Algebraic stability theorem for derived categories of zigzag persistence modules
aa r X i v : . [ m a t h . R T ] J un ALGEBRAIC STABILITY THEOREM FOR DERIVEDCATEGORIES OF ZIGZAG PERSISTENCE MODULES
YASUAKI HIRAOKA AND MICHIO YOSHIWAKI
Abstract.
The interleaving and bottleneck distances between ordinary per-sistence modules can be extended to the derived setting. Using these distances,we prove an algebraic stability theorem in the derived category of ordinarypersistence modules. It is well known that the derived categories of ordinaryand arbitrary zigzag persistence modules are equivalent. Through this de-rived equivalence, these distances can also be defined on the derived categoryof arbitrary zigzag persistence modules, and the algebraic stability theoremholds even in this setting. As a consequence, an algebraic stability theoremfor arbitrary zigzag persistence modules is proved. Introduction
Topological data analysis has recently become popular for studying the shapeof data in various research areas (Hiraoka et al. 2016; Lee et al. 2017; Oyama et al.2019; Saadatfar et al. 2017). In topological data analysis, one of the standardtools is persistent homology, the original concept for which was introduced byEdelsbrunner, Letscher, and Zomorodian (2000). For a filtration X : X ֒ → X ֒ → · · · ֒ → X n of topological spaces, the q -th persistent homology is defined by H q ( X ) : H q ( X ) → H q ( X ) → · · · → H q ( X n ) , where H q (-) is the q -th homology functor with a field coefficient. Persistent homol-ogy is utilized to study the persistence of topological features in the filtration X such as connected components, loops, voids, and so on, for each dimension q . Thealgebraic structure of persistent homology is expressed using the notion of (1D)persistence modules , which are representations of an equioriented A n -type quiver.This was pointed out by Carlsson and de Silva (2010).From Gabriel (1972) and the Krull-Schmidt Theorem, any persistence modulecan be uniquely decomposed into interval representations , which are exactly inde-composable representations in this setting. The endpoints of these interval repre-sentations define the birth-death parameters of the topological features, and thosetopological features are summarized in a barcode (or a persistence diagram ). Then,the persistence of a topological feature is expressed by the lifetime defined as thedifference between its death and birth parameters.Here, the Krull-Schmidt Theorem reduces the description of the category ofrepresentations of quivers into that of the full subcategory consisting of indecom-posable representations. To explicitly compute indecomposable representations, theAuslander-Reiten (AR) quiver was introduced (see Auslander et al. 1997) and hasbeen studied in representation theory of finite-dimensional algebras since the 1970s. Mathematics Subject Classification.
Key words and phrases.
Algebraic stability theorem, Derived category, Persistence module,Zigzag persistence module.
For details on the AR quiver, refer to Schiffler (2014, Section 3.1) (see Assem et al.2006, Chapter IV for a more general setting).From the viewpoint of AR theory, the barcode of a persistence module can bedefined as a map from the set of vertices in the AR quiver of the equioriented A n -type quiver to the integers, sending an interval representation to its multiplicity.In this sense, the AR quiver is hidden behind the barcode.Unlike ordinary homology, it is significant that a stability theorem holds for per-sistent homology, which was first proved by Cohen-Steiner, Edelsbrunner, and Harer(2007) for the persistent homology of the sublevel set filtration induced by an R -valued function. The theorem guarantees that the barcode is stable (precisely,1-Lipschitz) with respect to small changes in a given data set.The algebraic perspective on persistent homology allows for a generalization ofthe stability theorem, the so-called algebraic stability theorem (AST). Chazal et al.(2009) introduced the interleaving distance between persistence modules to weakenthe assumptions needed for the stability theorem, and then proved the AST byusing that distance. The AST guarantees that the barcode is stable with respectto small changes in the given persistence module. Following this algebraic gener-alization, Bauer and Lesnick (2014) provided a simpler proof of the AST via theinduced matching theorem (IMT) (see Theorem 2.8). It should be noted that theconverse of the AST also holds (Lesnick 2015), hence giving the isometry theoremfor persistence modules.Here, the representation of an A n -type quiver with alternating (resp. arbitrary)orientation is called a purely zigzag (resp. zigzag) persistence module herein, while1D persistence modules are said to be ordinary . Zigzag persistence modules canalso be applied to address characteristic topological features not captured by thetheory of ordinary persistence modules (Carlsson and de Silva 2010). For example,let us study time-series data given by a sequence X , · · · , X t , · · · , X T of topologicalspaces X t for each time t . In general, this sequence is not a filtration with respectto t , but we can consider the following zigzag diagram: X ֒ → X ∪ X ← ֓ X ֒ → · · · ֒ → X T − ∪ X T ← ֓ X T . By applying a homology functor H q (-) to this diagram, we obtain a purely zigzagpersistence module H q ( X ) → H q ( X ∪ X ) ← H q ( X ) → · · · → H q ( X T − ∪ X T ) ← H q ( X T ) . Recall that a purely zigzag persistence module can also be decomposed into in-terval representations. Hence, it has a well-defined barcode and the persistence oftopological features in the time-series data X , · · · , X T is encoded in the barcode.This generalization is enabled by the algebraic viewpoint of persistent homology asa representation of an A n -type quiver.It was proved in Botnan and Lesnick (2018) that an AST also holds for purelyzigzag persistence modules. Bjerkevik (2016) improved the theorem with a tightbound and provided an isometry theorem for purely zigzag persistence modules.Note that zigzag persistence modules in Botnan and Lesnick (2018); Bjerkevik(2016) are purely zigzag ones in our convention.In this paper, we first generalize the AST of the equioriented A n -type quiver intothe derived category and then show that this generalization naturally provides aproof of the AST for zigzag persistence modules. Botnan and Lesnick (2018) provedthe stability theorem for a class of modules, called block-decomposable 2D persis-tence modules, into which purely zigzag persistence modules can be embedded. Incontrast, our strategy focuses on the equivalence of derived categories of ordinaryand zigzag persistence modules (see Happel 1988). This enables us to obtain anAST for the wider class (i.e., arbitrary orientations) in a unified manner than the ERIVED AST 3 result of Botnan and Lesnick. In particular, through the derived equivalence, weshow the following statements:(a) The interleaving and bottleneck distances on the ordinary persistence mod-ules can be extended into its derived category (Definition 3.7 and Defini-tion 3.10).(b) By using these distances, the AST is generalized on the derived categoryof ordinary persistence modules (Theorem 4.1).(c) The derived equivalence naturally defines the distances on the derived cat-egory of zigzag persistence modules, and induces the AST on it (Defini-tion 4.2, Definition 4.5, and Theorem 4.6 with Proposition 4.12).(d) As a corollary, the AST on the zigzag persistence modules holds (Theo-rem 4.14).In the above, the isometry theorems also hold as in previous studies (Theo-rem 5.2, Corollary 5.3, and Theorem 5.4).Let us briefly address prominent issues in order to derive the above statements.(a) Recall that the derived category is defined by the Verdier localization of thehomotopy category of cochain complexes with quasi-isomorphisms as denominators.Then, cochain complexes of ordinary persistence modules can be uniquely decom-posed by their cohomologies in the derived category. This fact provides naturalextensions of the interleaving and bottleneck distances in the derived setting.(b) The derived interleaving and the derived bottleneck distances defined in (a)enable us to generalize the AST to the derived category of ordinary persistencemodules since these distances on cochain complexes are determined by their coho-mologies.(c) For a category D equivalent to the derived category of ordinary persistencemodules, the distances on D can be canonically obtained from the known ones onthe latter derived category. In fact, equivalences preserve isomorphisms and the in-decomposability of objects, and these properties are utilized to define the distance.As mentioned above, the derived categories of ordinary and zigzag persistence mod-ules are equivalent. Under the derived equivalence, the distances on the derivedcategory of zigzag persistence modules can be induced from the derived interleavingand the derived bottleneck distances defined in (a). Then, an AST for the derivedcategory of zigzag persistence modules follows from (b).(d) As a consequence of (a), (b), and (c), we obtain an AST for zigzag persistencemodules. Indeed, the category of zigzag persistence modules can be regarded as afull subcategory of its derived category.Finally, let us also note relationships between our induced distance and thedistance used in Botnan and Lesnick (2018). For direct calculation of our in-duced distance on the category of zigzag persistence modules, we need to fix aderived equivalence between derived categories of ordinary and zigzag persistencemodules. In this paper, we consider the derived equivalence given by a classi-cal tilting module (see Assem et al. 2006; Brenner and Butler 1980; Bongartz 1981;Happel and Ringel 1982). Indeed, for a classical tilting module T , the right derivedfunctor of the functor Hom( T, -) gives a derived equivalence (see Happel 1988). Byusing the derived equivalence, we can calculate our induced distance and will thenshow that this is incomparable with the distance in Botnan and Lesnick (2018).From the perspective of the sheaf theory, Kashiwara and Schapira (2018) firstdefined the derived interleaving distance in the setting of constructible sheaves.Berkouk and Ginot (2018) proved the isometry theorem by using that distance.Indeed, they introduced the bottleneck distance between graded barcodes and then YASUAKI HIRAOKA AND MICHIO YOSHIWAKI proved that this is equal to the derived interleaving distance in the sense of Kashi-wara and Schapira. Our future work is to investigate a relationship between theirdistance and our derived distance.The remainder of the paper is organized as follows. Section 2 reviews the basicconcepts of persistent homology from the viewpoint of representation theory andrecalls the AST in the ordinary setting. Section 3 reviews the basics of the derivedcategory and refines it for the category of persistence modules. Then we intro-duce the interleaving distance and the bottleneck distance in the derived setting.Section 4 proves the main results: the AST for the derived category of ordinarypersistence modules and that for zigzag persistence modules. Section 5 extends theresult of Section 4 to isometry theorems. In Section 6, we explicitly calculate ourinduced distance on zigzag persistence modules. Finally, Section 7 confirms thatthe distance of Botnan and Lesnick and our induced distance are incomparable inthe purely zigzag setting. 2.
Preliminaries
Quiver representations.
Throughout, k denotes an algebraically closed field,and all vector spaces, algebras, and linear maps are assumed to be finite-dimensional k -vector spaces, finite-dimensional k -algebras, and k -linear maps, respectively. Fur-thermore, all categories and functors are assumed to be additive.A quiver Q is a directed graph. Formally, a quiver Q is a quadruple Q =( Q , Q , s, t ) of sets Q of vertices and Q of arrows, and maps s, t : Q → Q . Wedraw an arrow α ∈ Q as α : 1 → s ( α ) = 1 , t ( α ) = 2 ∈ Q . The opposite quiver Q op of a quiver Q = ( Q , Q , s, t ) is Q op = ( Q , Q , t, s ). For example, the oppositequiver of 1 → ←
2. A quiver Q is finite if Q and Q are finite. Herein, onlyfinite quivers are considered, otherwise stated.A quiver morphism f from a quiver Q = ( Q , Q , s, t ) to a quiver Q ′ = ( Q ′ , Q ′ , s ′ , t ′ )is a pair f = ( f , f ) of maps f i : Q i → Q ′ i for i = 0 , s ′ ◦ f = f ◦ s and t ′ ◦ f = f ◦ t . For example, 1l Q = (1l Q , Q ) is a quiver morphism Q → Q ,which is called the identity morphism . A quiver morphism f : Q → Q ′ is calledan isomorphism if there is a quiver morphism g : Q ′ → Q such that f ◦ g = 1l Q ′ and g ◦ f = 1l Q . A quiver Q is isomorphic to a quiver Q ′ , denoted by Q ∼ = Q ′ , ifthere is an isomorphism from Q to Q ′ . For example, a quiver of the form 1 α −→ X β −→ Y . Indeed, we have an isomorphism f = ( f , f ) defined by f (1) = X, f (2) = Y , and f ( α ) = β .Here, we introduce the A n -type quiver A n ( a ) with orientation a , whose under-lying graph is the Dynkin diagram of type A : 1 — — · · · — n for n ∈ N . Then A n ( a ) is the quiver 1 ↔ ↔ · · · ↔ n, where ↔ means → or ← assigned by the orientation a . In this paper, the following A n -type quivers with certain orientations are frequently used. The A n -type quiverwith equi-orientation 1 → → · · · → n is called the equioriented A n -type quiver , which is denoted by A n (= A n ( e )), and an A n -type quiver with alternating orientation is called a purely zigzag A n -type quiver ,which is denoted by A n ( z ). Moreover, if the vertex 1 of a purely zigzag A n -typequiver Q is a sink vertex, Q is denoted by A n ( z ). Otherwise, it is denoted by A n ( z ). Namely, A n ( z ) is the following quiver:1 ← → ← · · · → n if n is odd, 1 ← → ← · · · ← n if n is even, ERIVED AST 5 and A n ( z ) is the following quiver:1 → ← → · · · ← n if n is odd, 1 → ← → · · · → n if n is even.A representation M of a quiver Q is a family of vector spaces M x at each vertex x ∈ Q and linear maps M α on each arrow α ∈ Q . For example, a representation M of the equioriented A n -type quiver A n : 1 α , −−−→ α , −−−→ · · · α n − ,n −−−−→ n has the following form: M M α , −−−−→ M M α , −−−−→ · · · M αn − ,n −−−−−−→ M n . A subrepresentation N of M is defined as a representation of Q such that N x ⊆ M x for each vertex x ∈ Q and N α = M α | N x for each arrow α : x → y ∈ Q . The direct sum M ⊕ N of representations M and N is defined by ( M ⊕ N ) x = M x ⊕ N x for each vertex x ∈ Q and ( M ⊕ N ) α = M α ⊕ N α for each arrow α ∈ Q . The dimension of M is defined by dim M := P x ∈ Q dim M x . All representations areassumed to be finite-dimensional , namely dim M < ∞ .Let M, N be representations of Q . Then a morphism f : M → N is a family oflinear maps f x : M x → N x on each vertex x ∈ Q such that the following diagramcommutes for any arrow α : x → y ∈ Q : M x f x / / M α (cid:15) (cid:15) N xN α (cid:15) (cid:15) M y f y / / N y . For example, 1l M = (1l M x ) x ∈ Q is a morphism M → M , which is called the identitymorphism . A morphism f : M → N is called an isomorphism if there is a morphism g : N → M such that f ◦ g = 1l N and g ◦ f = 1l M . A representation M is isomorphic to a representation N , denoted by M ∼ = N , if there is an isomorphism from M to N .Moreover, a non-zero representation M is said to be indecomposable if M ∼ = N ⊕ L implies N = 0 or L = 0.The abelian category of representations of Q is denoted by rep k Q . Note thatrep k Q is a Krull-Schmidt category (see Schiffler 2014, p.11, Theorem 1.2 for exam-ple). Indeed, for any M ∈ rep k Q , we have unique decomposition M ∼ = M ⊕ · · · ⊕ M s up to permutations and isomorphisms, where each M i is indecomposable.2.2. Persistence modules.
We call each M ∈ rep k A n , each N ∈ rep k A n ( z ), andeach L ∈ rep k A n ( a ) a (ordinary) persistence module , a purely zigzag persistencemodule , and a zigzag persistence module , respectively. In this subsection, we willdefine the internal morphisms of an ordinary persistence module and an endofunctorof the category of ordinary persistence modules in order to define the interleavingdistance.For any A n -type quiver A n ( a ), α x,y denotes the arrow between x and y with1 ≤ x < y ≤ n . Then the equioriented A n -type quiver A n is A n : 1 α , −−−−→ α , −−−−→ · · · α n − ,n −−−−→ n and a persistence module M has the form: M M α , −−−−−→ M M α , −−−−−→ · · · M αn − ,n −−−−−−→ M n . YASUAKI HIRAOKA AND MICHIO YOSHIWAKI
Moreover, when n is odd, the purely zigzag A n -type quiver A n ( z ) is1 α , ←−−−− α , −−−−→ · · · α n − ,n −−−−→ n and a purely zigzag persistence module M ∈ rep k A n ( z ) has the form: M M α , ←−−−−− M M α , −−−−−→ · · · M αn − ,n −−−−−−→ M n . In other cases, we can similarly describe the zigzag A n -type quivers and the zigzagpersistence modules. Definition 2.1.
Let
M, N be persistence modules and δ an integer.(1) For 1 ≤ s ≤ t ≤ n , the linear map φ M ( s, t ) : M s → M t is defined by φ M ( s, t ) = (cid:26) M s , s = tM α t − ,t ◦ · · · ◦ M α s,s +1 , otherwise . By definition, we have φ M ( s, t ) = φ M ( t − , t ) ◦ · · · ◦ φ M ( s, s + 1).(2) The δ -shift M ( δ ) of M is defined by( M ( δ )) x = (cid:26) M x + δ , ≤ x + δ ≤ n , otherwiseand ( M ( δ )) α x,x +1 = (cid:26) M α x + δ,x +1+ δ , ≤ x + δ ≤ x + 1 + δ ≤ n , otherwisefor each vertex x of A n . For a morphism f : M → N in rep k A n , the δ -shift f ( δ ) of f is defined by ( f ( δ )) x = (cid:26) f x + δ , ≤ x + δ ≤ n , otherwisefor each vertex x of A n . This defines the δ -shift functor ( δ ) : rep k A n → rep k A n .It should be noted that the δ -shift functor can only be defined in the equiorientedsetting.(3) The transition morphism φ δM : M → M ( δ ) in rep k A n is defined by ( φ δM ) x = φ M ( x, x + δ ) for each vertex x of A n . For any morphism f : M → N , we have thefollowing commutative diagram: M φ δM / / f (cid:15) (cid:15) M ( δ ) f ( δ ) (cid:15) (cid:15) N φ δN / / N ( δ ) . This defines a natural transformation φ δ : 1l → ( δ ) from the identity functor 1l tothe δ -shift functor ( δ ).(4) A persistence module M is δ -trivial if the transition morphism φ δM : M → M ( δ )is zero.In our setting, the functor ( δ ) is not an equivalence but an exact functor. Indeed,let M, N, L be persistence modules. A sequence0 → M → N → L → → M x → N x → L x → x of A n . This means that the sequence0 → M ( δ ) → N ( δ ) → L ( δ ) → ERIVED AST 7
Interleaving distance.
In this paper, a distance on a set X means an ex-tended pseudometric. Specifically, it is a function d : X × X → R ≥ ∪ {∞} suchthat, for every x, y, z ∈ X ,(1) d ( x, x ) = 0,(2) d ( x, y ) = d ( y, x ), and(3) d ( x, z ) ≤ d ( x, y ) + d ( y, z ) if d ( x, y ) , d ( y, z ) < ∞ .Let us recall the interleaving distance between persistence modules. Definition 2.2.
Let δ be a non-negative integer. Two persistence modules M and N are said to be δ -interleaved if there exist morphisms f : M → N ( δ ) and g : N → M ( δ ) such that the following diagrams commute: M φ δM / / f ! ! ❈❈❈❈❈❈❈❈❈ M (2 δ ) , N φ δN / / g ! ! ❈❈❈❈❈❈❈❈❈ N (2 δ ) .N ( δ ) g ( δ ) : : ✈✈✈✈✈✈✈✈✈ M ( δ ) f ( δ ) : : ✈✈✈✈✈✈✈✈✈ In this case, we call the pair of f : M → N ( δ ) and g : N → M ( δ ) a δ -interleavingpair . Moreover, we call a morphism f : M → N ( δ ) a δ -interleaving morphism ifthere is a morphism g : N → M ( δ ) such that the pair ( f, g ) is a δ -interleaving pair.For persistence modules M, N , the interleaving distance is defined asd I ( M, N ) := inf { δ ∈ Z ≥ | M and N are δ -interleaved } . We remark that in our setting, d I ( M, N ) = 0 if and only if M and N areisomorphic. Thus, the interleaving distance can measure how far these modules arefrom being isomorphic. We will extend this concept to the derived setting later (seeDefinition 3.7).2.4. Intervals and barcodes.
We recall that the category rep k A n ( a ) of zigzagpersistence modules is a Krull-Schmidt category, i.e., a representation of A n ( a ) isisomorphic to a direct sum of indecomposable representations. In this subsection,we discuss all indecomposable representations of A n ( a ). Definition 2.3.
For 1 ≤ b ≤ d ≤ n , the interval representation I [ b, d ] ∈ rep k A n ( a )is defined by ( I [ b, d ]) x := (cid:26) k , b ≤ x ≤ d , otherwiseand ( I [ b, d ]) α x,y := (cid:26) k , b ≤ x < y ≤ d , otherwise . Any interval representation is indecomposable. The converse also holds as fol-lows.
Theorem 2.4 (Gabriel 1972) . Any indecomposable representation of A n ( a ) is iso-morphic to an interval representation I [ b, d ] for some ≤ b ≤ d ≤ n . Thus, for a representation M of A n ( a ), we obtain the unique interval decompo-sition M ∼ = M ≤ b ≤ d ≤ n I [ b, d ] m ( b,d ) , leading to the definition of the barcode (or the persistence diagram ) B ( M ) of M by { ( b, d, m ) | ≤ b ≤ d ≤ n, ≤ m ≤ m ( b, d ) such that m ( b, d ) = 0 } . YASUAKI HIRAOKA AND MICHIO YOSHIWAKI
Below, we will use the same notation as in Bauer and Lesnick (2014). Recallthat a multiset is a pair (
S, m ) of a set S and a map m : S → Z > and that arepresentation Rep ( S, m ) of a multiset (
S, m ) is
Rep ( S, m ) = { ( s, n ) ∈ S × Z > | n ≤ m ( s ) } . If S is a totally ordered set, then a representation Rep ( S, m ) is also a totallyordered set with order obtained by restricting the lexicographic order on S × Z > to Rep ( S, m ).For a persistence module M , the barcode B ( M ) is regarded as a representationof a multiset ( I M , m ) of the set I M := { ( b, d ) | m ( b, d ) = 0 } and the map m : I M → Z > sending ( b, d ) to m ( b, d ). For simplicity, write an element ( b, d, m ) of B ( M ) as h b, d i , which is called an interval . For 1 ≤ b ≤ n , B ( M ) h b, - i denotes thesubset of B ( M ) consisting of the intervals h b, c i for some b ≤ c ≤ n , and B ( M ) h - ,d i denotes the subset of B ( M ) consisting of the intervals h c, d i for some 1 ≤ c ≤ d .Note that B ( M ) h b, - i , B ( M ) h - ,d i are regarded as totally ordered sets with the totalorder induced by the reverse inclusion relation on intervals. Indeed, if c < c ′ , then h b, c ′ i < h b, c i in B ( M ) h b, - i and h c, d i < h c ′ , d i in B ( M ) h - ,d i .From the perspective of AR theory, the barcode of a representation M of A n ( a )can be defined as a map Γ → Z sending an interval I [ b, d ] to its multiplicity m ( b, d )in the decomposition of M , where Γ is the set of all interval representations. Notethat Γ is the set of vertices of the AR quiver of A n ( a ) (for details, see Schiffler2014, Section 1.5 and 3.1, Chapter 7), and in this sense, AR quivers are hiddenbehind the barcodes. The AR quivers are important tools in the representationtheory of quivers. Indeed, under a certain assumption, the AR quiver can recoverthe category of representations. Example 2.5.
The AR quiver Γ( A ) of A isΓ( A ) = I [1 , I [2 , I [1 , I [3 , I [2 , I [1 , , while the AR quiver Γ( A ( z )) of A ( z ) : 1 ← → A ( z )) = I [1 , I [2 , I [1 , I [2 , I [3 , I [1 , . Matching and the bottleneck distance. A matching from a set S to a set T (written as σ : S T ) is a bijection σ : S ′ → T ′ for some subset S ′ of S andsome subset T ′ of T . For a matching σ : S T , we write S ′ as Coim σ and T ′ asIm σ .For totally ordered sets, a matching can be defined canonically as follows: let S = { S i | i = 1 , · · · , s } and T = { T i | i = 1 , · · · , t } be finite totally ordered setssuch that for a ≤ b , S a ≤ S b and T a ≤ T b . Then a canonical matching σ : S T is a matching σ given by σ ( S i ) = T i for i = 1 , · · · , min { s, t } . In this case, eitherIm σ = S or Coim σ = T is satisfied.We next define a δ -matching between barcodes. Definition 2.6.
Let δ be a non-negative integer. For a barcode B , let B δ be thesubset of B consisting of intervals h b, d i such that d − b ≥ δ . A δ -matching between ERIVED AST 9 barcodes B and B ′ is defined by a matching σ : B B ′ such that B δ ⊆ Coim σ, B ′ δ ⊆ Im σ, andfor all σ h b, d i = h b ′ , d ′ i , b ′ − δ ≤ b ≤ d ≤ d ′ + δ and b − δ ≤ b ′ ≤ d ′ ≤ d + δ. (1)Two barcodes B and B ′ are said to be δ -matched if there is a δ -matching between B and B ′ . Then the bottleneck distance is defined asd B ( B , B ′ ) := inf { δ ∈ Z ≥ | B and B ′ are δ -matched } . Note that equation (1) implies that the interval representations associated with h b, d i , h b ′ , d ′ i are δ -interleaved.2.6. Algebraic stability theorem.
In this subsection, we will explain the proofof an AST for rep k A n following Bauer and Lesnick (2014). Their strategy utilizesthe IMT. Definition 2.7.
Let f : M → N be a morphism in rep k A n . Then the inducedmatching B ( f ) : B ( M ) → B ( N ) is defined as follows :(1) when f is injective, B ( f ) is defined via the family of canonical matchingsfrom B ( M ) h - ,d i to B ( N ) h - ,d i .(2) when f is surjective, B ( f ) is defined via the family of canonical matchingsfrom B ( M ) h b, - i to B ( N ) h b, - i .(3) for any morphism f , f can be decomposed into the surjective morphism π : M → Im f and the injective morphism µ : Im f → N . Then B ( f ) := B ( µ ) ◦ B ( π ) by (1) and (2).This matching is what yields the IMT (see Bauer and Lesnick 2014, Theorem4.2). To state the IMT in our setting, we extend representations M in rep k A n tothose in rep k A ℓ for ℓ ≥ n as0 → · · · → → M → · · · → M n → → · · · → ∈ rep k A ℓ . Moreover, for a given representation M ∈ rep k A n and non-negative integer δ , themap r δM : B ( M ( δ )) → B ( M ) is defined by r δM h b, d i := h b + δ, d + δ i . In general, themap r δM is not bijective. However, we can take an integer ℓ ≥ n large enough suchthat M and ( M ( δ ))( − δ ) are isomorphic as representations of A ℓ . In this case, themap r δM is bijective.Then, the IMT is stated as follows: Theorem 2.8 (IMT) . Let f : M → N be a morphism in rep k A n . Assume that Ker f and Coker f are δ -trivial. Moreover, taking an integer ℓ ≥ n large enoughsuch that r δM is bijective, M, N are regarded as representations of A ℓ . Then B ( f ) ◦ r δM is a δ -matching B ( M ( δ )) B ( N ) . Let f : M → N ( δ ) be a δ -interleaving morphism. It is easily seen that Ker f and Coker f are 2 δ -trivial. Thus, Theorem 2.8 induces the following theorem (seeBauer and Lesnick 2014, Theorem 4.5): Theorem 2.9 (AST) . Let
M, N be persistence modules in rep k A n . Then d B ( B ( M ) , B ( N )) ≤ d I ( M, N ) . Proof.
Let f : M → N ( δ ) be a δ -interleaving morphism in rep k A n and ℓ ≥ n an integer large enough such that r δM and r δN are bijective. Then M and N areregarded as representations of A ℓ . Since Ker f and Coker f are 2 δ -trivial, r δN ◦ B ( f ) = r δN ◦ ( B ( f ) ◦ r δM ) ◦ ( r δM ) − : B ( M ) ∼ −→ B ( M ( δ )) B ( N ( δ )) ∼ −→ B ( N )is a δ -matching by Theorem 2.8, as desired. (cid:3) Derived categories
Definition and basic properties.
Let A be an abelian category. We startthis section with the definition of its derived category (see Happel 1988, Chapter I,3). Definition 3.1. (1) A cochain complex X • over A is a family X • = ( X i , d iX ) i ∈ Z of objects X i of A and morphisms d iX : X i → X i +1 in A satisfying d i +1 X ◦ d iX = 0.A cochain complex X • is said to be bounded if X i = 0 for | i | ≫
0. If X i = 0for i = l , then it is called a stalk complex concentrated at the l -th term . For eachcochain complex X • and each i ∈ Z , we have the i -th cohomology H i ( X • ) :=Ker d iX / Im d i − X .Let X • , Y • be cochain complexes over A . Then a cochain map f • : X • → Y • is afamily f • = ( f i ) i ∈ Z of morphisms f i : X i → Y i in A satisfying f i +1 ◦ d iX = d iY ◦ f i .We use C b ( A ) to denote the category of bounded cochain complexes and cochainmaps over A . Then the l -translation functor [ l ] : C b ( A ) → C b ( A ) is defined by X • [ l ] := (( X [ l ]) i , d iX [ l ] ) i ∈ Z with ( X [ l ]) i = X i + l , d iX [ l ] = ( − l d i + lX and ( f • [ l ]) i := f i + l for a cochain complex X • = ( X i , d iX ) i ∈ Z and a cochain map f • . The 0-translation functor [0] is exactly the identity functor.(2) A cochain map f • naturally induces the morphism H i ( f • ) : H i ( X • ) → H i ( Y • ) for each i . Then we have the i -th cohomology functor H i (-) : C b ( A ) → A .A cochain map f • is called a quasi-isomorphism if the induced morphisms H i ( f • )are isomorphisms for all i .(3) A cochain map f • : X • → Y • is said to be null-homotopic if there exists afamily ( h i ) i ∈ Z of morphisms h i : X i → Y i − such that f i = h i +1 ◦ d iX + d i − Y ◦ h i for each i .Let I be the ideal of C b ( A ) consisting of null-homotopic cochain maps. Then the bounded homotopy category K b ( A ) of A is defined as the quotient category of C b ( A )by the ideal I . Since a cochain map f • is null-homotopic if and only if f • [ l ] is so,we can extend the l -translation functor [ l ] to the setting of the bounded homotopycategory such that the following diagram commutes, where π : C b ( A ) → K b ( A ) isthe canonical quotient functor. C b ( A ) π / / [ l ] (cid:15) (cid:15) K b ( A ) [ l ] (cid:15) (cid:15) C b ( A ) π / / K b ( A )It is well-known that the homotopy category K b ( A ) with the 1-translation functor[1] forms a triangulated category (see Happel 1988).Moreover, if a cochain map f • is null-homotopic, then H i ( f • ) = 0. Thus, weobtain the i -th cohomology functor H i (-) : K b ( A ) → A such that the followingdiagram commutes: C b ( A ) π / / H i (cid:15) (cid:15) K b ( A ) H i z z ✉✉✉✉✉✉✉✉✉✉ A . A quasi-isomorphism in K b ( A ) is a morphism f • such that H i ( f • ) are isomorphismsfor all i .(4) The bounded derived category D b ( A ) of A is the triangulated category givenby the Verdier localization of the bounded homotopy category K b ( A ) with respectto the collection of all quasi-isomorphisms. Thus, a morphism in D b ( A ) is anisomorphism if and only if it is a quasi-isomorphism in K b ( A ). The construction is ERIVED AST 11 analogous to that of the localization of a ring. Indeed, a morphism from X • to Y • inthe derived category is represented by a pair ( f • , s • ) of a morphism f • : X • → Z • and a quasi-isomorphism Y • → Z • with some cochain complex Z • .By the universal property of the localization, we obtain the i -th cohomologyfunctor H i (-) : D b ( A ) → A such that the following diagram commutes, where ι : K b ( A ) → D b ( A ) is the canonical localization functor. K b ( A ) ι / / H i (cid:15) (cid:15) D b ( A ) H i z z ✉✉✉✉✉✉✉✉✉✉ A The abelian category A can be regarded as a full subcategory of its boundedderived category D b ( A ). Indeed, we have the fully faithful functor ξ from A to D b ( A ) sending X to the stalk complex · · · → → X → → · · · concentrated at the 0-th term. We denote this stalk complex as X [0], using the0-translation functor [0]. The stalk complex concentrated at the l -th term is writtenas X [ − l ].We use proj A to denote the full subcategory of A consisting of projective objects.An abelian category A is said to have enough projectives if for each object M ∈ A ,there exists an epimorphism P → M with P ∈ proj A . In this case, a projectiveresolution of M can be defined as a cochain complex P • : · · · → P → P over proj A satisfying H i ( P • ) ∼ = (cid:26) M, i = 00 , otherwise . In other words, we have an exact sequence · · · → P → P → M → . The projective dimension of M can be defined as the infimum of the lengths ofprojective resolutions of M , and the global dimension of A can be defined as thesupremum of all projective dimensions.Similarly, we use inj A to denote the full subcategory of A consisting of injectiveobjects, and we can dually consider the concept of having enough injectives andthe injective and global dimensions . Note that the global dimensions defined byprojective and injective dimensions coincide if the abelian category A has enoughprojectives and injectives.Then the following lemma is important for understanding the bounded derivedcategory (see Happel 1988, Chapter I, 3.3). Note that the categories C b ( A ) and K b ( A ) can be defined if A is an additive category, e.g., K b (proj A ) and K b (inj A ). Lemma 3.2.
Assume that A has enough projectives (resp. injectives) and has afinite global dimension. Then K b (proj A )( resp. K b (inj A )) ֒ → K b ( A ) ι −→ D b ( A ) isan equivalence as a triangulated category. This lemma states that an object of a derived category D b ( A ) is assumed to bea cochain complex over proj A (or inj A ), and a morphism in D b ( A ) can be writtenas a morphism in K b (proj A ) (or K b (inj A )). All representatives in a morphism in D b ( A ) from X • to Y • need some unknown cochain complex Z • (see Definition 3.1(4)). In contrast, a morphism in K b (proj A ) (or K b (inj A )) is concretely written asthe residue class of a cochain map. Thus, under the equivalence in Lemma 3.2, wecan well understand the bounded derived category D b ( A ). Let B be another abelian category. A is said to be derived equivalent to B if D b ( A ) and D b ( B ) are equivalent as triangulated categories.Let F : A → B be a functor. Any such functor yields the canonical functor C ( F ) : C b ( A ) → C b ( B ) given by C ( F )( X • ) := ( F X i , F d iX ) i ∈ Z for each cochaincomplex X • = ( X i , d iX ) i ∈ Z . Then C ( F ) naturally extends to the functor K ( F ) : K b ( A ) → K b ( B ). Moreover, we assume that F is exact. In this case, H i ◦ K ( F ) ∼ = F ◦ H i canonically. Thus, K ( F ) induces a functor D ( F ) : D b ( A ) → D b ( B ) givenby D ( F )( X • ) := ( F X i , F d iX ) i ∈ Z for each cochain complex X • = ( X i , d iX ) i ∈ Z . Weoften write C ( F ), K ( F ), D ( F ) as F in their respective contents and identify H i ◦ F with F ◦ H i . For example, since the δ -shift functor ( δ ) : rep k A n → rep k A n is exact,it induces a functor ( δ ) : D b (rep k A n ) → D b (rep k A n )via X • ( δ ) = ( X i ( δ ) , d iX ( δ )) i ∈ Z . Then H i ◦ ( δ ) is identified with ( δ ) ◦ H i .If A has enough injectives and F is left exact, then we can define the rightderived functor RF : D b ( A ) → D b ( B ) as RF( X • ) := F ( I • ) with X • ∼ = I • ∈ K b (inj A ). A typical example of a left exact functor is the Hom functor Hom( X, -)with X ∈ A . Then its right derived functor is denoted by RHom( X, -). Note that H i (RHom( X, -)) ∼ = Ext i ( X, -) for each i .3.2. Derived category of rep k A n ( a ) . We discuss some specific properties of D b (rep k A n ( a )) for an A n -type quiver A n ( a ) with arbitrary orientation a . Setproj A n ( a ) := proj(rep k A n ( a )). Note that rep k A n ( a ) has enough projectives andinjectives, and has global dimension 0 for n = 1 and 1 for n >
1. Then we alwayshave a projective resolution 0 → P → P → M → M ∈ rep k A n ( a ). In particular, any subrepresentation of a projec-tive representation is also projective in this setting. In addition, by Lemma 3.2, weobtain an equivalence between D b (rep k A n ( a )) and K b (proj A n ( a )).In the case of rep k A n ( a ), we have the following strong characterization of acochain complex in D b (rep k A n ( a )) by its cohomologies. Lemma 3.3.
For any cochain complex X • ∈ D b (rep k A n ( a )) , X • ∼ = M i ∈ Z H i ( X • )[ − i ] in D b (rep k A n ( a )) . More generally, X • ( δ ) ∼ = M i ∈ Z H i ( X • )( δ )[ − i ] . Proof.
We may assume that X i is projective for all i ∈ Z , since D b (rep k A n ( a )) ≃ K b (proj A n ( a )) by Lemma 3.2. The statement will be proved by induction on thelength of its non-zero terms. From the boundedness of X • , let us set s := min { i ∈ Z | X i = 0 } and t := max { i ∈ Z | X i = 0 } . Then the length ℓ ( X • ) of non-zeroterms of X • is defined by t − s + 1. Since the global dimension of rep k A n ( a ) is atmost 1, Im d t − X is projective. Hence the exact sequence0 → Ker d t − X → X t − → Im d t − X → d t − X is also projective. Thus, we have X • ∼ = Y • ⊕ Z • ,where the complexes Y • , Z • ∈ K b (proj A n ( a )) are given by Y • = · · · → X t − → Ker d t − X → Z • = · · · → → Im d t − X → X t . ERIVED AST 13
Here, Z • ∼ = H t ( X • )[ − t ] and H i ( Y • ) ∼ = ( H i ( X • ) , if i < t , if i ≥ t . The length ℓ ( Y • ) isless than the length ℓ ( X • ), so by induction Y • ∼ = M i A cochain complex X • ∈ D b (rep k A n ( a )) is indecomposable if andonly if X • is isomorphic to a stalk complex I [ b, d ][ − i ] : · · · → → I [ b, d ] → → · · · concentrated at the i -th term in D b (rep k A n ( a )) for some ≤ b ≤ d ≤ n and some i ∈ Z . Thus, any cochain complex X • is isomorphic to M b ≤ d,i ( I [ b, d ][ − i ]) m ( b,d,i ) , where the non-negative integer m ( b, d, i ) is the multiplicity of I [ b, d ][ − i ] . Since D b (rep k A n ( a )) is a Krull-Schmidt category (see Chen et al. 2008), theinterval decomposition in the corollary above is unique. By using this result, wepropose the ‘derived’ barcode. Definition 3.5. Let X • , Y • be cochain complexes of D b (rep k A n ( a )). Then the derived barcode B D ( X • ) is defined as B D ( X • ) := G i with H i ( X • ) =0 B ( H i ( X • ))where B ( H i ( X • )) is the ordinary barcode of H i ( X • ) (see the paragraph followingTheorem 2.4).As in Section 2.4, the AR quiver of D b (rep k A n ( a )) with arbitrary orientationscan be defined (see Happel 1988, Chapter I, 4 and 5). Similar to the case ofrep k A n ( a ), the derived barcode of X • can be defined as a map Γ → Z sending I [ b, d ][ − i ] to the multiplicity m ( b, d, i ), where Γ is the set of vertices of the ARquiver of D b (rep k A n ( a )). Thus, AR quivers are hidden behind the barcodes inthis setting, too. Moreover, the AR quiver of D b (rep k A n ( a )) consists of all shiftedcopies of the AR quiver Γ( A n ( a )) of A n ( a ). Example 3.6. The AR quiver Γ( D b (rep k A )) of D b (rep k A ) isΓ( D b (rep k A )) = · · · I [1 , − I [1 , I [3 , I [1 , − I [2 , I [1 , · · ·· · · I [3 , I [2 , I [1 , . More generally, the AR quiver Γ( D b (rep k A n )) of D b (rep k A n ) is as described inFigure 1. ・・・ ・・・ ・・・ ・・・ ・・・・・・ Γ (A n )[0] Γ (A n )[-1] Γ (A n )[1] Figure 1. AR quiver of D b (rep k A n )Moreover, the AR quiver Γ( D b (rep k A ( z ))) of D b (rep k A ( z )), where A ( z ) :1 ← → D b (rep k A ( z ))) = · · · I [1 , I [2 , I [3 , I [2 , − I [1 , I [2 , · · ·· · · I [3 , I [1 , I [1 , . Similarly to Figure 1, the AR quiver Γ( D b (rep k A n ( z ))) of D b (rep k A n ( z )) is givenby all shifted copies of the AR quiver of A n ( z ).3.3. Derived interleaving and bottleneck distances. In this subsection, wepropose distances on the derived category of persistence modules by extending theoriginal interleaving and bottleneck distances.Recall that the δ -shift functor ( δ ) : rep k A n → rep k A n induces a functor( δ ) : D b (rep k A n ) → D b (rep k A n )via X • ( δ ) = ( X i ( δ ) , d iX ( δ )) i ∈ Z since the functor ( δ ) is exact. Definition 3.7. Let X • , Y • be cochain complexes in D b (rep k A n ) and δ a non-negative integer. Then X • and Y • are said to be derived δ -interleaved if thereexist morphisms f • : X • → Y • ( δ ) and g • : Y • → X • ( δ ) such that for each i ∈ Z ,( H i ( f • ) , H i ( g • )) is a δ -interleaving pair between H i ( X • ) and H i ( Y • ) in the senseof Definition 2.2. Namely, the following diagrams commute for each i ∈ Z : H i ( X • ) φ δHi ( X • ) / / H i ( f • ) ●●●●●●●● H i ( X • )(2 δ ) , H i ( Y • ) φ δHi ( Y • ) / / H i ( g • ) ●●●●●●●● H i ( Y • )(2 δ ) H i ( Y • )( δ ) H i ( g • )( δ ) sssssssss H i ( X • )( δ ) H i ( f • )( δ ) ttttttttt In this case we also call the pair of f • : X • → Y • ( δ ) and g • : Y • → X • ( δ ) a derived δ -interleaving pair . Moreover, we call a morphism f • : X • → Y • ( δ ) a derived δ -interleaving morphism if there is a morphism g • : Y • → X • ( δ ) such that the pair( f • , g • ) is a derived δ -interleaving pair.For cochain complexes X • , Y • ∈ D b (rep k A n ), the derived interleaving distance is defined asd DI ( X • , Y • ) := inf { δ ∈ Z ≥ | X • and Y • are derived δ -interleaved } . Remark . (1) Similarly to the original setting, d DI ( X • , Y • ) = 0 for two cochaincomplexes X • , Y • ∈ D b (rep k A n ) if and only if X • and Y • are isomorphic in D b (rep k A n ). Thus, the derived interleaving distance can also measure how farthese complexes are from being isomorphic. ERIVED AST 15 (2) We define the derived interleaving distance independently of Berkouk’s (Berkouk2019). The derived interleaving distance of Berkouk is defined as a generalizationof the distance on the base abelian category. In contrast, ours is defined via thecharacterization of an object of the derived category by its cohomologies. It is alsoobvious that our definition can be generalized to arbitrary abelian categories A withsome natural transformation from the identity functor 1l to an exact endofunctor.Although these two ideas are different, it can be proved that the Berkouk inter-leaving implies our interleaving. The converse does not hold in general (e.g., forabelian categories having a higher global dimension). For rep k A n , these conceptscoincide since rep k A n has global dimension at most 1.Note that if X • and Y • are derived δ -interleaved, then H i ( X • ) and H i ( Y • ) are δ -interleaved for all i . It follows from Lemma 3.3 that the converse also holds. Corollary 3.9. Let X • , Y • be cochain complexes in D b (rep k A n ) . Then X • and Y • are derived δ -interleaved if and only if H i ( X • ) and H i ( Y • ) are δ -interleavedfor all i .Proof. For each i , let ( f i , g i ) be a δ -interleaved pair between H i ( X • ) and H i ( Y • ).Then the pair of ( f i [ − i ]) : M i H i ( X • )[ − i ] → M i H i ( Y • )( δ )[ − i ]and ( g i [ − i ]) : M i H i ( Y • )[ − i ] → M i H i ( X • )( δ )[ − i ]is a derived δ -interleaved pair. By Lemma 3.3, X • and Y • are derived δ -interleaved.Indeed, using the isomorphisms in Lemma 3.3, we can construct a derived δ -interleaving pair of morphisms f • : X • → M i H i ( X • )[ − i ] ( f i [ − i ]) −−−−−→ M i H i ( Y • )( δ )[ − i ] → Y • ( δ )and g • : Y • → M i H i ( Y • )[ − i ] ( g i [ − i ]) −−−−−→ M i H i ( X • )( δ )[ − i ] → X • ( δ )such that H i ( f • ) = f i and H i ( g • ) = g i for any i . (cid:3) Finally, we propose the ‘derived’ bottleneck distance between derived barcodesin the sense of Definition 3.5 in this setting. Definition 3.10. Let X • , Y • be cochain complexes of D b (rep k A n ). Two de-rived barcodes B D ( X • ) and B D ( Y • ) are said to be δ -matched if B ( H i ( X • )) and B ( H i ( Y • )) are δ -matched in the sense of Definition 2.6 for all i ∈ Z .For derived barcodes B D ( X • ) , B D ( Y • ), the derived bottleneck distance is definedas d DB ( B D ( X • ) , B D ( Y • )) := inf { δ ∈ Z ≥ | B D ( X • ) and B D ( Y • ) are δ -matched } . Main results In this section, we derive an AST for zigzag persistence modules from an ASTfor ordinary ones by using the derived category. We adopt a different approachfrom that of Botnan and Lesnick (2018) by considering that the distances on zigzagpersistence modules may be naturally induced by the known interleaving and bot-tleneck distances on ordinary ones using derived categories. This enables us toobtain an AST for a wider class compared to that of Botnan and Lesnick. AST for derived categories. We first prove an AST for derived categoriesof ordinary persistence modules. Theorem 4.1 (AST for derived categories) . Let X • , Y • be cochain complexes of D b (rep k A n ) . Then d DB ( B D ( X • ) , B D ( Y • )) ≤ d DI ( X • , Y • ) . Proof. Assume that X • and Y • are derived δ -interleaved. Then for all i ∈ Z , H i ( X • ) and H i ( Y • ) are δ -interleaved, and hence B ( H i ( X • )) and B ( H i ( Y • )) are δ -matched by Theorem 2.9. Thus, by definition, the inequality d DB ( B D ( X • ) , B D ( Y • )) ≤ d DI ( X • , Y • )holds. (cid:3) Next, we consider a derived equivalence. Let A be an abelian category. Assumethat there exists a derived equivalence E from D b ( A ) to D b (rep k A n ). Definition 4.2. Two objects X and Y of D b ( A ) are said to be δ -interleaved withrespect to E if E ( X ) and E ( Y ) are derived δ -interleaved in the sense of Defini-tion 3.7. The interleaving distance d E, A I ( X, Y ) with respect to E is defined as d E, A I ( X, Y ) := inf { δ ∈ Z ≥ | X and Y are δ -interleaved with respect to E } . Namely, d E, A I ( X, Y ) = d DI ( E ( X ) , E ( Y )) holds. Remark . By Remark 3.8 (1), d E, A I ( X, Y )=0 if and only if E ( X ) and E ( Y ) areisomorphic in D b (rep k A n ). Since E is an equivalence, this means that X and Y are isomorphic in D b ( A ). Thus, the interleaving distance defined as above can alsomeasure how far these objects are from being isomorphic. This justifies calling thedistance an interleaving distance. Remark . The δ -shift functor cannot be defined in the zigzag setting, so neitheris the original interleaving distance. One of the advantages of our approach isthat we can define the interleaving distance even in the zigzag setting through thederived equivalence.Since E is an equivalence, in particular, a fully faithful functor, X ∈ D b ( A ) isindecomposable if and only if E ( X ) ∈ D b (rep k A n ) is indecomposable. Hence, since D b (rep k A n ) is a Krull-Schmidt category, so is D b ( A ). Consequently, the derivedequivalence E induces a bijection between B D A ( X ) := { Z ∈ D b ( A ) | Z is indecomposable and a direct summand of X } and B D ( E ( X )) (see Definition 3.5). Then the following distance between B D A ( X )and B D A ( Y ) is naturally derived by passing through the derived equivalence E . Definition 4.5. For two objects X, Y of D b ( A ), B D A ( X ) and B D A ( Y ) are said tobe δ -matched with respect to E if B D ( E ( X )) and B D ( E ( Y )) are δ -matched in thesense of Definition 3.10. The bottleneck distance d E, A B ( B D A ( X ) , B D A ( Y )) with respectto E is defined as d E, A B ( B D A ( X ) , B D A ( Y )) := inf (cid:26) δ ∈ Z ≥ (cid:12)(cid:12)(cid:12)(cid:12) B D A ( X ) and B D A ( Y ) are δ -matched with respect to E (cid:27) . Namely, d E, A B ( B D A ( X ) , B D A ( Y )) = d DB ( B D ( E ( X )) , B D ( E ( Y ))) holds.In our convention, an AST states that the interleaving distance between objects X and Y gives an upper bound for the bottleneck distance between their barcodes.Thus, as a consequence of Theorem 4.1, Definition 4.2, and Definition 4.5, we havethe following AST for the derived category D b ( A ). ERIVED AST 17 Theorem 4.6. Let A be an abelian category and X, Y objects in D b ( A ) . Assumethat there exists a derived equivalence E from D b ( A ) to D b (rep k A n ) . Then d E, A B ( B D A ( X ) , B D A ( Y )) ≤ d E, A I ( X, Y ) . AST for zigzag persistence modules. We first discuss an AST for anabelian category A which is derived equivalent to rep k A n . Recall that A can beregarded as a full subcategory of D b ( A ). As a consequence of Theorem 4.6, we havethe following result. Corollary 4.7. Let A be an abelian category. Assume that A is derived equivalentto rep k A n . Then an AST also holds for A . Next, we will discuss an AST for zigzag persistence modules through a derivedequivalence given by a classical tilting module between the categories D b (rep k A n ( a ))and D b (rep k A n ) for an A n -type quiver A n ( a ) with arbitrary orientation a . Definition 4.8 (Assem et al. 2006; Brenner and Butler 1980; Bongartz 1981; Happel and Ringel1982) . Let T be a persistence module. Then T is called a classical tilting module ifit satisfies the following three conditions:(1) the projective dimension of T is at most 1,(2) Ext i ( T, T ) = 0 for all i > 0, and(3) T has exactly n non-isomorphic indecomposable direct summands.Bongartz (1981) proved that our classical tilting modules are equivalent to theoriginal ones (see Bongartz 1981 or Happel and Ringel 1982 for the original def-inition), which are exactly tilting modules with projective dimension at most 1in the sense of Miyashita (1986) (see also Happel 1988, p.118). Here, we recallthat Hom( T, -) is a functor from rep k A n to the module category mod End( T ) op ofthe endomorphism algebra on T . Then we have the following property of classicaltilting modules. Lemma 4.9 (Happel 1988, Chapter III) . Let T be a classical tilting module. As-sume that the endomorphism algebra B = End( T ) op is presented by the quiver Q B with no relations. Then the functor RHom( T, -) is a derived equivalence from D b (rep k A n ) to D b (rep k Q B ) . Now, we construct a classical tilting module whose endomorphism algebra ispresented by the quiver A n ( a ) (with no relations).Let τ, τ − be the AR translations in rep k A n (see Assem et al. 2006, ChapterIV.2, 2.3 Definition or Schiffler 2014, 2.3.3 for definition). For an indecomposablenon-projective (resp. non-injective) representation M , τ ( M ) is indecomposablenon-injective and τ − τ ( M ) ∼ = M (resp. τ − ( M ) is indecomposable non-projectiveand τ τ − ( M ) ∼ = M ) (see Assem et al. 2006, Chapter IV.2, 2.10 Proposition). The τ -orbit of M is the set of indecomposable representations of the form τ m ( M ) or τ − m ( M ) for some non-negative integer m , where τ − m := ( τ − ) m .Since the AR quiver of A n is finite and connected, there are finitely many τ -orbits. Moreover, each τ -orbit contains exactly one indecomposable projective rep-resentation (see Schiffler 2014, 3.1.2).Let O ( P i ) be the τ -orbit of the indecomposable projective representation P i cor-responding to the vertex i of A n . Note that there are n τ -orbits O ( P ) , · · · , O ( P n ),and that O ( P ) = { P } since P is projective-injective. Then, the set { O ( P i ) | i =1 , · · · , n } is just the set of all τ -orbits, the τ -orbit O ( P i ) is finite for any i , and anyindecomposable representation of A n belongs to the τ -orbit O ( P i ) for some i . A sec-tion of the AR quiver Γ( A n ) is a connected full subquiver formed by representativesof all τ -orbits O ( P i ). Example 4.10. In the AR quiver Γ( A )Γ( A ) = I [1 , I [2 , I [1 , I [3 , I [2 , I [1 , , the actions of τ are denoted by dashed arrows, meaning that there are three τ -orbits O ( I [3 , { I [3 , , I [2 , , I [1 , } , O ( I [2 , { I [2 , , I [1 , } , and O ( I [1 , { I [1 , } , of projective representations I [3 , , I [2 , I [1 , { I [1 , , I [2 , , I [2 , } is a section. These representations arewritten as rectangle-surrounded vertices in the foregoing AR quiver.More generally, it is easily understood that a section Σ in the AR quiver Γ( A n )of A n is described like the red line in Figure 2. Γ (A n ) Σ Figure 2. Example of a section Σ in the AR quiver Γ( A n ) of A n Fix a section Σ with vertices Σ = { X = P , · · · , X n } , where X i ∈ O ( P i ) foreach i . Note that P is the top vertex of the AR quiver Γ( A n ) of A n . Then wedefine the module T (Σ) as follows: T (Σ) = n M i =1 X i . Lemma 4.11 (Bongartz 1981, 2.6 Corollary) . For any section Σ , T (Σ) is a classicaltilting module in rep k A n . In this setting, P is a direct summand of any classical tilting module. More gen-erally, any projective-injective representation is a direct summand of any classicaltilting module.The endomorphism algebra End( T (Σ)) op is presented by the quiver Σ op . Bydefinition, every section is an A n ( a )-type quiver with some orientation a and any A n ( a )-type quiver appears as a section. Then, the following result is obtained as aconsequence of Lemma 4.9 and Lemma 4.11. Proposition 4.12. If a section Σ is isomorphic to A n ( a ) op , then the functor RHom( T (Σ) , -) is a derived equivalence from D b (rep k A n ) to D b (rep k A n ( a )) .Remark . Under the derived equivalence RHom( T (Σ) , -), the AR quiver Γ( A n ( a ))of A n ( a ) can be embedded into the AR quiver of D b (rep k A n ) as in Figure 3. ERIVED AST 19 Σ Γ (A n (a)) ・・・ (cid:2) (A n )[0] (cid:0) (A n )[1] Figure 3. The red-bordered polygon except for the broken polyg-onal line is the area of the AR quiver Γ( A n ( a )) of A n ( a ). Forexample, the AR quiver Γ( A ( z )) of A ( z ) was described as inExample 2.5In the case that A = rep k A n ( a ) and E is the quasi-inverse of RHom( T (Σ) , -)with Σ op ∼ = A n ( a ), we put d T,aI := d E, A I and d T,aB := d E, A B . Thus, we conclude thefollowing AST for rep k A n ( a ) with arbitrary orientations a . Note that B D A ( X ) inDefinition 4.5 and the ordinary barcode B ( X ) coincide for any X ∈ rep k A n ( a ). Theorem 4.14 (AST for zigzag) . Let X, Y be zigzag persistence modules in thecategory rep k A n ( a ) with arbitrary orientation a . Then d T,aB ( B ( X ) , B ( Y )) ≤ d T,aI ( X, Y ) . Proof. Since we have a derived equivalence RHom( T (Σ) , -) from D b (rep k A n ) to D b (rep k A n ( a )) by Proposition 4.12, the statement follows from Corollary 4.7. (cid:3) Isometry theorem In this section, we will prove an isometry theorem for the category rep k A n ( a ) ofzigzag persistence modules. Theorem 2.9 gives the inequality d B ≤ d I , which is apart of the following isometry theorem (see Bauer and Lesnick 2014, Theorem 3.1and Section B.1). Theorem 5.1 (Isometry theorem) . Let M, N be persistence modules. Then d B ( B ( M ) , B ( N )) = d I ( M, N ) . From Theorem 5.1, we obtain the isometry theorem for the derived category ofpersistence modules. Theorem 5.2 (Isometry theorem for derived categories) . Let X • , Y • be cochaincomplexes in D b (rep k A n ) . Then d DB ( B D ( X • ) , B D ( Y • )) = d DI ( X • , Y • ) . Proof. From Theorem 4.1, we have only to show d DB ( B D ( X • ) , B D ( Y • )) ≥ d DI ( X • , Y • ) . If B D ( X • ) and B D ( Y • ) are δ -matched, then there exists a δ -matching between B ( H i ( X • )) and B ( H i ( Y • )) for each i by definition. Then, by Theorem 5.1, thereexists a δ -interleaving pair ( f i , g i ) between H i ( X • ) and H i ( Y • ) for each i . ByCorollary 3.9, X • and Y • are derived δ -interleaved in D b (rep k A n ). (cid:3) As a consequence of Theorem 5.2, we can extend Theorem 4.6 and Corollary 4.7by Definition 4.2 and Definition 4.5. Corollary 5.3. Let A be an abelian category and X, Y objects in A or D b ( A ) .Assume that there exists a derived equivalence E from D b ( A ) to D b (rep k A n ) . Then d E, A B ( B D A ( X ) , B D A ( Y )) = d E, A I ( X, Y ) . Finally, we can extend Theorem 4.14 by Corollary 5.3 as follows. Theorem 5.4 (Isometry theorem for zigzag) . Let X, Y be zigzag persistence mod-ules in the category rep k A n ( a ) with arbitrary orientation a . Then d T,aB ( B ( X ) , B ( Y )) = d T,aI ( X, Y ) . Proof. Since we have a derived equivalence RHom( T (Σ) , -) from D b (rep k A n ) to D b (rep k A n ( a )) by Proposition 4.12, the statement follows from Corollary 5.3. (cid:3) The special case of Theorem 5.4 is exactly an isometry theorem for purely zigzagpersistence modules.6. Direct calculation of the induced distance on rep k A n ( a )We start this section with the following remark. Remark . There are actually multiple derived equivalences from D b (rep k A n )to D b (rep k A n ( a )). For example, all translations of a classical tilting module aretwo-sided tilting complexes, which give the derived equivalences (see Rickard 1991).The induced distance on rep k A n ( a ) by rep k A n depends on the choice of derivedequivalences. However, in all cases, the isometry theorem holds for rep k A n ( a ) byCorollary 5.3.The purpose of this section is to provide a direct calculation of the induceddistance on rep k A n ( a ). Let us fix a classical tilting module T := T (Σ) given by asection Σ such that Σ op ∼ = A n ( a ) (see Section 4).Let X be a representation of A n ( a ). By Proposition 4.12, there exists a uniquecochain complex M • ∈ D b (rep k A n ) up to isomorphism such that RHom( T, M • ) ∼ = X . This complex M • is called the corresponding complex of X .It follows from Theorem 5.4 that the interleaving d T,aI and bottleneck d T,aB coin-cide, so we put d a := d T,aI = d T,aB . By definition, we have d a ( X, Y ) = d DI ( M • , N • ) , where X, Y ∈ rep k A n ( a ) and M • , N • are the corresponding complexes of X, Y ,respectively (see Definition 3.7 and Definition 4.2). Thus, we need to deal with thecorresponding complexes to calculate d a on rep k A n ( a ). For this purpose, the classi-cal tilting torsion theory discussed below will be useful (for details, see Assem et al.2006, Chapter VI).Let T be the full subcategory of rep k A n consisting of representations V satisfyingExt ( T, V ) = 0 and F the full subcategory of rep k A n consisting of representations V satisfying Hom( T, V ) = 0. Moreover, let X be the full subcategory of rep k A n ( a )consisting of representations V satisfying T ⊗ V = 0 and Y the full subcategory ofrep k A n ( a ) consisting of representations V satisfying Tor ( T, V ) = 0. It is knownthat T , F , X , and Y are closed under taking extensions. Indeed, for a short exactsequence 0 → M → N → L → ,M, L ∈ T (resp. F , X , or Y ) implies N ∈ T (resp. F , X , or Y ). ERIVED AST 21 Remark . The pairs ( T , F ) and ( X , Y ) are the so-called torsion pairs (for defi-nition, see Assem et al. 2006, Chapter VI.1, 1.1 Definition). Moreover, these pairsare splitting , namely, for each indecomposable representation M ∈ rep k A n , we have M ∈ T or M ∈ F , and for each indecomposable representation N ∈ rep k A n ( a ), wehave N ∈ X or N ∈ Y (see also Assem et al. 2006, Chapter VI.5).The functor Hom( T, -) gives an equivalence from T to Y and the functor Ext ( T, -)gives an equivalence from F to X (see Assem et al. 2006, Chapter VI.3, 3.8 Theo-rem). From the perspective of the right derived functor RHom( T, -), these resultscan be combined as follows. Proposition 6.3. The derived equivalence RHom( T, -) induces the equivalencesfrom T [0] to Y [0] and from F [1] to X [0] .Proof. Note that we have H (RHom( T, -)) ∼ = Hom( T, -) and H (RHom( T, -)) ∼ = H (RHom( T, -[1]) ∼ = Ext ( T, -) . Then, the claim follows from Assem et al. (2006, Chapter VI.3, 3.8 Theorem). (cid:3) Proposition 6.3 states that for a given indecomposable representation X ∈ rep k A n ( a ), the corresponding complex M • is a stalk complex M • = L [0] or L [1]for some L ∈ T or F , respectively. The representation L is called the correspondingrepresentation of X .Furthermore, by using the AR quiver, Proposition 6.3 can be described as inFigure 4, where F = Hom( T. -) , G = Ext ( T, -) , and RF = RHom( T, -). ・・・ Γ (A n )[0] Γ (A n )[1] Σ TF ・・・ Γ (A n (a)) Y X F G F [1] Γ (D(rep k A n ))= Γ (D(rep k A n ( a )))= RF Figure 4. Correspondence between the AR quivers of rep k A n and rep k A n ( a ) in the derived categoryWe will calculate the derived interleaving distance between two stalk complexes. Proposition 6.4. Let M, N be representations of A n . For each pair i, j of integers, d DI ( M [ i ] , N [ j ]) = (cid:26) d I ( M, N ) , i = j max { d I ( M, , d I ( N, } , i = j . Proof. Note that H l ( M [ i ]) = (cid:26) M, i = − l , i = − l , H l ( N [ j ]) = (cid:26) N, j = − l , j = − l . Thus, in the case that i = j , M [ i ] and N [ j ] are derived δ -interleaved if and only if M and N are δ -interleaved by Corollary 3.9. Otherwise, M [ i ] and N [ j ] are derived δ -interleaved if and only if M and 0 are δ -interleaved and 0 and N are δ -interleavedby Corollary 3.9. (cid:3) As a consequence, for indecomposable representations, we have the followingcalculation of the derived interleaving distance by definition. Corollary 6.5. Let M = I [ x, y ] , N = I [ s, t ] be indecomposable representations of A n . For each pair i, j of integers, d DI ( M [ i ] , N [ j ]) = min (cid:26) max {| x − s | , | y − t |} , max {⌈ | y − x + 1 |⌉ , ⌈ | t − s + 1 |⌉} (cid:27) , i = j max {⌈ | y − x + 1 |⌉ , ⌈ | t − s + 1 |⌉} , i = j , where ⌈ - ⌉ is the ceiling function.Proof. Since d I ( M, N ) = min (cid:26) max {| x − s | , | y − t |} , max {⌈ | y − x + 1 |⌉ , ⌈ | t − s + 1 |⌉} (cid:27) , M = 0 , N = 0 ⌈ | y − x + 1 |⌉ , N = 0 ⌈ | t − s + 1 |⌉ , M = 0 , we obtain the desired statement by Proposition 6.4. (cid:3) By combining Proposition 6.3 and Proposition 6.4, we can calculate the distance d a on rep k A n ( a ) by the interleaving distance on rep k A n . Corollary 6.6. Let X, Y be indecomposable representations of A n ( a ) and M, N the corresponding representations of X, Y respectively. Then d a ( X, Y ) = (cid:26) d I ( M, N ) , if X, Y ∈ X or X, Y ∈ Y max { d I ( M, , d I ( N, } , otherwise. By Corollary 6.5 and Corollary 6.6, we can calculate the value d a ( X, Y ) con-cretely when we fix the orientation a .7. Comparison between the block distance and the induced distance Botnan and Lesnick (2018) proved an AST for purely zigzag persistence mod-ules. In that paper, they introduced the interleaving and bottleneck distances onpurely zigzag persistence modules, and Bjerkevik (2016) proved that those distancesactually coincide. Here, we call those distances the block distance , denoted by d BL ,following the paper Meehan and Meyer (2020).In this section, we will compare the distance d BL with our induced distance inthe purely zigzag setting.7.1. General correspondence. Let Z be the poset of integers with usual orderand Z op its opposite poset. As per Botnan and Lesnick (2018), let ZZ be thesubposet of the poset Z op × Z given by ZZ := { ( i, j ) | i ∈ Z , j ∈ { i, i − }} . ERIVED AST 23 Note that this can be expressed by the infinite purely zigzag quiver Q = ( i + 1 , i + 1)( i, i ) ( i + 1 , i ) o o O O ( i − , i − 1) ( i, i − o o O O , so that a (locally finite-dimensional) representation of ZZ is just that of the quiver Q . Then there are injections µ l : A ( z l ) → ZZ ( l = 1 , 2) defined by µ ( i ) = (cid:26) ( m, m ) , i = 2 m − m + 1 , m ) , i = 2 m , µ ( i ) = (cid:26) ( m, m ) , i = 2 m ( m + 1 , m ) , i = 2 m − . Moreover, in Botnan and Lesnick (2018), the intervals h b, d i ZZ ( b ≤ d ) of ZZ aredivided into the following 4 types: closed interval [ b, d ] ZZ := { ( i, j ) ∈ ZZ | ( b, b ) ≤ ( i, j ) ≤ ( d, d ) } , right-open interval [ b, d ) ZZ := { ( i, j ) ∈ ZZ | ( b, b ) ≤ ( i, j ) < ( d, d ) } , left-open interval ( b, d ] ZZ := { ( i, j ) ∈ ZZ | ( b, b ) < ( i, j ) ≤ ( d, d ) } , open interval ( b, d ) ZZ := { ( i, j ) ∈ ZZ | ( b, b ) < ( i, j ) < ( d, d ) } . We use I h b,d i ZZ to denote the interval representation of ZZ associated with theinterval h b, d i ZZ . Note that the interval representation I h b,d i ZZ of ZZ is uniquelydetermined by the interval h b, d i ZZ . Indeed, I h b,d i ZZ is the representation given by I h b,d i ZZ ( i,j ) = (cid:26) k , ( i, j ) ∈ h b, d i ZZ , otherwiseand is called a closed (resp. right-open, left-open, and open ) interval representationif h b, d i ZZ is closed (resp. right-open, left-open, and open). Then for i = 1 , 2, themap µ l induces the correspondence e µ l from the set of interval representations of A n ( z l ) to the subset of interval representations of ZZ . More precisely, we have thefollowing result. Proposition 7.1. (1) For any interval representation I [ s, t ] of A n ( z ) , (a) I [ s, t ] ∈ Y if and only if f µ ( I [ s, t ]) is closed or right-open. (b) I [ s, t ] ∈ X if and only if f µ ( I [ s, t ]) is open or left-open. (2) For any interval representation I [ s, t ] of A n ( z ) , (a) I [ s, t ] ∈ Y if and only if f µ ( I [ s, t ]) = I h b,d i ZZ is closed or right-open, or h b, d i ZZ is of the form h , - i ZZ . (b) I [ s, t ] ∈ X if and only if f µ ( I [ s, t ]) = I h b,d i ZZ is open or left-open exceptfor when h b, d i ZZ is of the form h , - i ZZ . To prove this proposition, we need the following lemma. Lemma 7.2. (1) For any interval representation I [ s, t ] of A n ( z ) , (a) I [ s, t ] ∈ Y if s is odd, (b) I [ s, t ] ∈ X otherwise. (2) For any interval representation I [ s, t ] of A n ( z ) , (a) I [ s, t ] ∈ Y if s is or even, (b) I [ s, t ] ∈ X otherwise.Proof. (1). For any pair 1 ≤ s ≤ t ≤ n , there is an indecomposable stalk complex Y s,t in T [0] or in F [1] of A n such that RHom( T, Y s,t ) ∼ = I [ s, t ] by Proposition 6.3.Let Q ( s ) = Hom( T, X s ) ∈ Y be the indecomposable projective representation of A n ( z ) corresponding to the vertex 1 ≤ s ≤ n . Note that Q ( s ) ∼ = I [ s, s ] if s is odd, Q ( n ) ∼ = I [ n − , n ] if n is even, and Q ( s ) ∼ = I [ s − , s + 1] otherwise. Then we haveHom( X , Y ,t ) ∼ = Hom( Q (1) , I [1 , t ]) = 0 for any t . Since X = P (1) is projective-injective and a direct summand of T , Hom( X , Y ,t ) = 0 implies that Y ,t ∈ T [0]and hence I [1 , t ] ∈ Y by Proposition 6.3. For any odd integer s > 1, we have anexact sequence 0 → I [ s, s ] → I [ s, t ] ⊕ I [1 , s ] → I [1 , t ] → . Since Y is closed under taking extensions, I [ s, s ] , I [1 , t ] ∈ Y implies I [ s, t ] ∈ Y .Statement (b) then follows from (a) and the fact that the torsion pair ( X , Y ) issplitting.(2). First, we have I [ s, s ] ∈ Y when s is even and I [1 , t ] ∈ Y as in the proof of (1).Next, for any even integer s > 1, we similarly have an exact sequence0 → I [ s, s ] → I [ s, t ] ⊕ I [1 , s ] → I [1 , t ] → . Since Y is closed under taking extensions, I [ s, s ] , I [1 , t ] ∈ Y implies I [ s, t ] ∈ Y .Statement (b) then follows from (a) and the fact that the torsion pair ( X , Y ) issplitting. (cid:3) As a consequence of Lemma 7.2, Proposition 7.1 is immediately obtained. Proof of Proposition 7.1. (1). I [ s, t ] ∈ Y if and only if s is odd. In this case, µ ( s ) = ( m, m ) when we write s = 2 m − 1. Then, s is odd if and only if f µ ( I [ s, t ])is closed or right-open, as desired.(2). Similar to the proof of (1). (cid:3) Remark . Proposition 7.1 tells us that the AR quiver of A n ( z l ) ( l = 1 , 2) can bedivided into 2 areas consisting of 4 kinds of intervals in the sense of Botnan and Lesnick(2018) with respect to classical tilting torsion theory.7.2. Comparison. Using the results in Subsection 7.1, we will directly comparethe block distance d BL with our induced distance. For this purpose, we considerthe following orientation: A n ( z ) : 1 ← → · · · → n, where n is odd. In this case, we denote the induced distance by d z instead of d a (see Section 6). By Proposition 7.1, the interval representations I [ s, t ] of A n ( z )can be divided into 4 kinds of representations I h b,d i ZZ . More precisely, we have thefollowing correspondence between ( s, t ) and ( b, d ): closed interval [ b, d ] ZZ ( s = 2 b − , t = 2 d − , right-open interval [ b, d ) ZZ ( s = 2 b − , t = 2 d − , left-open interval ( b, d ] ZZ ( s = 2 b, t = 2 d − , open interval ( b, d ) ZZ ( s = 2 b, t = 2 d − . Since 1 ≤ s ≤ t ≤ n , we have 1 ≤ b ≤ d ≤ ⌈ n ⌉ . In this setting, by Proposition 7.1, I [ s, t ] ∈ Y if and only if f µ ( I [ s, t ]) is closed or right-open. We use Y c , Y co to denotethe sets of interval representations I [ s, t ] ∈ Y which correspond to closed or right-open interval representations of ZZ , respectively. Similarly, we use X o , X oc to denotethe sets of interval representations I [ s, t ] ∈ X which correspond to open or left-openinterval representations of ZZ , respectively. ERIVED AST 25 From the proof of Proposition 7.1, we recall that s is odd if and only if f µ ( I [ s, t ]) isclosed or right-open, and that t is odd if and only if f µ ( I [ s, t ]) is closed or left-open.Let I be an interval representation of A n ( z ), and let us set S I := { s = 1 , · · · , n | Hom( I [ s, s ] , I ) = 0 or Hom( I, I [ s, s ]) = 0 } . In this case, I [ s, s ] is simple projective if s is odd, and is simple injective otherwise.Consequently, Hom( I, I [ s, s ]) = 0 if s is odd, and Hom( I [ s, s ] , I ) = 0 otherwise.Thus, we have I = I [ s, t ] with s := min S I and t := max S I . Note that simple pro-jective representations are source vertices and simple injective representations aresink vertices in the AR quiver (see Assem et al. 2006, Chapter IV.3, 3.6 Corollary).Then Y c , Y co , X o , and X oc can be described in the AR quiver Γ( A n ( z )) of A n ( z )as in Figure 5: Γ (A n (z )) ・・・ Y c Y co X o X oc Figure 5. Division of the AR quiver Γ( A n ( z )) of A n ( z )It is noteworthy that Meehan and Meyer (2020) give the same division of theAR quiver of purely zigzag persistence modules as our model.The division gives us the following correspondence between the interval repre-sentation of A n and ZZ . Lemma 7.4. Let I [ s, t ] be an interval representation of A n ( z ) . Then for theinterval representation I h b,d i ZZ := f µ ( I [ s, t ]) of ZZ , we have the corresponding rep-resentation I [ x, y ] ∈ rep k A n of I [ s, t ] , where ( x, y ) is given by the following: ( x, y ) = ( b, n − d + 1) h b, d i ZZ = [ b, d ] ZZ ( s = 2 b − , t = 2 d − , ( x, y ) = ( b, d − h b, d i ZZ = [ b, d ) ZZ ( s = 2 b − , t = 2 d − , ( x, y ) = ( n − d + 2 , n − b + 1) h b, d i ZZ = ( b, d ] ZZ ( s = 2 b, t = 2 d − , ( x, y ) = ( d, n − b + 1) h b, d i ZZ = ( b, d ) ZZ ( s = 2 b, t = 2 d − . Proof. The endpoint formulas can be easily calculated by Proposition 6.3 and theabove discussion. (cid:3) By Botnan and Lesnick (2018, Lemma 3.1, Lemma 4.1), we have the followingcalculation of d BL for the 4 kinds of representations above. Proposition 7.5 (Botnan and Lesnick 2018, Lemma 3.1, Lemma 4.1) . Let h b, d i ZZ , h e, f i ZZ be intervals of ZZ . Then the following holds. d BL ( I h b,d i ZZ , 0) = ∞ , h b, d i ZZ is closed | d − b | , h b, d i ZZ is half-open | d − b | , h b, d i ZZ is open . Moreover, if h b, d i ZZ , h e, f i ZZ have the same type, then d BL ( I h b,d i ZZ , I h e,f i ZZ ) = min (cid:26) max {| b − e | , | d − f |} , max { d BL ( I h b,d i ZZ , , d BL ( I h e,f i ZZ , } (cid:27) . Otherwise, d BL ( I h b,d i ZZ , I h e,f i ZZ ) = max { d BL ( I h b,d i ZZ , , d BL ( I h e,f i ZZ , } . Then, Proposition 7.1 and Proposition 7.5 lead to the following. Proposition 7.6. Let I [ s, t ] , I [ u, v ] be interval representations of A n ( z ) . For inter-val representations I h b,d i ZZ := f µ ( I [ s, t ]) , I h e,f i ZZ := f µ ( I [ u, v ]) of ZZ , the followinginequalities hold: (1) d BL ( I h b,d i ZZ , ≤ d z ( I [ s, t ] , if I [ s, t ] ∈ Y co , X o , X oc , (2) d z I ( I [ s, t ] , < d BL ( I h b,d i ZZ , 0) = ∞ , if I [ s, t ] ∈ Y c , (3) d BL ( I h b,d i ZZ , I h e,f i ZZ ) ≤ d z ( I [ s, t ] , I [ u, v ]) if I [ s, t ] , I [ u, v ] ∈ Y co , X o or X oc ,or if I [ s, t ] ∈ Y co , I [ u, v ] ∈ X , and (4) d z ( I [ s, t ] , I [ u, v ]) ≤ d BL ( I h b,d i ZZ , I h e,f i ZZ ) if I [ s, t ] ∈ Y c .Proof. In each case, the value of d BL can be calculated by Proposition 7.5. Onthe other hand, the value of d z in each case can be caluclated by Lemma 7.4,Corollary 6.5, and Corollary 6.6.Then we have d z ( I [ s, t ] , 0) = (cid:26) ⌈ | d − b |⌉ , I [ s, t ] ∈ Y co or X oc ⌈ | n − ( b + d ) + 2 |⌉ , I [ s, t ] ∈ Y c or X o . When 1 ≤ b ≤ d ≤ ⌈ n ⌉ , it is easy to check the inequality | d − b | < | n − ( b + d ) + 2 | .Thus, we obtain inequality (1). Since d z ( I [ s, t ] , < ∞ always holds, we obtaininequality (2).Moreover, by the symmetry of the distance, d := d z ( I [ s, t ] , I [ u, v ]) can be calcu-lated as follows.(a) if I [ s, t ] , I [ u, v ] ∈ Y c , then d = min { max {| b − e | , | d − f |} , max {⌈ | n − ( b + d ) + 2 |⌉ , ⌈ | n − ( e + f ) + 2 |⌉}} , (b) if I [ s, t ] , I [ u, v ] ∈ Y co or I [ s, t ] , I [ u, v ] ∈ X oc , then d = min { max {| b − e | , | d − f |} , max {⌈ | d − b |⌉ , ⌈ | f − e |⌉}} , (c) if I [ s, t ] , I [ u, v ] ∈ X o , then d = min { max {| b − e | , | d − f |} , max {⌈ | n − ( b + d ) + 2 |⌉ , ⌈ | n − ( e + f ) + 2 |⌉}} , (d) if I [ s, t ] ∈ Y c , I [ u, v ] ∈ Y co , then d = min { max {| b − e | , | n − ( d + f ) + 2 |} , max {⌈ | n − ( b + d ) + 2 |⌉ , ⌈ | f − e |⌉}} , (e) if I [ s, t ] ∈ X oc , I [ u, v ] ∈ X o , then d = min { max {| b − e | , | n − ( d + f ) + 2 |} , max {⌈ | d − b |⌉ , ⌈ | n − ( e + f ) + 2 |⌉}} , and(f) if I [ s, t ] ∈ Y , I [ u, v ] ∈ X , then d = max { d aI ( I [ s, t ] , , d aI ( I [ u, v ] , } . ERIVED AST 27 Since the inequality | b − d | < | n − ( b + d ) + 2 | (1 ≤ b ≤ d ≤ ⌈ n ⌉ ) holds, byinequality (1), inequality d BL ( I h b,d i ZZ , I h e,f i ZZ ) ≤ d z ( I [ s, t ] , I [ u, v ])holds in cases (b), (c), and (f) except for when I [ s, t ] ∈ Y c in case (f). Thus, weobtain inequality (3).In case (f), if I [ s, t ] ∈ Y c , then d BL ( I [ b,d ] ZZ , 0) = ∞ , and hence it is obvious that d z ( I [ s, t ] , I [ u, v ]) < d BL ( I [ b,d ] ZZ , I [ e,f ] ZZ ) . In case (a), since I [ s, t ] , I [ u, v ] ∈ Y c , d BL ( I [ b,d ] ZZ , 0) = d BL ( I [ e,f ] ZZ , 0) = ∞ . Thenby definition, the inequality d z ( I [ s, t ] , I [ u, v ]) ≤ d BL ( I [ b,d ] ZZ , I [ e,f ] ZZ ) = max {| b − e | , | d − f |} holds.In case (d), since I [ s, t ] ∈ Y c , I [ u, v ] ∈ Y co , d BL ( I [ b,d ] ZZ , I [ e,f ) ZZ ) = max { d BL ( I [ b,d ] ZZ , , d BL ( I [ e,f ) ZZ , } = ∞ . Then it is obvious that d z ( I [ s, t ] , I [ u, v ]) < d BL ( I [ b,d ] ZZ , I [ e,f ] ZZ ).Thus, we obtain inequality (4). (cid:3) Remark . In Proposition 7.6, the case in which I [ s, t ] ∈ X oc , I [ u, v ] ∈ X o remains.In this case, we have d BL ( I ( b,d ] ZZ , I ( e,f ) ZZ ) = max { d BL ( I ( b,d ] ZZ , , d BL ( I ( e,f ) ZZ , } = max { | d − b | , | f − e |} , and d BL and d z are incomparable for large n . For example, in the case that n = 7, we consider representations I [2 , I [2 , I [1 , 2] of A n ( z ). Then wehave f µ ( I [2 , I (1 , ZZ , f µ ( I [2 , I (1 , ZZ , and f µ ( I [2 , I (1 , ZZ . By Propo-sition 7.5, Lemma 7.4, Corollary 6.5, and Corollary 6.6, the inequalities d BL ( I (1 , ZZ , I (1 , ZZ ) ≥ d z ( I [2 , , I [2 , d BL ( I (1 , ZZ , I (1 , ZZ ) ≤ d z ( I [2 , , I [2 , d BL in Botnan and Lesnick (2018) and ourinduced distance d z are incomparable. Indeed, Proposition 7.6 (3) and (4) informus that the inequality for comparing the distances is dependent on interval type. Acknowledgements We are grateful to Kosuke Sakurai, who was a Master’s student of the firstauthor at Tohoku University, for his cooperation in proving the converse AST inour derived setting. The second author also thanks Emerson G. Escolar and KillianF. Meehan for their helpful discussions.The first author was partially supported by JST CREST Mathematics (15656429)and JSPS Grant-in-Aid for Scientific Research (A) (JP20H00119). The secondauthor was partially supported by JSPS Grant-in-Aid for Scientific Research (C)(JP20K03760) and Osaka City University Advanced Mathematical Institute (MEXTJoint Usage/Research Center on Mathematics and Theoretical Physics). Conflict of interest On behalf of all authors, the corresponding author states that there is no conflictof interest. 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