Noncommutative tensor triangulated categories and coherent frames

We develop a point-free approach for constructing the Nakano-Vashaw-Yakimov-Balmer spectrum of a noncommutative tensor triangulated category under some mild assumptions. In particular, we provide a conceptual way of classifying radical thick tensor ideals of a noncommutative tensor triangulated category using frame theoretic methods, recovering the universal support data in the process. We further show that there is a homeomorphism between the spectral space of radical thick tensor ideals of a noncommutative tensor triangulated category and the collection of open subsets of its spectrum in the Hochster dual topology.


Introduction
The subject of tensor triangular geometry has been an active area of research for the past two decades and has touched a wide range of areas in mathematics including algebraic geometry, modular representation theory, stable homotopy theory, noncommutative topology, to name a few.The subject involves the study of triangulated categories with a given biexact symmetric monoidal functor called the tensor.Balmer [1,2] showed that the triangulated category of perfect complexes over a scheme X along with the derived tensor functor contains enough data to reconstruct X, establishing that the subject is rich enough to be studied.Associated to a tensor triangulated category C, Balmer [2] constructed a locally ringed space Spec C called the spectrum of C which carries the geometric essence of the tensor triangulated category.For example, Spec Dperf(X) ∼ = X for a quasi-compact, quasi-separated scheme X (see [2,8]).The construction of the spectrum involves constructing a space out of prime thick tensor ideals of the tensor triangulated category.Balmer showed that this satisfies the correct universal property among all spaces which act as targets for support data.In the case of modular representation theory, where the relevant tensor triangulated category is the stable module category of modules over a finite group scheme G, the spectrum recovers the projective support variety.In addition to capturing the underlying scheme, the subject of tensor triangular geometry further lifts to this abstraction, the notions of finite étale maps [5], the Chow group and intersection theory [4,14,15], Grothendieck-Neeman duality, Wirthmüller isomorphism [6] among others.The theory also detects (the failure of) Gersten conjecture for singular schemes [3].This demonstrates the richness of the theory.However, all tensor structures on triangulated categories need not be symmetric.The basic examples being stable module categories over Hopf algebras.To study these Nakano, Vashaw and Yakimov [17] introduced a noncommutative version of tensor triangulated categories and, extending Balmer's theory, constructed a topological space in terms of prime thick tensor ideals.This construction is also universal in a manner parallel to Balmer's construction.Nakano et al prove that in this case again, the space corresponds to the projective support variety described in terms of the cohomology of the Hopf algebra.Following the success of the theories a natural direction of exploration will be to understand the construction of spectrum itself to gain a better insight into the structure of the tensor triangulated category.A conceptual and formal way of constructing the spectrum was described by Kock and Pitsch [16] using the language of frames and locales (point free topology).Frames are complete lattices where finite meet distributes over arbitrary joins (see [12] or section 2.2).A typical example of a frame is the lattice of open subsets of a topological space.The essence of point free approach to toplogy is to reduce the study of topology to the study of these frames.In this approach, one constructs the topological space in terms of the frame of open sets instead of starting with a set of points.Spectral spaces are topological spaces homeomorphic to the spectrum of a commutative ring.The spectral spaces correspond to coherent frames (see Definition 2.6) in the point free approach.Kock and Pitsch gave a point free description of the Balmer spectrum of a (commutative) tensor triangulated category C as the Hochster dual of the spectral space associated to the coherent frame of radical thick tensor ideals of the tensor triangulated category.They also define a notion of support taking values in a frame and prove that the coherent frame of radical thick tensor ideals is universal as a target for such supports.Kock and Pitsch's paper shows that one can arrive at various results about the Balmer spectrum (including the sheaf of rings) from this viewpoint.In this paper, we explore the noncommutative Balmer spectrum studied by Nakano et al from a frame theoretic viewpoint.However, in this case the arguments have to be modified substantially due to the lack of commutativity of tensor and consequently the failure of the construction of the radical of an ideal by adjoining n-th roots (where n is a natural number).For the modified arguments to work, one needs to restrict to a class of noncommuative tensor triangulated categories, where the prime ideals (defined in terms of thick tensor ideals as IJ ⊆ P =⇒ (I ⊆ P ) or (J ⊆ P )) are complete primes (defined in terms of objects, i.e., A ⊗ B ∈ P =⇒ (A ∈ P ) or (B ∈ P )).As Nakano et al [18] shows there is a rich class of examples where this holds, for instance the stable module category of any finite dimensional Hopf algebra.Following Kock and Pitsch, we extend the notion of a frame theoretic support data to the noncommuative setup and prove the relevant universal properties to recover the spectrum.Let K be a (noncommuative) tensor triangulated category.The radical thick tensor ideals form a coherent frame (Theorem 3.6) and the association of a ∈ Ob(K) to the smallest radical thick tensor ideal containing a gives a universal frametheoretic support datum (Theorem 3.10) giving us a classification of radical thick tensor ideals (Theorem 3.11).The relation of this construction with the Nakano-Vashaw-Yakimov-Balmer spectrum is clarified in Corollary 3.12.Nakano et al's construction of the universal support data taking values in the Balmer spectrum is recovered in a frame theoretic way in Proposition 3.20.Finally, extending a result by Banerjee [7], we also show that there is a homeomorphism between the set of radical thick tensor ideals and the set of closed subsets of the spectrum with quasi-compact complements under the proper notions of topologies on these sets (see Theorem 3.22).

Preliminaries 2.1 Noncommutative tensor triangulated category and support
A general noncommutative theory of tensor triangular geometry was introduced by Nakano, Vashaw and Yakimov in [17].They further studied support maps and its connection with tensor product in the setup of noncommutative tensor triangular geometry in their next paper [18].In this section, we recall some definitions and results from [17] and [18].A noncommutative tensor triangulated category, as introduced in [17], is a triangulated category K with a biexact monoidal structure.Throughout this paper K will denote an essentially small noncommutative tensor triangulated category.
Definition 2.1 ([17], §1.2). (1) A thick tensor ideal of K is a full triangulated subcategory I of K such that it contains all direct summands of its objects and for any A ∈ Ob(I), we have A ⊗ B, B ⊗ A ∈ Ob(I) for all B ∈ Ob(K).
(2) A prime ideal of K is a proper thick tensor ideal P such that for all thick tensor ideals I and J of K, we have I ⊗ J ⊆ P =⇒ I ⊆ P or J ⊆ P. We denote by Spc(K) the collection of all prime ideals of K.
(3) A completely prime ideal of K is a proper thick tensor ideal P such that A ⊗ B ∈ P =⇒ A ∈ P or B ∈ P for all A, B ∈ Ob(K).
Definition 2.2 ([17], §1.2).The noncommutative Balmer spectrum Spc(K) of K is the topological space of prime ideals of K endowed with the Zariski topology which is given by closed sets of the form for all subsets S of K.
Let X be a topological space and let X cl (X) denote the collection of all closed subsets of X.
We recall (see, [17,Lemma 4.1.2])that the restriction of the map V (as in Definition 2.2) to the objects of K gives a support datum K −→ X cl (Spc(K)) Equivalently, for any support datum σ satisfying the above condition, there is a unique continuous map ).This map is precisely given by

Coherent frames and support
In this section, we recall some definition and results from [12], [13] and [16].
Definition 2.5.A frame is a complete lattice which satisfies the infinite distributive law: A frame map is a lattice map that preserves arbitrary joins.The category of frames and frame maps is denoted by Frm.
There is a pair of adjoint functors between the category of topological spaces Top and the opposite category of frames Frm op which we now recall ([12,§II.1.4]).The open sets of any topological space form a frame with join operation given by union of open sets and finite meet given by intersection.This gives a functor Top −→ Frm op which has a right adjoint, the functor of points.A point of a frame is a frame map x : F −→ {0, 1} where {0, 1} is the Boolean algebra of two elements (with 0 < 1).The set of points of any frame form a topological space whose open sets are given by sets of the form {x : F −→ {0, 1} | x(u) = 1} for any u ∈ F and this gives the functor Frm op −→ Top.
We recall from [12, §II.3.1] that an element a of a frame F is called finite if for every subset S ⊆ F with a ≤ ∨ s∈S s, there exists a finite subset S ′ ⊆ S with a ≤ ∨ s∈S ′ s.Definition 2.6 ( [12], §II.3.2).A frame is called coherent if every element of the frame can be expressed as a join of finite elements and the finite elements form a sublattice (equivalently, 1 is finite and the meet of two finite elements is finite).
Spectral spaces, introduced by Hochster in [11], are topological spaces homeomorphic to the spectrum of a commutative ring.A spectral map between spectral spaces is a continuous map such that the inverse image of a quasi-compact open is quasi-compact.Every coherent frame corresponds uniquely to a spectral space.In fact, we have the following theorem: Theorem 2.7 ( [13]).The category of spectral spaces and spectral maps is contravariantly equivalent to the category of coherent frames and coherent maps.
For a spectral space X, Hochster [11] considered a new topology on X by taking as basic open subsets the closed sets with quasi-compact complements.The space so obtained is called the Hochster dual of X and it is denoted by X ∨ .He showed that the Hochster dual X ∨ of any spectral space X is also a spectral space and that X ∨ ∨ = X.Motivated by this, the Hochster dual of a coherent frame is defined as follows: Definition 2.8 ([16], Definition 1.2.4).The Hochster dual of a coherent frame F is its join completion.
We recall that an ideal of a frame (in general for any lattice) is a down-set, closed under finite joins.An ideal I of a frame F is called prime if 1 / ∈ I and if a ∧ b ∈ I implies either a ∈ I or b ∈ I.The points of a frame F correspond bijectively to prime ideals of F .Indeed, any point x : F −→ {0, 1} corresponds to the prime ideal x −1 (0).Moreover, in any frame, every prime ideal P is principal and the generating element is u P := ∨ b∈P b.We have The generating element u P of a prime ideal P is called a prime element.Therefore, we have the following natural bijections points ↔ prime ideals ↔ prime elements Let (T , ⊗, 1) be a (commutative) tensor triangulated category.We recall the definition of support on (T , ⊗, 1) from [16, §3.2]: Definition 2.9.A support on (T , ⊗, 1) is a pair (F, d) where F is a frame and d : A morphism of supports from (F, d) to (F ′ , d ′ ) is a frame map F −→ F ′ compatible with the maps d and d ′ .
3 Frames, Hochster duality and noncommutative tensor triangulated category Assumption : All primes of K are complete primes.
One has a vast repertoire of examples where this assumption holds, and a detailed description of the current knowledge about this can be found in the introduction of [18] Definition 3.1.Let S be a set of objects in a noncommutative tensor triangulated category K.We define G(S) to be the set of objects of K which are of the following forms: (1) an iterated suspension or desuspension of an object in S, (2) or a finite sum of objects in S, (3) or objects of the form s ⊗ t and t ⊗ s with s ∈ S and t ∈ K, (4) or an extension of two objects in S, (5) or a direct summand of an object in S.
If I is a thick tensor ideal containing S, then clearly G(S) ⊆ I. Hence, by induction, G ω (S) := n∈N G n (S) ⊆ I.
It may be easily verified that G ω (S) is itself a thick tensor ideal and therefore it is the smallest thick tensor ideal containing S. We will denote it by S .
Recall that the radical of an ideal of a noncommutative ring is defined as the intersection of all the prime ideals containing it.In the same spirit, we give the following definition.Proof.Let S := {k ∈ K | k ⊗n ∈ I for some n ∈ N}.Clearly, S ⊆ P for all prime ideals of K such that P ⊇ I. Hence, S ⊆ √ I. Given t / ∈ S , consider the collection Ω of all ideals J ⊇ I such that J ∩ {t ⊗n | n ∈ N} = ∅.Clearly, I ∈ Ω and the set Ω can be partially ordered by inclusion and any chain in Ω has an upper bound in Ω.Therefore, by Zorn's Lemma there exists a maximal element, say M, in Ω.Thus, M ⊇ I and M ∩ {t ⊗n | n ∈ N} = ∅.It is now enough to show that M is prime to prove that t / ∈ √ I. Let k, k ′ ∈ K be such that k ⊗ k ′ ∈ M. It may be easily verified using Lemma 3.3 that M, k ⊗ k ′ ⊆ M and obviously we have M, k M ⊆ M. Therefore, again applying Lemma 3.3 we obtain M, k M, k ′ ⊆ M. Suppose, if possible, k, k ′ / ∈ M. M is maximal, we must have t ⊗n ∈ M, k and t ⊗m ∈ M, k ′ for some n, m ∈ N.This implies t ⊗(n+m) ∈ M, k M, k ′ ⊆ M which gives the required contradiction.Lemma 3.5.Let Rad K denote the poset of radical ideals of a noncommutative tensor triangulated category K satisfying Assumption.Then, Rad K is a frame with the following meet and join operations: for any two radical thick tensor ideals I 1 and I 2 and for any set of radical thick tensor ideals {I j } j∈J .
Proof.By definition, j∈J I j is a radical thick tensor ideal.Also, given a set of radical thick tensor ideals {I i } i∈I , we have for every i ∈ I. Thus, we have i∈I I i = i∈I I i .Hence, Rad K is a complete lattice.Let us now verify that i∈I We clearly have i∈I (J I i ) ⊆ J ( i∈I I i ).Now, let x ∈ J ( i∈I I i ).We define Since, by assumption, all primes are complete, we have x ∈ i∈I (J I i ) and this completes the proof.We will now give a proof of our claim.First, we show that C x is a thick subcategory of K. Let k 1 , k 2 ∈ K be such that k 1 ⊕k 2 ∈ C x .Then since i∈I (J I i ) is a thick subcategory, we have We will now show that C x is radical.Clearly, C x ⊆ √ C x .Now, if possiblle, let t ∈ √ C x = Cx⊆P P be such that t / ∈ C x .This implies either x ⊗ s ⊗ t or t ⊗ s ⊗ x does not belong to i∈I (J I i ) for some s ∈ K. Without loss of generality, suppose x ⊗ s ⊗ t / ∈ i∈I (J I i ) = ∪i∈I (J∩Ii)⊆P P.Then, there exists some prime ideal P 0 ⊇ i∈I (J I i ) such that x ⊗ s ⊗ t / ∈ P 0 .This implies x / ∈ P 0 and t / ∈ P 0 .For k ∈ C x , we have Hence k ∈ P 0 showing that C x ⊆ P 0 .Therefore, t ∈ √ C x ⊆ P 0 .This gives the required contradiction.Therefore, C x is radical.It is clear that I i ⊆ C x for all i ∈ I. Hence, i∈I I i ⊆ C x and since C x is radical, we must have Theorem 3.6.The poset of radical thick tensor ideals Rad K of a noncommutative tensor triangulated category K satisfying Assumption forms a coherent frame.
Proof.By Lemma 3.5, we know that Rad K a frame.It is now enough to check that an element of the frame Rad K is finite if and only if it is a principal radical thick tensor ideal i.e., of the form √ k for some k ∈ K. Let I be a finite element of the frame Rad K .Then, clearly we have We need to check that I is a finite radical thick tensor ideal.Assume that where the J λ are radical thick tensor ideals.Then in particular k 0 ∈ λ∈Λ J λ .Let us denote λ∈Λ J λ by S. Thus, by Proposition 3.4, Then, we have k ⊗ni i ∈ λ∈Λ J λ for some n i ∈ N for each i = 1, . . ., r.Let the finitely many elements of λ∈Λ J λ involved in the iterative construction of k ⊗ni i , i = 1, . . ., r be x 1 , . . ., x m .Suppose {x 1 , . . ., x m } ⊆ J λ1 ∪ . . .∪ J λν for some ν ∈ N. Thus, for each i = 1, . . ., r, we have Proof.(1) Recall that for any frame F , the frame-theoretic points of F correspond bijectively to the prime ideals of F and the prime ideals in turn are in natural bijection with the prime elements of F .Now, we put F = Zar(K).For any point x : Zar(K) −→ {0, 1} of Zar(K), the corresponding prime ideal of Zar(K) is given by p x := x −1 (0) and the corresponding prime element of Zar(K) is given by I x := I∈px I. We also know p x = (I) x = {I ∈ Zar(K) | I ⊆ I x }.We will now show that I x is prime.Let J 1 and J 2 be thick tensor ideals such that J 1 ⊗ J 2 ⊆ I x .Clearly, (J 1 ∩ J 2 ) ⊗2 ⊆ J 1 ⊗ J 2 ⊆ I x .Since I x is radical, we have J 1 ∩ J 2 ⊆ I x by Proposition 3.4.Therefore, we have J 1 ∧ J 2 = J 1 ∩ J 2 ∈ (I x ) = p x in the frame Zar(K).Since p x is a prime ideal of Zar(K) we must have J 1 ∈ p x or J 2 ∈ p x .In other words, we have J 1 ⊆ I x or J 2 ⊆ I x which proves that I x is prime.Thus, we obtain the following well defined map of sets It may be easily verified that y : Zar(K) −→ {0, 1} is a morphism of frames which shows that y is a frametheoretic point of Zar(K).Since p y := y −1 (0) = (P), it follows that y maps to P under (3.0.3). ( The open set corresponding to the finite element √ k of the coherent frame Zar(K) is {x : 3) is a bijection, it follows that the set {x : From the frame theoretic support data, one can reconstruct the support data V : K −→ X cl (Spc(K)) described by Nakano, Vashaw and Yakimov [18,Definition 2.3.1].This is described below in terms of a functorial equivalence between the frame theoretic support data and the support data taking values in closed subsets of Spc(K).Construction 3.13.We briefly recall the construction of a topological support data corresponding to a frame theoretic support data.Suppose that d : K → F is a frame-theoretic support data and that F is coherent.Let X F be the spectral space corresponding to F (see Theorem 2.7).We know that the points in X F correspond to frame maps p : F → {0, 1} and the topology consists of open sets U where the open sets are closed subsets of X F with quasi-compact complement.Consider the assignment σ This is well defined (see remark 3.14).
One also has a reverse construction.By the Lemma in [12, page 41], the closed subsets of Y F are in one-to-one correspondence with elements of F .Thus, given a support σ : K −→ X cl (Y F ), one can define for a ∈ K, d(a) to be the element of F corresponding to the closed subset σ(a).When σ satisfies the tensor product property, d turns out to be a frame theoretic support in the sense of definition 3.8.
Remark 3.14.In the argument below we shall need the fact that the support σ(a) constructed above are closed subsets of Y F .This is standard and can be seen as follows.The space Y F is denoted by (Spec F ) inv in [9].The subsets of the form {p : F −→ {0, 1} | p(a) = 1} are quasi-compact by [9, 2.2.3(c)] as Spec F inherits the subspace topology from 2 F .Notation 3.15.Let F be the category of support data for K taking values in coherent frames as in definition 3.8.Let S be the category of support data for K taking values in spectral topological spaces, along with the restriction that the support data should have the tensor product property.In other words, an object (X, σ) in S is a support data σ : K −→ X cl (X), where X is a spectral topological space and σ in additon to being a support data as defined in Definition 2.3, satisfies σ(a bijective correspondence with radical ideals of R which can be viewed as a nullstellensatz-like result.A topological enhancement of this nullstellensatz-like result was provided by Finocchiaro, Fontana and Spirito in [10] where they showed that this bijective correspondence can be promoted to a homeomorphism.In [7], Banerjee provided a similar "topological nullstellensatz"-like result for a (commutative) tensor triangulated category.Our Theorem 3.20 could be seen as a "topological nullstellensatz" for a noncommutative tensor triangulated category.

Definition 3 . 2 .Proposition 3 . 4 .
We define the radical closure of a thick tensor ideal I of a noncommutative tensor triangulated category K by √ I := I⊆P P where P denotes the prime ideals of K.If I is a thick tensor ideal such that I = √ I, we call I radical.Clearly, any prime ideal is radical.It is also clear that if I is a thick tensor ideal, then √ I is a radical thick tensor ideal.For any set of objects S, let √ S denote the radical of the thick tensor ideal S .Lemma 3.3.Let I and J be two thick tensor ideals and let S be a set of objects of K.Then, {t⊗ s | t ∈ I, s ∈ S} ⊆ J implies I ⊗ S ⊆ J. Proof.By definition, S = n∈N G n (S) and G 0 (S) = S. So, by assumption, I ⊗ G 0 (S) ⊆ J. Suppose I ⊗ G m (S) ⊆ J for some 0 = m ∈ N. We will now show that I ⊗ G m+1 (S) ⊆ J i.e., {t ⊗ x | t ∈ I} ⊆ J (3.0.1) for any x ∈ G m+1 (S).Obviously, (3.0.1) is satisfied if x is a finite sum of objects in G m (S).If x is an iterated suspension or desuspension of an object in G m (S) or if x is an extension of two objects in G m (S), then (3.0.1) holds since ⊗ is biexact.If x is a direct summand of an object in G m (S), then (3.0.1) holds since J is thick.If x is of the form s ⊗ k or k ⊗ s for any s ∈ G m (S) and k ∈ K, then clearly (3.0.1) holds since I and J are ideals.Thus, by induction, we obtain I ⊗ G n (S) ⊆ J for all n ∈ N. It follows that I ⊗ S ⊆ J. Let I be a thick tensor ideal of K.Then, √ I = {k ∈ K | k ⊗n ∈ I for some n ∈ N} .
{frame-theoretic points of Zar(K)} −→ {prime thick tensor ideals in K} x → I x (3.0.3)If I x = I y , then we have x −1 (0) = p x = (I x ) = (I y ) = p y = y −1 (0).Clearly, this implies x = y showing that (3.0.3) is an injection.To show surjection, let P be any prime thick tensor ideal of K.It may be easily verified that (P) := {I ∈ Zar(K) | I ⊆ P} defines a prime ideal of the frame Zar(K).Now, we define y : Zar(K) −→ {0, 1} by y(I) := 0 if I ∈ (P) 1 otherwise corresponds bijectively to the set of prime thick tensor ideals {P ∈ Spc(K) | k / ∈ P}.Corollary 3.12.Let K be a noncommutative tensor triangulated category satisfying Assumption.The noncommutative Balmer's spectrum Spc(K) of K is the Hochster dual of the Zariski spectrum Spec Zar (K).Proof.The topology of Spc(K) is given by open sets which are complements of the sets of the form {P ∈ Spc(K) | k / ∈ P}.The result therefore follows from Theorem 3.11.