nilpotent differentials d 1 , . . . , d t depending on the prime factors of n, in contrast to [Mir15], thus getting rid of the restriction on n having to be the product of two odd primes. In contrast to [Mir15] we only employ a single Z-grading rather than a bigrading. This requires us to use a filtration condition on modules in the ideal I k .
A negative result from [Bic03, Proposition 5] is the non-existence of a quasi-triangular structure on the Hopf algebra Apqq describing n-complexes. In our setup, we show that instead of a quasi-triangular structure, there exist weak replacements given by functorial isomorphisms V b q W -W b q´1 V . For n " 2, these satisfy the axioms of a braiding, but for other values of n no analogue of the braiding axioms could be identified. We plan to explore this structure in subsequent works.
For further investigation, we would like to construct module categories over O n , developing triangulated analogues parallel to the abelian theory of [EGNO15]. We will also seek interesting algebra objects in O n , in a similar way as done in [KQ15,EQ16] over the homotopy category of pcomplexes. The Grothendieck groups of such algebra objects would then give rise to interesting modules over O n . It would also be an interesting problem to combine the recent categorification of fractional integers due to Khovanov-Tian [KT17] in order to categorify the algebra O n " 1 n ‰ , over which the extended 3-dimensional Witten-Reshetikhin-Turaev TQFT lives.

THE STABLE CATEGORY
2.1. Notation. We start by fixing some conventions concerning Z-graded vector spaces over a ground field k. Let us denote the category of finite-dimensional Z-graded vector spaces by gvec.
Let M " ' iPZ M i and N " ' jPZ N j be Z-graded vector spaces over k. We set M b k N , or simply M b N , to be the graded vector space For any integer k P Z, we denote by M tku the graded vector space M with its grading shifted down by k: pM tkuq i " M i`k . The morphisms space Hom 0 k pM, N q consists of homogeneous k-linear maps from M to N : Writing Hom i k pM, N q :" Hom 0 k pM, N tiuq " f : M ÝÑ Nˇˇf pM j q Ď N i`j ( , we set the graded hom space to be Hom ‚ k pM, N q :" If no confusion can be caused, we will simplify Hom ‚ k pM, N q to Hom ‚ pM, N q. A special case is the graded dual M˚" Hom ‚ pM, kq.
Given three Z-graded vector spaces M , L and K, the following easily proven tensor-hom adjunction will be used. There are isomorphisms of graded vector spaces, natural in M, L, K: where f P Hom ‚ pM b L, Kq, m P M and l P L are arbitrary elements. We will also require (unbalanced) q-integers. In particular, for a formal variable ν, we define polynomials Given q P k, we set rns q to be the value of rns ν evaluated at ν " q. For a Z-graded vector space M , denote by dim ν pM q " ÿ iPZ dim k pM i qν i the graded dimension of M . We abbreviate, for f pνq "

Stable module categories.
Let H be a Z-graded self-injective algebra over a field k. We denote by H-gmod the category of finite-dimensional Z-graded modules over H, with morphisms of degree zero. For ease of notation, we will drop mentioning "graded" in what follows if no confusion can arise. Note that, as H is self-injective, a graded H-module is injective if and only if it is projective. The (graded) stable category of finite-dimensional H-modules, denoted by H-gmod, is the categorical quotient of the category H-gmod by the class of (graded) projective-injective objects. More precisely, recall that a degree-zero morphism is (homogeneous) null-homotopic if it factors through a projective-injective H-module. For any two H-modules M, N P H-gmod, let us denote the space of null-homotopic morphisms in H-gmod by I 0 H pM, N q. It is readily seen that, the collection of I 0 H pM, N q's ranging over all M, N P H-gmod constitute an ideal in H-gmod. Then H-gmod has the same objects as H-gmod, and the morphism space between two objects M, N P H-gmod is by definition the quotient Hom H-gmod pM, N q :" It is a classical theorem that H-gmod is triangulated, see [Hap87,Theorem 9.4] and [Hel60]. The shift functor r1s : H-gmod ÝÑ H-gmod is defined as follows. For any M P H-gmod, choose an injective envelope I M for M in H-gmod and let K M be the cokernel of the embedding map ρ M : Then M r1s :" K M . The inverse functor r´1s can be defined similarly by taking a projective cover and the corresponding kernel of the canonical epimorphism.
Let us also recall how distinguished triangles are defined in the stable category. Let f : M ÝÑ N be a morphism in H-gmod. Consider the diagram (2.4) where the left-hand square is a push-out. One declares to be a standard distinguished triangle. Then any triangle in H-gmod isomorphic to a standard one is called a distinguished triangle. We refer the reader to Happel's book [Hap88] for more details on this fundamental construction.
As for graded vector spaces, we set  [LS69]). We will use adapted Sweedler's notation that If M and N are H-modules, then H acts on M b N by, for any x P M , y P N and h P H, We equip M˚" Hom ‚ pM, kq with the dual H-module structure (2.10) ph¨f qpxq :" f pS´1phqxq, for any f P M˚and x P M . Notice that the grading on M˚is given by It is easily checked that there is an isomorphism of graded H-modules Furthermore, it is easy to check that the natural adjunction maps commute with the H-actions, where te i u is a homogeneous basis for M and tei u is the dual basis.
Remark 2.1. We remark that there is an alternative way to introduce internal homs for H-gmod, by using Hom ‚ pM, N q -N b M˚. In this case, the module structure is given by Note that under this action, M˚is left dual to M , whereas in equation (2.12), M˚plays the role of a right dual. With the alternative convention, the modified form of the tensor-hom adjuction (c.f. equation (2.1)) The H-invariants discussed in Lemma 2.3 below will be naturally isomorphic for the two versions of internal homs.
By a classical result of Larson-Sweedler [LS69], H is (graded) Frobenius, and, in particular, it is (graded) self-injective. Let Λ be a fixed non-zero left integral in H, i.e., an element in H such that for all h P H, one has (2.15) hΛ " ǫphqΛ.
The element is unique up to a non-zero scalar, and hence a homogeneous element using that the multiplication on H and ǫ are degree-preserving maps. Denote the degree of Λ by degpΛq :" ℓ.
Then for any H-module M , we have a canonical embedding of M into the injective H-module M b H: because of the following result.
Lemma 2.2. Let H be a (graded) Hopf algebra and M a (graded) H-module. Then there is an isomorphism of tensor products of (graded) H-modules Here M 0 stands for the vector space M endowed with the trivial H-module structure for any h P H and m 0 P M 0 . In particular, M b H is projective and injective.
Proof. It is an easy exercise to check that the inverse of φ M is given by For the last statement, the projectivity of M 0 b H is clear. For injectivity, one uses the wellknown fact that a (possibly infinite) direct sum of injective H-modules remains injective if and only if H is Noetherian.
Despite the fact that the module M b Htℓu is usually larger than the injective envelope of M , the functoriality of this canonical map in M will allow us to understand the morphism spaces in the stable category more explicitly.
Recall that, for any H-module M , the space M H of H-invariants in M consists of (2.17) M H :" tm P M |hm " ǫphqm, @h P Hu.
Lemma 2.3. The space of H-invariants in Hom ‚ pM, N q coincides with the space of H-module maps between M and N : so that h¨f " ǫphqf . Here, in the last equality, we have used the fact that, for any element h P H, the identity ř h S´1ph 2 qh 1 " ǫphq holds. On the other hand, if f P Hom k pM, N q H , then The lemma now follows.
The lemma can be rephrased as saying that the category H-gmod is an enriched category over itself.

Lemma 2.4. An H-module homomorphism f : M ÝÑ N factors through a projective-injective Hmodule if and only if there is an H-module map g making the following diagram commute:
Proof. It suffices to prove the result when N is projective-injective. In this case, consider the following commutative diagram.
Proof. If f " Λ¨g for some k-linear g : M ÝÑ N , we will extend g to an H-linear map It will then follow by construction that f " p g˝ρ M . Indeed, we check that p g is H-linear. For any x, h P H and m P M , we have that Here in the fourth equality, we have used the fact that, for any element x P H, it holds that ř x S´1px 2 qx 1 " ǫpxq.
Conversely, if f factors as a composition of H-linear maps so that f pmq " p gpm b Λq for any m P M , we then define a k-linear map g : M ÝÑ N by gpmq :" p gpm b 1q. It remains to verify that Λ¨g " f . To do this, we compute, for any m P M , The result follows.
which is natural in both M and N .
Proof. It suffices to show the statement in degree zero. By Lemma 2.3, the numerator in the equality above coincides with the space of H-intertwining maps. Combining Lemma 2.4 and 2.5, one sees that the space of maps between two H-modules that factor through projectiveinjective modules coincides with I ‚ H pM, N q -Λ¨Hom ‚´ℓ pM, N q. The theorem follows.
The theorem implies that the stable category H-gmod for a finite-dimensional Hopf algebra is equipped with an internal Hom, which is no other than the space of graded vector space homomorphisms Hom ‚ pM, N q.
Corollary 2.7. The graded tensor-hom adjunction holds in H-gmod. That is, for any M , N and L in H-gmod, there is an isomorphism of graded vector spaces

In particular, there is an isomorphism of ungraded vector spaces
Hom H-gmod pM, N q -Hom H-gmod pk, Hom ‚ pM, N qq functorial in M and N .
Proof. This follows from taking the canonical isomorphism of H-modules (upgraded from the vector space version (2.1)) and applying the theorem to both sides. The second equation is then established by taking L " k and taking degree zero parts on both sides in the first equation.
Remark 2.8. We will be applying the results in this section to a slightly more general situation than graded Hopf algebras in what follows. In particular, we will be studying graded vector spaces with a non-trivial braiding, and H being a Hopf algebra object in this braided category. The results of this section hold without any changes as long as H is also a Frobenius algebra object.
3. THE HOPF ALGEBRA H n AND ITS BOSONIZATION 3.1. Braided vector spaces. The category gvec of finite-dimensional Z-graded vector spaces is naturally a symmetric monoidal category with the symmetric braiding τ pv b wq " w b v. For the purpose of this paper, we will consider a non-symmetric braiding on this category. Fix a natural number N ě 2 and let k be a field of any characteristic which contains a primitive N th root of unity q. Given two graded vector spaces V, W define the Z-linear map where v, w are homogeneous elements. It follows that Ψ defines a braiding on the category of Z-graded vector spaces. We denote the braided monoidal category thus obtained by gvec q (in contrast to the symmetric monoidal category gvec).
Via a form of Tannakian reconstruction, the category gvec is equivalent to the category of finite-dimensional comodules over the group algebra kC, where C " xKy is the free abelian group generated by K. The Hopf algebra kC can be equipped with a dual R-matrix R : kC b kC Ñ k defined by We denote the category of finite-dimensional C-comodules with braiding obtained from R by C-comod q . Hence there is an equivalence of braided monoidal categories kC-comod q » gvec q .
3.2. Graded rational modules. Let H be a Hopf algebra object in gvec q . We want to study the category of H-modules in gvec q in terms of graded modules over a k-Hopf algebra. For this, we first pass from gvec q to a braided category of modules over the group algebra of a finite cyclic group. Let C N denote the finite group C{pK N q and let π N : C Ñ C N be the canonical quotient homomorphism of groups. Then there is an induced Hopf algebra morphism kC Ñ kC N , which, in turn, produces a functor of monoidal categories where δ denotes the left coaction on V . The dual R-matrix R on kC induces a dual R-matrix on kC N so that pπ N q˚becomes a functor of braided monoidal categories For the next result, note that N must be invertible in k since, as the polynomial f pxq " x N´1 does not have multiple roots in k, its formal derivative equals N x N´1 ‰ 0.
Proposition 3.1. There is an equivalence of braided monoidal categories kC N -comod q » kC N -mod q . Here, the latter is the braided monoidal category of kC N -modules with braiding given by the R-matrix Proof. Denote by krC N s the algebra of k-linear functions C N Ñ k. This is a Hopf algebra, dual to the group algebra kC N . Consider the basis tδ i | 0 ď i ď N´1u for krC N s, where δ i pK j q " δ i,j ; we also denote δ k " δ l if k " l mod N . The relations, and structural morphisms ∆, ǫ, and S of the Hopf algebra structure for krC N s, are given by An explicit Hopf algebra pairing p , q : krC N s b kC N Ñ k is given by pδ i , K j q " δ i,j . This nondegenerate Hopf algebra pairing defines, as krC N s is finite-dimensional and (co)commutative, an equivalence of monoidal categories where for a homogeneous element v of degree i we define the action δ j¨v " δ i,j v.
Under the pairing p , q the dual R-matrix RpK i , K j q " q ij for the group algebra kC N induces on krC N s the universal R-matrix Hence, denoting the obtained braided monoidal category of krC N s-module by krC N s-mod q , we obtain an equivalence of braided monoidal categories kC N -comod q » krC N s-mod q . Note also that, since the polynomial f pxq " x N´1 splits over k, krC N s is isomorphic to kC N as a Hopf algebra, although not canonically. An isomorphism kC N Ñ krC N s is given by sending K to the group like element ř i q i δ i . Since δ i , i " 0, . . . , N are mutually orthogonal idempotents, one has`ř i q i δ i˘k " ř i q ik δ i . The inverse is given by sending δ j to 1 N ř i q´i j K i . The above isomorphism of Hopf algebras kC N -krC N s makes kC N a quasi-triangular Hopf algebra with universal R-matrix given as in equation (3.2). Indeed, we compute that applying the above isomorphism to the universal R-matrix of kC N from equation (3.2) gives which is the universal R-matrix of krC N s from equation (3.4). See [Maj95, Example 2.1.6] for a direct proof of this quasi-triangular Hopf algebra structure.
The convolution inverse R´˚is given by In any braided monoidal category B, we can form the braided tensor product D 1 b D 2 of two algebra objects D 1 , D 2 in B. The product m D 1 bD 2 is given by Tensor products of coalgebra objects are defined similarly. We can also define bialgebra (or Hopf algebra objects) in B. These are sometimes called braided Hopf algebras, see e.g. [Maj95, Definition 9.4.5]. The crucial point is that a bialgebra B in B is both an algebra and coalgebra in B such that ∆ and ǫ are morphisms of algebras, i.e.
Let H be a braided Hopf algebra in gvec q . Then the image of H under the composite functor is a braided Hopf algebra in kC N -mod q . By slight abuse of notation, this image is also denoted by H. We may now consider the Radford-Majid biproduct ( [Rad85], also called the bosonization [Maj95, Theorem 9.4.12]) H ⋊ kC N . By construction, there is an equivalence of categories where the latter denotes the category of modules over H within the braided monoidal category kC N -mod q . That is, the morphisms of the H-module structure are all morphisms in this category, cf. [Maj95, Section 9.4]. The monoidal functor P therefore restricts to a monoidal functor Note that, in addition, H is a graded k-algebra, and the bosonization H ⋊ kC N is a graded Hopf algebra, where deg K " 0. Thus, we can consider graded modules over H ⋊ kC N , and the essential image of the functor P H is contained in H ⋊ kC N -gmod.
We denote the category of rational graded H ⋊ kC N -modules, together with morphisms of graded H ⋊ kC N -modules, by H ⋊ kC N -rmod.
Working with rational graded modules we obtain a characterization of the braided monoidal category H-gmod pkC-gmod q q in terms of modules over the finite-dimensional Hopf algebra H ⋊ kC N :

Proposition 3.3. An H ⋊ kC N -module V is in the essential image of the monoidal functor P H if and only if V is a rational graded module.
Proof. Let V be an H ⋊ kC N -module in the essential image of P H . Then, in particular, V is a graded H ⋊ kC N -module. For a vector v P V i we have that Hence, V is a rational graded module. Conversely, let W be a rational graded module over H ⋊ kC N . Then W is graded, and hence a kC-comodule. Using that H ãÑ H ⋊ kC N is a graded subalgebra, W becomes a graded H-module, denoted by W 1 . We have to show that P H pW 1 q and W are isomorphic as graded H ⋊ kC N -modules. By construction, they are the same graded H-modules, and for a vector w P P H pW 1i q, K¨w " q i¨w . As W is rational graded, the same formula describes the C N -action on W . It follows that P H pW 1 q and W are also isomorphic as H ⋊ kC N -modules.
It follows that, as a full subcategory of H ⋊ kC N -gmod, H-gmod is closed under tensor products and extension. As all rational C N -modules are graded modules, all constructions from Section 2.1 can be applied to rational graded C N -modules. In particular, the internal graded hom Hom ‚ pM, N q of two rational graded modules is itself a rational graded module. Notation 3.4. This section shows that the category H-gmod of graded H-modules has a tensor product which can either be computed using the coproduct within gvec q or, equivalently, the coproduct of the bosonization by Proposition 3.3. In Section 4, we will simply denote the resulting monoidal category by H-gmod.
3.3. A braided Hopf algebra. We first fix some notation and assumptions. Let n ě 2 be a positive integer, and factorize n " p a 1 1 . . . p at t as a product of distinct prime powers. Denote by m " p 1 . . . p t the radical of n and define N :" n 2 {m. Set n k :" n{p k , m k :" m{p k .
We assume the ground field k contains a primitive N th root of unity q. Then we denote ξ :" q n{m , which is a primitive nth root of unity, and ξ k :" ξ m k " q n k .
Definition 3.5. Let H n be the k-algebra which is graded by setting degpd k q " n k for all 1 ď k ď t.
Define the comultiplication map ∆ : H n ÝÑ H n b H n on generators by and set the counit and antipode maps to be (3.8) ǫ : H n ÝÑ k, ǫpd k q " 0, (3.9) S : H n ÝÑ H op n , Spd k q "´d k , for all 1 ď k ď t.
Lemma 3.7. The above definitions of ∆, ǫ, and S uniquely extend to give H n the structure of a primitively generated Hopf algebra object in the braided category gvec q of q-vector spaces.
Proof. It is well-known that the free k-algebra kxd 1 , . . . , d t y extends to give the structure of a primitively generated braided Hopf algebra in the braided category of q-vector spaces in a unique way. The conditions from equation (3.6) inductively imply that ∆pd a k q " It hence remains to check that the ideal generated by rd k , d l s for l ‰ k and d p k k is a Hopf ideal. This follows as the generators are primitive elements: Here, we have used that ξ n k l " ξ m l n k " q n k n l " 1, and that ξ n k k " ξ m k n k " q n 2 k is a primitive p k -th root of unity.
Remark 3.8. The braided Hopf algebra H n can be constructed as the Nichols algebra over the Yetter-Drinfeld module V " Span k td 1 , . . . , d t u over the group C N (see e.g. [AS02] for this construction). The C N -coaction δ on V is given by δpd k q " K n k b d k , and the C N -action is given by K¨d k " ξ k d k . The Yetter-Drinfeld braiding Ψ V of V determines the relations in the Nichols algebra BpV q " H n . Note that, for distinct indices k, l " 1, . . . , t, This implies that in the Nichols algebra H n the relations rd k , d l s " 0 hold.
Using that ξ n k k is a primitive p k th root of unity in k, this computation of the braiding implies that in the Nichols algebra, d p k k " 0. These are the only relations (cf. [AS02, Theorem 4.3]). This construction as a Nichols algebra proves that H n is a braided Hopf algebra in kC N -gmod q which is generated by primitive elements.
This construction of H n further implies that H n is self-dual as a braided Hopf algebra. That is, there is a non-degenerate Hopf pairing x-, -y : H n b H n Ñ k, defined on generators by xd k , d l y " δ k,l in the category gvec q (see [Lus93, Proposition 1.2.3]).
By construction, H n is a commutative algebra. Note that, even though Ψ∆pd k q " ∆pd k q for all generators, H n is not braided cocommutative in gvec q . This follows using [Sch98, Corollary 5], since Ψ 2 Hn,Hn ‰ Id b Id.
Remark 3.9. The element Λ :" d p 1´1 1¨¨¨d pt´1 t has the property that hΛ " ǫphqΛ, @h P H n .
That is, Λ is an integral element for the braided Hopf algebra H n (as in [BKLT00, Definition 3.1]), cf. also Lemma 3.12 below. Note that Trphq " xh, Λy, @h P H n , (3.12) with respect to the integral and trace map from Lemma 3.6. We denote the degree of the integral Λ by 3.4. The bosonization of H n . In order to study modules over H n in terms of rational graded modules, we consider the bosonization H n ⋊kC N . Using Section 3.2, H n is a Hopf algebra object in C N -mod q . Hence, we can form the bosonization H n ⋊ kC N [Rad85].
Lemma 3.11. The Hopf algebra H n ⋊kC N is generated by the elements d 1 , . . . , d t and K as a k-algebra, subject to the algebra relations The coproduct, antipode and counit are given on the generators by Proof. This follows using [Maj95, Theorem 9.4.12].
Inductively, we obtain the formula ∆pd a k q " for any integer a ě 0. Using K n k d l " d l K n k for k ‰ l, we derive a more general formula. For this, given a t-tupel of non-negative integers a " pa 1 , . . . , a t q P N t 0 , we write d a " d a 1 1 . . . d at t and K a " K a 1 n 1 . . . K atnt . Then where the sum is taken over all b " pb 1 , . . . , b t q P N t 0 such that b k ď a k for all k, and a´b " pa 1´b1 , . . . , a t´bt q P N t 0 .
Lemma 3.12. The element Λ 1 " Proof. We have to show that hΛ 1 " ǫphqΛ 1 for all h P H n ⋊ kC N . It suffices to check the property on generators, on which it is evident.
Note that the trace map Tr from Lemma 3.6 is related to Λ in the following way. First, there is a non-degenerate Hopf pairing x , y : pH n ⋊ kC N q b pH n ⋊ kC N q Ñ k obtained by extending the pairing x , y from Remark 3.8 via Thus, H n ⋊ kC N is self-dual as a Hopf algebra. Following [LS69], we obtain another, so-called right orthogonal, pairing p , q on pH n ⋊ kC N q b pH n ⋊ kC N q by the formula Restricting p , q to H n b H n gives the pairing given by Trpd a¨db q which makes H n a Frobenius algebra.
3.5. A spherical structure. In this section we show that the category H n -gmod, viewed as a monoidal category using the tensor product structure of H n ⋊ kC N -rmod, is a spherical monoidal category (cf. [BW99, Section 2] or [EGNO15, Section 4.7]).
(ii) Conjugating by ω implements S 2 . That is, for any h P H n ⋊ kC N , Proof. This follows from a simple computation using Lemma 3.11.
The lemma shows that H n ⋊ kC N is almost a spherical Hopf algebra, with only condition (5) of [BW99, Definition 3.1] missing. However, working with the full subcategory H n -gmod, this condition will always hold to give the following result: Proposition 3.14. The monoidal category H n -gmod is a spherical category.
Proof. This follows using [BW99,Theorem 3.6]. In fact, the conditions from Lemma 3.13 give that H n ⋊kC N -mod is a pivotal category [BW99, Definition 2.1]. We observe that for V a rational graded H n -module, ω acts by Thus, for any graded morphism θ : V Ñ V of H n -modules, ωθ " θω. This shows that H n ⋊ kC N -rmod is a spherical category.
3.6. A weak replacement for the braiding. In general, the category H n ⋊ kC N -gmod and its subcategory H n -gmod are not braided monoidal. This agrees with the observation of [Bic03, Proposition 5] that the category of n-complexes is not braided monoidal (unless q " q´1).
A further observation is that, as an algebra, H n does not depend on the parameter q. The coproduct and H n -module structure, however, are dependent on q, manifested in the use of the braiding in gvec q . For any choice of a primitive N th root of unity, we have two different coproducts on H n ⋊kC N -the coproduct ∆ " ∆ q from Lemma 3.11, and its opposite coproduct ∆ op " ∆ op q . The tensor product obtained from the former is denoted by b " b q for the purpose of this section. Note the symmetry that are distinct coproducts for bosonizations of H n , utilizing b q or b q´1 , respectively.
In this section, we describe a weaker symmetry that is present in place of a quasi-triangular structure on H n ⋊kC N . A quasi-triangular structure would give natural isomorphisms V bW -W b V. Instead, we obtain the following.
Proposition 3.15. There are natural isomorphisms of graded H n -modules Proof. The proposition can be checked by a direct computation that Ψ V,W intertwines with the action of the H n generators d k , k " 1, . . . , t. More intrinsically, consider the universal R-matrix R for kC N from equation (3.2). Now, R is a right 2-cycle for kC N , and also for H n ⋊ kC N which contains kC N as a Hopf subalgebra. Hence, we can consider the Drinfeld twist ∆ R q " R´˚∆ q R of the coproduct of H n ⋊ kC N [Dri89]. We compute that The result follows.

A TENSOR IDEAL IN H n -gmod
4.1. The category of H n -modules. We use the same notation as in the previous sections, and work with the category H n -gmod of finite-dimensional graded H n -modules. This category has internal homs Hom ‚ pV, W q -V˚b W . The differential d k P H n acts on an element f : V Ñ W , for v homogeneous of degree i, by pd k¨f qpvq " ξ´i k pd k f pvq´f pd k vqq, (4.1) as in equation (2.11) 1 , where ξ k " q n k . Hence d k¨f " 0 if and only if f pd k vq " d k f pvq for all v P V . In particular, a linear map is graded H n -invariant if and only if it is of degree zero and commutes with all differentials. In this way, the category of H n -modules is enriched over itself.
As H n is naturally a Z-graded algebra, we have the grading shift functors on H n -gmod tku : H n -gmod ÝÑ H n -gmod for all k P Z. Equivalently, consider the modules kt˘1u, which are one-dimensional over k, with generators 1 sitting respectively in Z-degrees¯1.
This shows that the isomorphism V b kt˘1u Ñ V t˘1u sending v i b 1 to v i commutes with the d k -action, for v i b 1 has degree i¯1.
Lemma 4.1. For any two H n -modules V, W , there are natural isomorphisms of H n -modules Proof. We show the t1u case. Since W t1u -W b kt1u, the first isomorphism is easy. To establish the second isomorphism, we consider the isomorphisms of H n -modules which reduces the problem to showing that kt1u b W -W b kt1u.
1 Note that this formula differs slightly from [Kap96, equation (1.14)]. A formula similar to that of Kapranov is obtained by using the alternative internal hom from Remark 2.1. In this case, we would obtain pd k f qpvq " d k f pvq´ξ degpf q k f pd k vq. The results of this section apply using either convention.

R. Laugwitz and Y. Qi
Denote by 1 a generator of kt1u which lives in degree´1. We define the map r V , for any homogeneous v i P V i , by It follows that, for any k " 1, . . . , t, proving that r V is a morphism of H n -modules. Naturality is clear as any morphism f : V Ñ W of H n -modules preserves the grading, and hence The grading shift t´1u is similar, and one just replaces q by q´1 in the above computations.
Proof. The first equation (4.2) is a repeated application of the previous Lemma 4.1.
Using equation (2.12) and the first part of the corollary, we have the chain of isomorphisms of H n -modules The last isomorphism in the second equality (4.3) is established in a similar way.
As a special case, we can consider n " p a . In this case, we can fully classify indecomposable modules over H n . Any indecomposable H n -module is isomorphic to a grading shift of a quotient module H n {pd l 1 q, for l " 0, . . . , p 1´1 . Such a simple classification is not possible in the presence of more than two distinct prime factors in n.

4.2.
The tensor ideals I k . Once again, fix a positive integer n and its prime decomposition n " p a 1 1¨¨¨p at Lemma 4.3. The left regular module is, up to isomorphism and grading shift, the only indecomposable projective-injective H n -module. Its graded dimension equals Proof. This follows since H n is a graded Frobenius local algebra (Lemma 3.6), and thus is graded self-injective. The graded dimension computation is an easy exercise.
Definition 4.4. For each prime factor p k of n, we define a p k -dimensional graded H n -module V k by V k :" Ind Hn Further, if t ą 1, we denote Observe that the H n -module V k is a p k -fold extension of the trivial H n -module k by itself: We further observe that V k is isomorphic as an H n -module, up to grading shift, to the submod- . Similarly, W k is isomorphic to a grading shift of the submodule generated by d p k´1 k . It follows similarly to Lemma 4.3 that dim ν pV k q " 1´ν n 1´ν n k , dim ν pW k q " ź l‰k 1´ν n 1´ν n l . (4.5) The module V k is free when viewed as an H k n -module, while W k is free as an p H k n module. In particular, W k is free as an H l n -module for all l ‰ k. Definition 4.5. Assume that t ą 1. For any k " 1, . . . , t, we let I k be the full subcategory of modules in H n -gmod consisting of direct summands of H n -modules V of the following form: (i) V is equipped with a finite-step filtration by H n -submodules: 0 " F 0 Ă F 1 Ă F 2 Ă¨¨¨Ă F r " V . (ii) Each of the subquotient modules F i {F i´1 (i " 1, . . . , r) is isomorphic to W k up to a grading shift. If t " 1, so that n " p a 1 1 , we denote I 1 :" I Hn , the full subcategory of graded projective-injective H n -modules, cf. Section 2.2.
Lemma 4.6. The ideal I k is closed under extensions. More precisely, if U , V and W fit into a short exact sequence of H n -modules 0 ÝÑ U ÝÑ W π ÝÑ V ÝÑ 0 with U, V being in I k , then W also lies in I k .
Proof. The case when t " 1 is clear, so we assume t ą 1. Assume given such a short exact sequence of H n -modules such that U 1 , V 1 be H n -modules satisfying that U ' U 1 and V ' V 1 are equipped with filtrations F 1 Ă¨¨¨Ă F r and F 1 1 Ă¨¨¨Ă F 1 s as in Definition 4.5. Then we have a short exact sequence and W ' U 1 ' V 1 is equipped with a filtration 0 Ă F 1 Ă¨¨¨Ă F r " U Ă π´1pF 1 1 q Ă¨¨¨Ă π´1pF 1 s q " W, which satisfies the hypothesis of Definition 4.5. Hence W , as a direct summand of W ' U 1 ' V 1 , is contained in I k .
Lemma 4.7. The ideal I k is closed under forming duals and tensor products with arbitrary objects in H n -gmod. Consequently, I k is a two-sided tensor ideal in H n -gmod.
Proof. The case t " 1 follows from [Kho16, Proposition 2]. Hence, we assume t ą 1. If V is a direct summand of an object W of I k with a filtration F ‚ , then W˚is equipped with the dual filtration F‚ , which is readily checked to satisfy the conditions of Definition 4.5. Hence V˚, as a direct summand of W˚, is an object in I k .
Suppose V P I k and U is any H n -module. The module U has a nontrivial socle since H n is a graded local algebra. Choose ktsu lying inside the socle of U , which gives us a short exact sequence of H n -modules 0 ÝÑ ktsu ÝÑ U ÝÑŪ ÝÑ 0. Tensoring, for instance, on the left with V , we obtain By induction on dimpU q, we may assume that V bŪ P I k (the case dimpU q " 1 is the assumption that V P I k ). Now the previous lemma applies and shows that V b U P I k .
It follows that the internal homs also preserve the ideals I k .
Corollary 4.8. Let U be an H n -module in the ideal I k and V be an arbitrary finite-dimensional H nmodule. Then both Hom ‚ pU, V q and Hom ‚ pV, U q are objects of I k .
Proof. This follows from Lemma 4.7 and the isomorphism of graded H n -modules Hom ‚ pU, V q -U˚b V from equation (2.12).
Remark 4.9. We note that the category I k is the smallest subcategory of H n -gmod closed under grading shifts, extensions, and direct summands that contains the objects W k . We conjecture that any object in I k in fact has a filtration as in Definition 4.5.
Lemma 4.10. The class of projective-injective objects of H n -gmod is contained in each I k , for k " 1, . . . , t.
Proof. This follows since we have Example 4.11. Let n " 2 a¨3b , with a, b ě 1. Then d 1 raises degrees by n 1 " 2 a´1 3 b , and d 2 raises degrees by n 2 " 2 a 3 b´1 . We note that V k " W k in the case of only two distinct prime factors. Let us consider the following module V with the non-zero differential acting by identity maps indicated on the arrows: The module V is contained in the ideal I 2 (note that p 2 " 3 here). Note that V does not split as a direct sum of shifts of W 2 , but we see that there is a short exact sequence of H n -modules 0 ÝÑ W 2 tn 2´n1 u ÝÑ V ÝÑ W 2 ÝÑ 0.
If n " 2 a¨3b¨5c , there exist various non-split extensions in I 1 . For example, consider the module W , where we omit the degree shifts, Proof. We first show the statement for V " W k and W " W l , k ‰ l. According to Theorem 2.6, Hom ‚ Hn-gmod pW k , W l q " Since k ‰ l, we can equip W k with a filtration of H n -modules whose successive quotients are grading shifts of the module V l . Hence Wk also has such a filtration. Next, we observe that the tensor product V l tsu b W l is free over H n . Inductively, it follows from the exactness of b that Wk bW l has a (split) resolution by free H n -modules and is hence free. Therefore, Λ¨pWk bW l q " pWk b W l q Hn and we have shown that Hom ‚ Hn-gmod pW k , W l q " t0u. Using Corollary 4.2, we can replace W k , W l by grading shifts. Thus, the statement holds for all modules V in I k and W in I l that have filtrations as in Definition 4.5. If U V is a direct summand of V and U W a direct summand of W and f : U V Ñ U W an H n -module morphism. Then f extends by zeros to a H n -morphism V Ñ W , which is null homotopic by the above. Hence, f is also null-homotopic, and the statement is proved for general objects in I k and I l .

CATEGORIFYING CYCLOTOMIC RINGS
In this section, we construct a tensor triangulated category O n , whose Grothendieck ring is isomorphic to the cyclotomic ring O n at an nth root of unity. 5.1. A triangulated quotient category. Consider the stable category H n -gmod from Section 2 which is tensor triangulated. Let us denote by I the full subcategory consisting of objects that are isomorphic to those of I under the natural quotient functor H n -gmod ÝÑ H n -gmod. Thus I is a strictly full subcategory of H n -gmod. Our first goal is to show that I is a thick triangulated subcategory in H n -gmod. To do this we first exhibit some preparatory results.
Lemma 5.1. The subcategory I is closed under the tensor product action by H n -gmod. More precisely, if U is an object of I and V P H n -gmod, then both V b U and U b V are in I. Consequently, I constitutes a tensor ideal in H n -gmod.
Proof. We may take U to be the image of an object of I under the quotient functor. The lemma is then a consequence of Lemma 4.13 of the previous subsection.
Corollary 5.2. The subcategory I is closed under the homological shifts of H n -gmod.
Proof. This follows from the previous Lemma and the fact that for any object U P H n -gmod. Proof. Using Corollary 5.2 and the fact that any distinguished triangle is isomorphic to a standard distinguished triangle, we are reduced to showing that, if U , V are objects of I and f : U ÝÑ V is a map of H n -modules, then the cone C f of f is also in I.
There exist direct sum decompositions U -' t k"1 X k and V -' t l"1 Y l , with X k , Y k P I k . Under these isomorphisms, f " pf kl q is a matrix of H n -module maps, where f kl " π Y l f ι X k for the canonical inclusion ι X k : X k Ñ U and projection π Y k : V Ñ Y k . It follows from Proposition 4.16 that the images of the components f kl are zero in H n -gmod. Hence, we may replace f by the diagonal H n -module map f 1 " pδ k,l f kk q which has an isomorphic cone in H n -gmod. by C, fits into the diagram Thus the dashed arrow exists by the exactness of Hom Hn-gmod pX, -q applied to the distinguished A standard distinguished triangle in O n is the image of a distinguished triangle in H n -gmod, and any triangle of O n isomorphic to a standard distinguished triangle is called a distinguished triangle.

5.2.
Tensor triangulated structure. Our goal in this part is to establish the triangulated tensor category structure on O n which is inherited from that of H n -gmod under localization.
Lemma 5.8. The following functors on H n -gmod descend to (bi-)exact functors on O n : (1) The tensor product p-b -q : H n -gmodˆH n -gmod ÝÑ H n -gmod.
Proof. The tensor product functor b on H n -gmod is bi-exact [Kho16]. Thus, for (1), it suffices to show that it preserves the class of quasi-isomorphisms. Let s : M Ñ M 1 be a quasiisomorphism in H n -gmod that arises from an actual H n -module map s : M Ñ M 1 . Replacing s by ps, ρ M q : M ÝÑ M 1 ' M b Htℓu if necessary, we may assume from the start that s is injective. Thus C :" cokerpsq is isomorphic to a module in I in H n -gmod, and a direct sum of C by some projective-injective H n -module belongs to I. Since I is closed under summands (Lemma 4.15), we may assume C is also in I. Tensoring the exact sequence 0 ÝÑ M s ÝÑ M 1 ÝÑ C ÝÑ 0 with any module N on the left, we have a short exact sequence By Lemma 4.13, N b C P I, and hence Id N b s descends to a quasi-isomorphism in H n -gmod. The case of tensoring on the right is similar, and this finishes the proof of (1).
Part (4) is clear since the dual of any object in I is also in I by definition. Now part (2) and (3) are easy consequences of (1) and (4) because of Corollary 4.2 and the isomorphism Hom ‚ pM, N q -M˚b N of equation (2.12).
We are now ready to establish a tensor-hom adjunction in our category O n . Here we have adopted the conventional notation The left-most and, by inductive hypothesis, the right-most vertical arrow are isomorphisms of Ext-groups. The theorem then follows from the usual "two-out-of-three" properties for distinguished triangles in triangulated categories.
They hold because the multiplicative group of mth roots of unity is partitioned, by the order of the root of unity, into the divisors d of m. The product of all ν´q, where q is a primitive dth root of unity, is equal to Φ d pνq. It follows that, if d ‰ m, then d is not divisible by at least one of the distinct primes p k , and thus Φ d pνq does not divide rms ν {rm{p k s ν . On the other hand, Φ m pνq clearly divides each rms ν {rm{p k s ν , k " 1, . . . , t. Hence the greatest common divisor of all polynomials rms ν {rm{p k s ν is precisely Φ m pνq, establishing equation (5.4). Equation (5.5) is easy since, for a prime p, Φ p pνq " pν p´1 q{pν´1q, so that Φ p k pν m{p k q " ν m´1 ν m{p k´1 " ν m´1 ν´1 ν m{p k´1 ν´1 " rms ν rm{p k s ν .
In the following, we denote by K 0 pH n -gmodq the Grothendieck group of the stable category of H n -modules. Given an object V , we denote its class in the Grothendieck group by rV s. Recall that this is the abelian group generated by symbols of isomorphism classes of objects in H n -gmod, subject to relations rU s´rW s`rV s " 0 whenever is a distinguished triangle.
The monoidal structure of H n gives K 0 pH n -gmodq a ring structure, and the Z-grading shift introduced in Section 4.1 gives it the structure of a left and right Zrν, ν´1s-algebra, such that the left and right module structure coincide using the natural isomorphism from Lemma 4.1.
The tensor product on H n -gmod descends to the multiplication on the Grothendieck group level, while the grading shift functor t1u descends to multiplication by ν.
Proof. The Grothendieck ring K 0 pH n -gmodq is generated, as a Zrν, ν´1s-module, by the class of the only simple H n -module k, which is one-dimensional. The only relations imposed on the symbol of the simple module arise from graded dimensions of projective-injective H n -modules. The result thus follows from Lemma 4.3.
In contrast, the Verdier quotient category O n categorifies the cyclotomic ring O n .
Theorem 5.15. The Grothendieck ring of O n is isomorphic to the ring of cyclotomic integers K 0 pO n q -Zrν, ν´1s pΦ n pνqq .
Proof. We have an exact sequence of triangulated categories where the first containment is fully faithful and idempotent complete (Lemma 5.4). It follows from well-known facts on K-theory of exact sequence of triangulated categories that K 0 pO n q " K 0 pH n -gmod{Iq " K 0 pH n -gmodq{K 0 pIq