aa r X i v : . [ m a t h . QA ] N ov ALGEBRA DEPTH INTENSOR CATEGORIES
LARS KADISON
IN MEMORY OF DANIEL KASTLER
Abstract.
Study of the quotient module of a finite-dimensional Hopf subal-gebra pair in order to compute its depth yields a relative Maschke Theorem, inwhich semisimple extension is characterized as being separable, and is there-fore an ordinary Frobenius extension. We study the core Hopf ideal of a Hopfsubalgebra, noting that the length of the annihilator chain of tensor powersof the quotient module is linearly related to the depth, if the Hopf algebrais semisimple. A tensor categorical definition of depth is introduced, and asummary from this new point of view of previous results are included. It isshown in a last section that the depth, Bratteli diagram and relative cyclichomology of algebra extensions are Morita invariants. Introduction and Preliminaries
Sometimes it is useful to classify numbers with the same prime factors together.Similarly, it is useful to classify together finite-dimensional modules over a finite-dimensional algebra with isomorphic indecomposable summands - two such mod-ules, which have the same indecomposables but perhaps with different nonzeromultiplicities, are said to be similar. Since an abelian category has direct sum ⊕ that work as usual, similarity of two objects X, Y , denoted by X ∼ Y , is definedby X ⊕ ∗ ∼ = n · Y , i.e., “ X divides a multiple of Y ,” and Y ⊕ ∗ ∼ = m · X (or briefly Y | m · X ) for some multiplicities m, n ∈ N . In the presence of a uniqueness theo-rem for indecomposables that includes X, Y , they share isomorphic indecomposablesummands. Also, the endomorphism rings of X and Y are Morita equivalent in aparticularly transparent way [1, 20]. For example, one may introduce the theory ofbasic algebras without complications using the regular representation and a similardirect sum of projective indecomposables with constant multiplicity one.A special type of abelian category is a tensor category, which has a tensor product ⊗ satisfying the usual distributive, associative and unital laws up to natural iso-morphism. An algebra A may then be defined in terms of multiplication A ⊗ A → A as usual. Define the minimum depth of A to be the least 2 n + 1 = 1 , , , . . . suchthat A ⊗ ( n ) = A ⊗ · · · ⊗ A ( n times A ) is similar to A ⊗ ( n +1) , which simplifies to A ⊗ ( n +1) | q · A ⊗ ( n ) for some q ∈ N , since A ⊗ ( n ) | A ⊗ ( n +1) follows from applying themultiplication and unit. This definition applied to an algebra A in the category ofbimodules over a ring B with tensor ⊗ = ⊗ B , recovers the minimum odd depth of Mathematics Subject Classification.
Key words and phrases. subgroup depth, Morita equivalent ring extensions, Frobenius exten-sion, semisimple extension, tensor category, core Hopf ideals, relative Maschke theorem.
IN MEMORY OF DANIEL KASTLER the ring extension B → A [2], where it is applied to finite group algebra extensionsto recover (together with minimum even depth) subgroup depth [7]. Interestingvalues of subgroup depth have been computed in [7, 2, 11, 13, 14, 18], where sub-group depth less than 3 are normal subgroups [3, 4, 25, 28, 26]. Several propertiesof subgroup depth extend to Hopf subalgebra (and left coideal subalgebra) pairssuch as a characterization of normality [3] and unchanged minimum even depthwhen factoring out the subgroup core [2, 16].The main problem in the area is the one formulated in [2, p. 259] for a finite-dimensional Hopf subalgebra pair R ⊆ H , where d ( R, H ) denotes the minimumdepth.
Problem 1.1. Is d ( R, H ) < ∞ ? There are examples in subfactor theory by Haagerup of infinite depth, althoughnot anwering the problem. We bring up three other equivalent problems below.In the opposite tensor category, algebra becomes a notion of coalgebra with thesame definition of depth. In the tensor category of bimodules over B , a coalgebra inthis sense is a B -coring. Applying the definition of depth to the Sweedler coring ofa ring extension, one recovers the minimum h-depth of the ring extension as definedin [29]. The minimum h-depth of a Hopf subalgebra pair R ⊆ H is shown in [31] tobe precisely determined by the depth of their quotient module Q H = H/R + H inthe finite tensor category of finite-dimensional H -modules [12]. In turn the depthof Q is determined precisely by the length of the descending chain of annihilatorideals of the tensor powers of Q , if the Hopf algebra is semisimple, as provenin Theorem 3.14. The quotient module Q has many uses, including the followingequivalent reformulation of the problem above, either as an H - or R -module isoclassin the respective representation ring (see [31] or Section 3, the notion below isalgebraic element in a ring). Problem 1.2. Is Q an algebraic module? For example, a finite group algebra extension has quotient module Q equal to apermutation module, which is algebraic [13, Ch. 9]. The question in general is onlyinteresting for the projective-free summands of Q , since projectives form a finiterank ideal in the representation ring [15]. If either R or H has finite representationtype (e.g., is semisimple, Nakayama serial), Q is similarly algebraic. Example 4.6computes a finite depth where both Hopf algebras are of infinite representationtype.In Section 4, we study depth of a non-normal subalgebra in a factorisable Hopfalgebra in terms of entwined subalgebras such as a matched pair of Hopf algebras. InSection 3, we prove a relative Maschke theorem characterizing semisimple extensionof finite-dimensional Hopf algebras as a separable extension; as a corollary, theseare ordinary (or untwisted) Frobenius extensions. We also define and study thecore Hopf ideal of a Hopf subalgebra, which extends to Hopf algebras the usualnotion of core of a subgroup pair of finite groups. We note that the length ofthe annihilator chain of tensor powers of the quotient module is linearly related tothe depth if the Hopf algebra is semisimple, improving on some results in [15]. InSection 5, we make a categorical study of a Morita equivalence of noncommutativering extensions. We show that depth and relative cyclic homology of a ring extensionare Morita invariants, as is the inclusion matrix of a semisimple complex algebraextension. LGEBRA DEPTH IN TENSOR CATEGORIES 3
Similar modules.
Let A be a ring. Two left A -modules, A N and A M , aresaid to be similar ([1], or H-equivalent [20]) denoted by A M ∼ A N if two conditionsare met. First, for some positive integer r , N is isomorphic to a direct summandin the direct sum of r copies of M , denoted by A N ⊕ ∗ ∼ = r · A M ⇔ N | r · M ⇔ ∃ f i ∈ Hom ( A M, A N ) , g i ∈ Hom ( A N, A M ) : r X i =1 f i ◦ g i = id N (1)Second, symmetrically there is s ∈ Z + such that M | s · N . (Say that M and N are dissimilar if neither condition M | s · N or N | r · M holds.) It is easy to extendthis definition of similarity to similarity of two objects in an abelian category, andto show that it is an equivalence relation. Example 1.3.
Suppose A is an artinian ring, with indecomposable A -modules { P α | α ∈ I } (representatives from each isomorphism class for some index set I ). ByKrull-Schmidt finitely generated modules M A and N A have a unique factorizationinto a direct sum of multiples of finitely many indecomposable module components.Denote the indecomposable constituents of M A by Indec ( M ) = { P α | [ P α , M ] = 0 } where [ P α , M ] is the number of factors in M isomorphic to P α . Note that M | q · N for some positive q if and only if Indec ( M ) ⊆ Indec ( N ). It follows that M ∼ N iff Indec ( M ) = Indec ( N ).Suppose A A = n P ⊕ · · · ⊕ n r P r is the decomposition of the regular moduleinto its projective indecomposables. Let P A = P ⊕ · · · ⊕ P r . Then P A and A A aresimilar (and call P the basic A -module in the similarity class of A ). Then A andEnd P A are Morita equivalent. The algebra End P A is of course the basic algebraof A .Suppose A is a semisimple ring. Then P i = S i are simple modules. Note thatthe annihilator ideal Ann S i is a maximal ideal in A ; denote it by I i . Note thatAnn ( n i · S i ) = I i , Ann ( n i · S i ⊕ n j · S j ) = I i ∩ I j , and any ideal I is uniquelyAnn ( S i ⊕ · · · ⊕ S i s ) for the 2 r integer subsets, 1 ≤ i < · · · < i s ≤ r . Proposition 1.4.
If two modules are similar, then their annihilator ideals areequal. Conversely, if A is a semisimple ring, two finitely generated modules withequal annihilator ideals are similar.Proof. Given modules M and N , if M ֒ → N , then Ann N ⊆ Ann M . It followsthat M | r · N implies that Ann N ⊆ Ann M . Hence, M ∼ N ⇒ Ann M = Ann N .Suppose A is a semisimple ring; we use the notation in the example. If M and N are finitely generated A -modules such that Ann M = Ann N is the ideal I in A ,then I = I i ∩ · · · ∩ I i s for some integers 1 ≤ i < · · · < i s ≤ r . It follows that S i ⊕ · · · ⊕ S i s is the basic module in the similarity class of both M and N ; inparticular, M ∼ N . (cid:3) Example 1.5.
Suppose R is an artinian ring that is not semisimple and with twoadditional indecomposable modules I , I that are not projective and not isomor-phic. Then the modules M = R ⊕ I and N = R ⊕ I are both faithful generators,but dissimilar by Krull-Schmidt. This contradicts the converse of the propositionfor more general rings. (Without dissimilarity, one additional nonprojective inde-composable would suffice.) LARS KADISON
IN MEMORY OF DANIEL KASTLER
Subring depth.
Throughout this section, let A be a unital associative ringand B ⊆ A a subring where 1 B = 1 A ; more generally, it suffices to assume B → A is a unital ring homomorphism, called a ring extension, although we suppress thisoption notationally. Note the natural bimodules B A B obtained by restriction ofthe natural A - A -bimodule (briefly A -bimodule) A , also to the natural bimodules B A A , A A B or B A B , which are referred to with no further ado. Let A ⊗ B ( n ) denote A ⊗ B · · · ⊗ B A ( n times A , n ∈ N ), where A ⊗ B = B . For n ≥
1, the A ⊗ B ( n ) has a natural A -bimodule structure which restricts to B - A -, A - B - and B -bimodulestructures occuring in the next definition. Note that A ⊗ B ( n ) | A ⊗ B ( n +1) automat-ically occurs in any case for n ≥
2, since A → A ⊗ B A given by a a ⊗ B n = 1 and A -bimodules, this is the separability conditionon A ⊇ B ; otherwise, A | A ⊗ B A as A - B - or B - A -bimodules (via the split epi a ⊗ B a ′ aa ′ ). Definition 1.6.
The subring B ⊆ A has depth 2 n + 1 ≥ B -bimodules A ⊗ B ( n ) ∼ A ⊗ B ( n +1) . The subring B ⊆ A has left (respectively, right) depth2 n ≥ A ⊗ B ( n ) ∼ A ⊗ B ( n +1) as B - A -bimodules (respectively, A - B -bimodules).Equivalently, A ⊇ B has depth 2 n + 1 ≥
1, or left depth 2 n ≥
2, if A ⊗ B ( n +1) ⊕ ∗ ∼ = q · A ⊗ B ( n ) (2)as B - B -bimodules, or B - A -bimodules, respectively. Right depth 2 n is defined sim-ilarly in terms of A - B -bimodules.It is clear that if B ⊆ A has either left or right depth 2 n , it has depth 2 n + 1 byrestricting the similarity condition to B -bimodules. If B ⊆ A has depth 2 n +1, it hasdepth 2 n +2 by tensoring the similarity by −⊗ B A or A ⊗ B − . The minimum depth isdenoted by d ( B, A ); if B ⊆ A has no finite depth, write d ( B, A ) = ∞ . We similarlydefine minimum odd depth d odd ( B, A ) and minimum even depth d even ( B, A ).A subring B ⊆ A has h-depth n − A - A -bimodules ( n = 1 , , , . . . ). Note that B has h-depth 2 n − A implies thatit has h-depth 2 n + 1 (also that it has depth 2 n ). Thus define the minimum h-depth d h ( B, A ) (and set this equal to ∞ if no such n ∈ N exists). Note that h-depth 1is the Azumaya-like condition of Hirata in [20]. The notion of h-depth is studiedin [29]; by elementary considerations the inequality | d h ( B, A ) − d ( B, A ) | ≤ Depth of algebras and coalgebras in tensor categories
In this section, we define depth of algebras and coalgebras in tensor categories.When applied to algebras and coalgebras in a bimodule tensor category, this defi-nition recovers minimum odd depth defined in [7] and h-depth defined in [30]. Inparticular, a coalgebra in bimodule tensor category is a coring, with depth definedin [16]. An algebra or coalgebra in a finite tensor category is an H -module algebraor H -module coalgebra with depth defined in [31].2.1. Tensor Category.
By a tensor category ( M , ⊗ , ) we mean an abelian cat-egory M with unit object ∈ Ob( M ) and tensor product ⊗ : M × M → M ,an additive bifunctor (satisfying distributive laws w.r.t. ⊕ ) with associativity con-straint , a natural isomorphism α X,Y,Z : ( X ⊗ Y ) ⊗ Z ∼ −→ X ⊗ ( Y ⊗ Z ) , X, Y, Z ∈ M LGEBRA DEPTH IN TENSOR CATEGORIES 5 satisfying the pentagon axiom (a commutative pentagon with 4 arbitrary objectsin a tensor product grouped together in different ways, see for example [41, (2.3)]),and unit constraints , natural isomorphisms ℓ, r such that ℓ X : ⊗ X ∼ −→ X, r X : X ⊗ ∼ −→ X, X ∈ M satisfy the triangle axiom (a commutative triangle with the unit object betweentwo other arbitrary objects in a tensor product associated in two ways using α, ℓ, r ,[41, (2.4)]). The Coherence Theorem of MacLane states that every diagram con-structed from associativity and unit constraints commutes. (Here we are makingno requirement of left and right duals satisfying rigidity axioms.)A tensor functor between tensor categories ( M , ⊗ , ) and ( M ′ , ⊗ ′ , ′ ) is a func-tor F : M → M ′ such that for every X, Y ∈ Ob( M ), there are isomorphisms J X,Y : F ( X ) ⊗ ′ F ( Y ) ∼ −→ F ( X ⊗ Y ) defining a natural isomorphism, and φ : ′ ∼ −→ F ( )is an isomorphism satisfying a commutative hexagon and two commutative rectan-gles, see for example [41, (2.12),(2.13),(2.14)]. If F is an equivalence of categories,the tensor categories M , M ′ are said to be tensor equivalent . Example 2.1.
Let R be a ring, and R M R denote the category of R - R -bimodulesand their bimodule homorphisms (denoted by Hom R − R ( X, Y ) or Hom ( R X R , R Y R )).Note that R M R has a tensor product ⊗ R and unit object R R R , the natural bimod-ule structure on R itself. For example, ℓ X : R ⊗ R X ∼ −→ X is the well-knownnatural isomorphism. This makes ( R M R , ⊗ R , R R R ) into a tensor category.Let A, R are rings, M A , M R their categories of right modules and homomor-phisms. Recall that A and R are Morita equivalent rings if R ∼ = End P A for someprogenerator A -module P , if and only if the categories M R and M A are equiv-alent, via the additive functor − ⊗ R P . The inverse bimodule of P is denotedwithout ambiguity by P ∗ ∼ = Hom ( P A , A A ), since Hom ( P A , A A ) ∼ = Hom ( R P, R R )as A - R -bimodules (by a theorem of Morita [39]). Lemma 2.2.
Suppose T : M R ∼ −→ M A is an equivalence of categories given by T ( X ) = X ⊗ R P A . Then the categories R M R and A M A are tensor equivalent via F ( R Y R ) = P ∗ ⊗ R Y ⊗ R P .Proof. The proof follows from F ( X ⊗ R Y ) = P ∗ ⊗ R X ⊗ R Y ⊗ R P ∼ = P ∗ ⊗ R X ⊗ R R ⊗ R Y ⊗ R P ∼ = P ∗ ⊗ R X ⊗ R P ⊗ A P ∗ ⊗ R Y ⊗ R P ∼ = F ( X ) ⊗ A F ( Y ) . Also F ( R R R ) ∼ = A A A . The functor F is an equivalence with inverse functor F − ( A Z A ) = P ⊗ A Z ⊗ A P ∗ . (cid:3) In a tensor category ( M , ⊗ , M ), one says ( B, m, u ) is an algebra in M if themultiplication m : B ⊗ B → B , a morphism in M , satisfies a commutative pentagon[41, 3.9] w.r.t. associativity isomorphism α A,A,A and “the unit” u : 1 M → A , amorphism in M , satisfies two commutative rectangles [41, 3.10] w.r.t. the naturalisomorphisms ℓ A , r A in the notation of Subsection 2.1. ( Coalgebra ( B, ∆ , ε ) isdefined dually by coassociative comultiplication ∆ : B → B ⊗ B and counit ε : B → M satisfying the counit diagrams.) That B ⊗ ( n ) | B ⊗ ( n +1) for n ≥ Definition 2.3.
Let B be an algebra (or coalgebra) in a tensor category M . Define B to have depth 1 if B ∼ M . Define B to have depth 2 n + 1 ( n ≥
1) if B ⊗ ( n +1) | q · LARS KADISON
IN MEMORY OF DANIEL KASTLER B ⊗ ( n ) for some q ∈ N ( ⇔ B ⊗ ( n ) ∼ B ⊗ ( n +1) ) ; in this case, B also has depth2 n + 3 , n + 5 , . . . by tensoring repeatedly by − ⊗ B . If there is a finite n ∈ N likethis, let d ( B, M ) denote the minimum depth (an odd number); otherwise, write d ( B, M ) = ∞ . Example 2.4.
Let A be a ring, with tensor category of bimodules A M A . Analgebra B (or monoid) in A M A has unit mapping u : A → B and multiplication B ⊗ A B → B satisfying associativity and unital axioms as usual. This is equiv-alently a ring extension. The depth just defined is the minimum odd depth; i.e., d ( B, A M A ) = d odd ( A, B ), which is obvious from Definition 1.6 (with role reversal).
Remark 2.5.
The reference [41, 3.8] also sketches the definition of modules andbimodules over such algebras, as well as Morita equivalence between two such al-gebras. For example, a left module over algebra A in tensor category B M B is an A - B -bimodule N as an exercise in applying these ideas. The category A M B isequivalent to the category A M of left modules over A . If A ′ is another algebra in B M B Morita equivalent in the sense of [41], then A ′ M B is equivalent to A M B .This is the case if the ring extensions B → A and B → A ′ are Morita equivalent inthe sense of Section 5, cf. Diagram (34). Example 2.6.
Let B = C be an A -coring; i.e., a coalgebra (or comonoid) in thetensor category A M A . Dual to algebra, there is a comultiplication ∆ : C → C ⊗ A C and counit ε : C → A , both A - A -bimodule homomorphisms, satisfying coassociativ-ity and counit diagrams [5]. The definition of minimum depth d ( C , A M A ) coincideswith the depth d ( C , A ) of corings defined in [16, 2.1]: d ( C , A M A ) = d ( C , A ).Let A ⊇ B be a ring extension, and C = A ⊗ B A its Sweedler A -coring, withcomultiplication simplifying to A ⊗ B (2) → A ⊗ B (3) , a ⊗ B a a ⊗ B ⊗ B a ,and counit ε C : A ⊗ B A → A , a ⊗ B a a a ( a , a ∈ A ). Comparing withDefinition 1.6 and applying cancellations of the type X ⊗ A A ∼ = X , we see thatcoring depth of C recovers h-depth of the ring extension: d ( C , A M A ) = d h ( B, A ).Suppose k is a field, the ground field below for all algebras, coalgebras, modulesand unadorned tensor products in finite tensor categories (including the tensorcategory of finite-dimensional vector spaces, Vect k ). Example 2.7.
Let H be a finite-dimensional Hopf k -algebra; its category of finite-dimensional modules M H is a finite tensor category [12]. The tensor ⊗ = ⊗ k isdefined by the diagonal action, where V ⊗ W : ( v ⊗ w ) · h = vh (1) ⊗ wh (2) . Theunit module is k ε where ε : H → k is the counit. An algebra A in M H is a right H -module algebra, which the reader may check satisfies the (measuring) axioms( ab ) .h = ( a.h (1) )( b.h (2) ) and 1 A .h = 1 A ε ( h ) for all a, b ∈ A and h ∈ H . A coalgebra C in M H is a right H -module coalgebra ( C, ∆ , ε C ) satisfying∆( ch ) = c (1) h (1) ⊗ c (2) h (2) , ε C ( ch ) = ε C ( c ) ε ( h ) (3)for all c ∈ C, h ∈ H .The depth d ( A, M H ) and d ( C, M H ) is a linear rescaling of the minimum depthof any object in M H defined in [31, 15, 16], not an important difference, thoughslightly more convenient in formulas given below. Example 2.8.
Continuing with H , the category of right H -comodules M H is atensor category, where X, Y ∈ M H has tensor product X ⊗ Y as linear space withcomultiplication x ⊗ y x (0) ⊗ y (0) ⊗ x (1) y (1) . The unit module is k with coaction LGEBRA DEPTH IN TENSOR CATEGORIES 7 k H . An algebra A in M H has multiplication m : A ⊗ A → A and unit k → A right H -comodule morphisms. This condition is equivalent to the coaction of A , ρ A : A → A ⊗ H , being an algebra homomorphism (w.r.t. the tensor algebra). Thus A is a right H -comodule algebra. See for example [36].3. Entwining structures
In this section we summarise the equalities and inequalities obtained in [16] and[15] between depths of entwined corings and factorisable algebras on the one hand(in the “difficult” tensor bimodule category) and depth of an H -module coalgebraor algebra on the other hand (in a more manageable finite tensor category [12]). Westudy the quotient module Q of a finite-dimensional Hopf subalgebra pair R ⊆ H in terms of core Hopf ideals, duals and Frobenius extensions, and under conditionsof semisimplicity, relative or not.Recall that an entwining structure of an algebra A and coalgebra C is given bya linear mapping ψ : C ⊗ A → A ⊗ C (called the entwining mapping) satisfyingtwo commutative pentagons and two triangles (a bow-tie diagram on [5, p. 324]).Equivalently, ( A ⊗ C, id A ⊗ ∆ C , id A ⊗ ε C ) is an A -coring with respect to the A -bimodule structure a ( a ′ ⊗ c ) a ′′ = aa ′ ψ ( c ⊗ a ′′ ) (or conversely defining ψ ( c ⊗ a ) =(1 A ⊗ c ) a ) (details in [5, 32.6] or [9, Theorem 2.8.1]).In more detail, an entwining structure mapping ψ : C ⊗ A → A ⊗ C takes valuesusually denoted by ψ ( c ⊗ a ) = a α ⊗ c α = a β ⊗ c β , suppressing linear sums of rankone tensors, and satisfies the axioms: (for all a, b ∈ A, c ∈ C )(1) ψ ( c ⊗ ab ) = a α b β ⊗ c αβ ;(2) ψ ( c ⊗ A ) = 1 A ⊗ c ;(3) a α ⊗ ∆ C ( c α ) = a αβ ⊗ c (1) β ⊗ c (2) α (4) a α ε C ( c α ) = aε C ( c ),which is equivalent to two commutative pentagons (for axioms 1 and 3) and twocommutative triangles (for axioms 2 and 4), in an exercise.3.1. Doi-Koppinen entwinings [5, 9] . Let H be a finite-dimensional Hopf alge-bra. Suppose A is an algebra in the tensor category of right H -comodules, equiva-lently, A is a right H -comodule algebra. Moreover, let ( C, ∆ C , ε C ) be a coalgebrain the tensor category M H , right H -module coalgebra as noted in the exampleabove in Section 2. Of course, if H = k is the trivial one-dimensional Hopf algebra, A may be any k -algebra and C any k -coalgebra. Example 3.1.
The Hopf algebra H is right H -comodule algebra over itself, where ρ = ∆. Given a Hopf subalgebra R ⊆ H the quotient module Q defined as Q = H/R + H . Note that Q is a right H -module coalgebra. So is ( H, ∆ , ε ) trivially aright H -module coalgebra. The canonical epimorphism H → Q denoted by h h is an epi of right H -module coalgebras. The module Q H is cyclic with generator1 H .The mapping ψ : C ⊗ A → A ⊗ C defined by ψ ( c ⊗ a ) = a (0) ⊗ ca (1) is anentwining (the Doi-Koppinen entwining [5, 33.4], [9, 2.1]). From the equivalence ofcorings with entwinings, it follows that A ⊗ C has A -coring structure a ( a ′ ⊗ c ) a ′′ = aa ′ a ′′ (0) ⊗ ca ′′ (1) (4)which defines the bimodule A ( A ⊗ C ) A . The coproduct is given by id A ⊗ ∆ C andthe counit by id A ⊗ ε C . LARS KADISON
IN MEMORY OF DANIEL KASTLER
Note that Eq. (4) above, and Eq. (5) below, exhibit the category M A as a modulecategory over M H [12]. Proposition 3.2. [16, Prop. 4.2]
The depth of the A -coring A ⊗ C (of a Doi-Koppinen entwining) and the depth of the H -module coalgebra C are related by d ( A ⊗ C, A M A ) ≤ d ( C, M H ) .Proof. One notes that ( A ⊗ C ) ⊗ A ( n ) ∼ = A ⊗ C ⊗ ( n ) as A - A -bimodules via cancellationsof the type X ⊗ A A ∼ = X . Keeping track of the right A -module structure on A ⊗ C ⊗ ( n ) , one shows that it is given by( a ⊗ c ⊗ · · · ⊗ c n ) b = ab (0) ⊗ c b (1) ⊗ · · · ⊗ c n b ( n ) . (5)If d ( C, M H ) = n , then C ⊗ ( n ) ∼ C ⊗ ( n +1) in the finite tensor category M H . Apply-ing an additive functor, it follows that A ⊗ C ⊗ ( n ) ∼ A ⊗ C ⊗ ( n +1) as A -bimodules.Then applying the isomorphism just above and Definition 2.3 obtains the inequalityin the proposition. (cid:3) For example, if A = H , and C a right H -module coalgebra, the Doi-Koppinenentwining mapping ψ : C ⊗ H → H ⊗ C is of course ψ ( c ⊗ h ) = h (1) ⊗ ch (2) . Theassociated H -coring H ⊗ C has coproduct id H ⊗ ∆ C and counit id H ⊗ ε C with H -bimodule structure: ( x, y, h ∈ H, c ∈ C ) x ( h ⊗ c ) y = xhy (1) ⊗ cy (2) (6) Corollary 3.3. [16, Prop. 3.2]
The depth of the H -coring H ⊗ C and the depth ofthe H -module coalgebra C are related by d ( H ⊗ C, H ) = d ( C, M H ) .Proof. This follows immediately from the proposition, but the proof reverses asfollows. If d ( H ⊗ C, H M H ) = 2 n + 1, so that H ⊗ C ⊗ ( n ) ∼ H ⊗ C ⊗ ( n +1) as H - H -bimodules, apply the additive functor k ⊗ H − to the similarity and obtainthe similarity of right H -modules, C ⊗ ( n ) ∼ C ⊗ ( n +1) . Thus d ( C, M H ) ≤ d ( H ⊗ C, H M H ) as well. (cid:3) The corollary applies as follows. Let K ⊆ H be a left coideal subalgebra of afinite-dimensional Hopf algebra; i.e., ∆( K ) ⊆ H ⊗ K . Let K + denote the kernelof the counit restricted to K . Then K + H is a right H -submodule of H and acoideal by a short computation given in [5, 34.2]. Thus Q := H/K + H is a right H -module coalgebra (with a right H -module coalgebra epimorphism H → Q givenby h h + K + H := h ). The H -coring H ⊗ Q has grouplike element 1 H ⊗ H ; infact, [5, 34.2] together with [46] shows that this coring is Galois: H ⊗ K H ∼ = −→ H ⊗ Q (7)via x ⊗ R y xy (1) ⊗ y (2) , an H - H -bimodule isomorphism. That H K is faithfullyflat follows from Skryabin’s Theorem [46] that K is a Frobenius algebra and H K is free. Note that an inverse to (7) is given by x ⊗ z xS ( z (1) ) ⊗ K z (2) for all x, z ∈ H .From Proposition 3.3, Eq. (7) and Example 2.6 we note the first statement below.The second statement is proven similarly as shown in [31]. Corollary 3.4. [16, Corollary 3.3][31, Theorem 5.1]
The h-depth of K ⊆ H isrelated to the depth of Q in M H by d h ( K, H ) = d ( Q, M H ) . (8) LGEBRA DEPTH IN TENSOR CATEGORIES 9 If R is a Hopf subalgebra of H , the following holds: d even ( R, H ) = d ( Q, M R ) + 1 (9)The following is of use to computing depth graphically from a bicolored graph incase R and H are semisimple C -algebras. Let U denote the functor of restriction-induction, i.e., U = Ind HR Res HR : M H → M H . Proposition 3.5.
The depth d ( Q, M H ) = 2 n + 1 is the least n for which U n ( k ) ∼ U n +1 ( k ) .Proof. Recall that Q ∼ = k ⊗ R H and for any module M H , U ( M ) ∼ = M ⊗ Q (tensorin M H ) [31]. It follows by induction that Q ⊗ ( n ) ∼ = U n ( k ). (cid:3) Note that decomposing Q into its projective-free direct summand Q and pro-jective summand Q , such that Q = Q ⊕ Q , leads to the following from the factthat projectives form an ideal in the Green ring of H . Proposition 3.6.
The depth of the Hopf subalgebra, d h ( R, H ) < ∞ if and only ifthe module depth d ( Q , M H ) < ∞ .Proof. For the statement and proof of this proposition, we apply the extendeddefinition of module depth of any finitely generated module X ∈ M H in terms ofthe depth n condition, T n ( X ) ∼ T n +1 ( X ) where T n ( X ) = X ⊕ · · · ⊕ X ⊗ ( n ) [31].Since T n ( X ) | T n +1 ( X ), any projective module Y has finite depth, as there are afinite number of isoclasses of projective indecomposables. But Y ⊗ M is projectiveas well for any M ∈ M H . Then Q ⊗ ( n ) = Q ⊗ ( n )0 ⊕ Q ⊗ ( n )1 ⊕ mixed terms of Q , Q ,which are all projective. Thus d h ( R, H ) < ∞ ⇔ Q ⊗ ( n ) ∼ Q ⊗ ( n +1) as H -modules forsome n ∈ N , which implies that the summand Q has finite depth by [31, Lemma4.4]. Conversely, if T n ( Q ) ∼ T n +1 ( Q ) as H -modules, from T i ( Q ) | T i +1 ( Q ), weobtain that T n + m ( Q ) ∼ T n + m +1 ( Q ), equivalently Q ⊗ ( n + m ) ∼ Q ⊗ ( n + m +1) , where m is the number of distinct isoclasses of projective indecomposables. (cid:3) Semisimple and separable extensions.
Recall that any ring extension A ⊇ B is said to be a right semisimple extension if any right A -module N isrelative projective, i.e., N | N ⊗ B A as A -modules. More strongly, a ring extension A ⊇ B is said to be a separable extension if for any right A -module M , the mul-tiplication epimorphism µ M : M ⊗ B A → M splits [19], which also generalizes thestraightforward notion of left semisimple extension. The following theorem is a rel-ative Maschke theorem characterizing semisimple extensions of finite-dimensionalHopf algebras R ⊆ H . We freely use the notation Q = H/R + H and ground field k developed above. Theorem 3.7.
The Hopf subalgebra pair R ⊆ H is a right (or left) semisimpleextension ⇔ k H | Q H ⇔ k H is R -relative projective ⇔ there is q ∈ Q such that ε Q ( q ) = 0 and qh = qε ( h ) for every h ∈ H ⇔ ∃ s ∈ H : sH + ⊆ R + H and ε ( s ) = 1 ⇔ H is a separable extension of R .Proof. The counit of Q , given by ε Q ( h ) = ε ( h ) for h ∈ H , is always R -split by1 H . If all modules are relative projective, it follows that ε Q H -splits, so k H isisomorphic to a direct summand of Q H . Conversely, if Q H ∼ = k H ⊕ Q ′ H , then any H -module N satisfies by [31, Lemma 3.1] N ⊗ R H ∼ = N . ⊗ Q . ∼ = N ⊕ ( N . ⊗ Q ′ . ) IN MEMORY OF DANIEL KASTLER since N . ⊗ k . ∼ = N H . Thus, N and all H -modules are relative projective.If ε Q : Q → k is split by an H -module mapping k H → Q H , where 1 q underthis mapping, then q satisfies the integral-like condition of the theorem as well as ε Q ( q ) = 1. Moreover, q = s = 0, satisfies ε ( s ) = 1 and sh − sε ( h ) ∈ R + H for all h ∈ H , but all elements of H + are of the form h − ε ( h )1 H .If an element s ∈ H exists satisfying the conditions of the theorem, for any H -module M , the epi µ M : M ⊗ R H → M is split by m mS ( s (1) ) ⊗ R s . This is alsoseen from a commutative triangle using M ⊗ R H ∼ = −→ M . ⊗ Q . and the mappings in[31, Lemma 3.1]. Note that S ( s (1) ) ⊗ R s is a separability element, for given any h ∈ H , sh = ε ( h ) s − P i x i h i for some x i ∈ R + , h i ∈ H . Applying π ( S ⊗ id)∆(where π : H ⊗ H → H ⊗ R H is the canonical epimorphism) to this equation: S ( h (1) ) S ( s (1) ) ⊗ R s (2) h (2) = ε ( h ) S ( s (1) ) ⊗ R s (2) − X i S ( h i (1) ) S ( x i (1) ) ⊗ R x i (2) h i (2) = ε ( h ) S ( s (1) ) ⊗ R s (2) . Then hS ( s (1) ) ⊗ R s = S ( s (1) ) ⊗ R s (2) h for all h ∈ H follows from a standardapplication of h (1) S ( h (2) ) ⊗ h (3) = 1 ⊗ h . (cid:3) Note that if R = k H , the theorem recovers the extended Maschke’s theorem forHopf algebras (e.g., [39, Ch. 2]), since R + = { } , Q = H and q or s are integralelements of H with nonzero counit. For example, if Q ⊗ ( n ) is projective as an H - or R -module for any n ∈ N , it follows from this theorem that R is semisimple, since k R | Q | · · · | Q ⊗ ( n ) .Let t R , t H denote nonzero right integrals in R, H , respectively, for the proof ofthe corollary below.
Corollary 3.8.
Suppose H ⊇ R is a semisimple extension of finite-dimensionalHopf algebras. Then (1) the modular functions of H and R satisfy m H | R = m R ; (2) the Nakayama automorphisms of H and R satisfy η H | R = η R ; (3) the extension H ⊇ R is an ordinary Frobenius extension.Proof. Suppose s ∈ H satisfies the conditions of the theorem, ε ( s ) = 1 and sH + ⊆ R + H . By [31, Lemma 3.2]. the quotient module Q ∼ −→ t R H, which sends q = s t R s . Then t R sH + ∈ t R R + H = { } , i.e., t R s is a nonzerointegral in H . Without loss of generality, set t H = t R s . Then for all r ∈ R , m H ( r ) t H = rt H = rt R s = m R ( r ) t H , from which it follows that m H restricts on R to the modular function of R , m R .Recall that finite-dimensional Hopf subalgebra pairs such as H ⊇ R are β -Frobenius extensions (Fischman-Montgomery-Schneider) with β ( r ) = r ↼ m H ∗ m − R = η R ( η − H ( r )) . See [24] or [45] for textbook coverages of the full details. Consequently, η H ( r ) = η R ( r ), m H ( r ) = m R ( r ) and β ( r ) = r for all r ∈ R . (cid:3) The hypothesis of semisimplicity that removes the twist in the Frobenius exten-sion of Hopf algebra substantially uncomplicates the associated induction theory.
LGEBRA DEPTH IN TENSOR CATEGORIES 11
Depth of Hopf subalgebras from right or left quotient modules.
Let R ⊆ H be a Hopf subalgebra pair where H is finite-dimensional, and R + = ker ε ∩ R .The right quotient H -module Q := H/R + H controls induction of right H -modulerestricted to R -modules as follows: ∀ M ∈ M H ,M ⊗ R H ∼ = −→ M . ⊗ Q . , m ⊗ R h mh (1) ⊗ h (2) (10)with inverse mapping given by m ⊗ h mS ( h (1) ) ⊗ R h (2) where S : H → H denotesthe antipode of H . At the same time, the k -dual of the left quotient H -module Q := H/HR + controls the coinduction of right H -modules restricted to R -modulesin a somewhat similar way: ∀ M ∈ M H ,M . ⊗ Q ∗ . ∼ = −→ Hom ( H R , M R ) , m ⊗ q ∗ ( h mh (1) q ∗ ( h (2) )) (11)Both Eqs. (10) and (11) are first recorded in [47, Ulbrich]; we use the notation forcosets h for both coset spaces Q and Q .The following is then a consequence of Eqs. (10) and (11). As mentioned above, H ⊇ R is always a twisted (“beta”) Frobenius extension, with a twist automor-phism β : R → R given by a relative modular function or a relative Nakayamaautomorphism. If the twist is trivially the identity on R , the Hopf subalgebra isan ordinary Frobenius extension: see subsection 5.1 of this paper for the definition.This hypothesis on H ⊇ R allows us to prove the following. Proposition 3.9. If H ⊇ R is a Frobenius extension, then Q ∗ ∼ = Q as right H -modules.Proof. This follows from the characterization of Frobenius extension: for each right R -module N , N ⊗ R H ∼ = Hom ( H R , N R ) . (12)Now apply this and the display equations above to N = M = k ε . (cid:3) Recall that H and R are Frobenius algebras: let A be any Frobenius algebra.Then there are one-to-one correspondences of right ideals with left ideals of A viathe correspondence I ℓ ( I ) := { a ∈ A : aI = 0 } for every right ideal I of A ,and inverse correspondence J r ( J ) := { a ∈ A : Ja = 0 } for every left ideal J of A . The following comes from the basic fact that ℓ ( I ) ∼ = Hom (( A/I ) A , A A ) and r ( J ) ∼ = Hom ( A ( A/J ) , A A ). See [34, Lam II]. Proposition 3.10.
Let t R denote a nonzero right integral in R , a Hopf subalgebraof H as above. Then ℓ ( R + H ) = Ht R , Hom ( H ( H/Ht R ) , H H ) ∼ = R + H and Hom ( Q H , H H ) ∼ = Ht R . If H is a symmetric algebra, the k -duals Q ∗ ∼ = Ht R and Q ∗ ∼ = t R H .Proof. Note that Ht R R + H = 0. From [31, 3.2] Q ∼ = t R H and dim Q = dim H/ dim R .By definition of Q , dim Q = dim H − dim R + H ; similarlydim Ht R = dim Q = dim H/ dim R. For a Frobenius algebra A , we know that dim ℓ ( I ) = dim A − dim I [34]. Setting A = H , it follows from dimensionality that Ht R = ℓ ( R + H ). The next two isomorphismsare applications of r ( ℓ ( I ) = I and ℓ ( r ( J ) = J . The last statement follows fromHom ( M A , A A ) ∼ = M ∗ IN MEMORY OF DANIEL KASTLER as left A -modules, for every A -module M , for a symmetric algebra A (and a similarstatement for left A -modules, see [34]). (cid:3) The equivalent problems in Section 1 have a third equivalent formulation basedon elementary considerations using Eq. (1):
Problem 3.11.
Is there an n ∈ N such that the composition Hom ( Q ⊗ ( n ) , Q ⊗ ( n +1) ) ⊗ End Q ⊗ ( n ) Hom ( Q ⊗ ( n +1) , Q ⊗ ( n ) ) −→ End Q ⊗ ( n +1) is surjective? Either R -modules or H -modules suffice above. If we assume that H ⊇ R is anordinary Frobenius extension however, the following interesting isomorphisms ofHom-groups over H exist. Note that for any H -module M , there is a subring pairEnd M H ⊆ End M R . Proposition 3.12.
There are
End Q ⊗ ( n ) H := E -module isomorphisms, Hom ( Q ⊗ ( n ) H , Q ⊗ ( n +1) H ) ∼ = End Q ⊗ ( n ) R ∼ = Hom ( Q ⊗ ( n +1) H , Q ⊗ ( n ) H ) (right and left E -modules respectively).Proof. The second isomorphism follows from Eq. (10) and the hom-tensor adjointisomorphism [1, 20.6]. The first isomorphism requires additionally the fact for anyFrobenius extension H ⊇ R with modules M H and N R :Hom ( M H , N ⊗ R H H ) ∼ = Hom ( M R , N R ) (13)which follows from a natural isomorphism Hom ( H R , N R ) ∼ = N ⊗ R H as right H -modules, and the hom-tensor adjoint isomorphism. (cid:3) It is worth remarking that the tensor powers of Q are also H -module coalgebraquotients, since they are pullbacks via ∆ n : H → H ⊗ ( n ) of the quotient module ofthe Hopf subalgebra pair R ⊗ ( n ) ⊆ H ⊗ ( n ) , which is isomorphic to Q ⊗ ( n ) as H ⊗ ( n ) -modules.3.4. Core Hopf ideals of a Hopf subalgebra pair.
Let R ⊆ H be a finite-dimensional Hopf subalgebra pair. We continue the study begun in [15] relatingthe depth of a quotient module Q to its descending chain of annihilator ideals ofits tensor powers:Ann Q ⊇ Ann ( Q ⊗ Q ) ⊇ · · · ⊇ Ann Q ⊗ ( n ) ⊇ · · · . (14)The chain of ideals are either contained in R + or H + depending on whether Q is considered an R -module or H -module (as in Corollary 3.4). By classical theoryrecapitulated in [15, Section 4], for some n ∈ N we have Ann Q ⊗ ( n ) = Ann Q ⊗ ( n + m ) for all integers m ≥
1: this ideal I is a Hopf ideal, indeed the maximal Hopfideal contained in Ann Q . Let ℓ Q denote the least n for which this stabilizationof the descending chain of annihilator ideals takes place; call ℓ Q the length of theannihilator chain of tensor powers of the quotient module. This may be nuancedby ℓ Q R or ℓ Q H depending on which module Q is being considered: since for anymodule M H we have Ann M R = Ann M H ∩ R , it follows that ℓ Q R ≤ ℓ Q H . (15) LGEBRA DEPTH IN TENSOR CATEGORIES 13
Let S , . . . , S t be the simple composition factors of Q or one of its tensor powers;by elementary considerations with the composition series of Q ⊗ i , we note that I ⊆ ∩ tj =1 Ann S j , (16)in particular, if some Q ⊗ i contains all simples (of R or H ), I ⊆ J ω , the (Chen-Hiss[8]) Hopf radical ideal, since J ω is the maximal Hopf ideal in the radical which isthe intersection of the annihilator ideals of all simples. If one simple is projective,the corresponding J ω = 0 by a result in [8], whence Q is conditionally faithful, i.e., Q ⊗ ( n ) is faithful for some n ∈ N [15].Recall that the core of a subgroup U ≤ G is N := ∩ g ∈ G gU g − , and is themaximal normal subgroup of G contained in U . Proposition 3.13.
Suppose H is a group algebra kG and R is a group algebra kU ,where U ≤ G is a subgroup pair. Then I is determined by the core N as follows: I H = kN + H and I R = kN + R .Proof. Note that kN + H = HkN + is a Hopf ideal since N is normal in G . Anarbitrary element in Q is the coset U g annihilated by 1 − n for any n ∈ N , since N ⊆ U . Then KN + H ⊆ I , since I is maximal Hopf ideal in the annihilator of Q .Conversely, the Hopf ideal I = k ˜ N + H for some normal subgroup ˜ N ⊳ G by a resultin [43]. Since 1 − ˜ n annihilates each U g , it follows that ˜ N ⊆ U , whence ˜ N = N bymaximality. (cid:3) Due to the proposition, we propose calling the pair of Hopf ideals I = Ann Q ⊗ ℓ QH and I ∩ R = Ann Q ⊗ ℓ QR the core Hopf ideals of the Hopf subalgebra R ⊆ H .Note that [15, Prop. 4.3] is equivalent to the inequality2 ℓ Q R + 1 < d even ( R, H ) , (17)true without further conditions on H and R , since the even depth of Q , determinedfrom similarity of tensor powers of Q as R -modules, results in equal annihilatorideals: see the first statement in Proposition 1.4. Similarly, considering the H -module Q and h-depth instead, we note that2 ℓ Q H + 1 ≤ d h ( R, H ) (18)Now we make use of the second statement in Proposition 1.4:
Theorem 3.14.
Suppose R is a semisimple Hopf algebra, then d even ( R, H ) = 2 ℓ Q R + 2 . If moreover H is semisimple, then d h ( R, H ) = 2 ℓ Q H + 1 .Proof. Semisimple rings satisfy the equal-annihilator-similar-module condition inProposition 1.4. The definition 2.3 of depth of Q depends on similarity of tensorpowers of Q and involves a rescaling of 1 plus a factor of 2 with respect to ℓ Q .The rest follows from the inequalities (17) and (18); see also [31, Theorem 5.1] for d even ( R, H ) = d ( Q, M R ) + 1. (cid:3) For a semisimple Hopf subalgebra pair, also note the equalities that follow fromDef. 2.3 and Prop. 1.4: d ( Q, M H ) = 2 ℓ Q H + 1 (19) d ( Q, M R ) = 2 ℓ Q R + 1 . (20) IN MEMORY OF DANIEL KASTLER
For semisimple Hopf algebra-subalgebra pairs, these formulas put the length ℓ Q of the annihilator chain of tensor powers of Q in close relation to diameter of samecolored points in the bicolored graph [7] as well as the base size or minimal numberof “conjugates” of the Hopf subalgebra intersecting in the core, cf. [14, 7].A general finite-dimensional Hopf subalgebra pair R ⊆ H may sometimes reduceto the hypothesis of the previous theorem via the following proposition, whichextends [16, Corollary 4.13] from the core of a subgroup-group algebra pair. Proposition 3.15.
Suppose I denotes the maximal Hopf ideal in the annihila-tor ideal of Q = H/R + H ; let J = R ∩ I denote the restricted Hopf ideal in R .Then h-depth d h ( R, H ) = d h ( R/J, H/I ) . Similarly, minimum even depth satisfies d even ( R, H ) = d even ( R/J, H/I ) .Proof. Note that d h ( R, H ) = d ( Q, M H ) by Corollary 3.4, and d ( Q, M H ) = d ( Q, M H/I )by [16, Lemma 1.5]. Note that
R/J ֒ → H/I is a Hopf subalgebra pair with quotientmodule isomorphic to Q by a Noether isomorphism theorem. Then d h ( R/J, H/I ) = d ( Q, M H/I ). (cid:3) Quotient module for the permutation group series.
It is interesting atthis point to compute the quotient module Q for the inclusion C S n ⊆ C S n +1 ofpermutation group algebras. Notice that the proposition below implies that thecharacter χ Q = χ + χ t , where χ is the principal character and χ t is the characterof the standard irreducible representation ( n, Proposition 3.16.
The quotient module Q = C [ S n /S n +1 ] is isomorphic to thestandard representation of S n +1 on C n +1 .Proof. Recall the Artin presentation of S n +1 with generators σ i = ( i i + 1) for i = 1 , . . . , n and relations σ i σ i +1 σ i = σ i +1 σ i σ i +1 , σ i σ j = σ j σ i , σ i = 1for all | i − j | ≥
2. Note that σ , . . . , σ n − ∈ S n . An ordered basis for Q is given by h S n σ n σ n − · · · σ , S n σ n · · · σ , . . . , S n σ n , S n i This ordered basis maps onto the ordered basis h e , . . . , e n +1 i of the S n +1 represen-tation space C n +1 via the canonical order-preserving mapping. This mapping isan S n +1 -module isomorphism, since σ i exchanges e i and e i +1 as it does S n σ n · · · σ i and S n σ n · · · σ i +1 , respectively, (here we use σ i = 1), and it leaves the other ba-sis elements fixed, since σ i commutes with σ i +2 and/or σ i − (here we also use σ i σ i − σ i = σ i − σ i σ i − ) etc. while σ i ∈ S n for i < n . In more detail, note that( S n σ n · · · σ i σ i − ) σ i = S n σ n · · · σ i +1 σ i − σ i σ i − = S n σ n · · · σ i − The rest of the proof is routine. (A second proof follows from Q ∼ = U ( ) ∼ = Ind S n +1 S n and Young diagram branching rule of adding a box.) (cid:3) Since S n ⊆ S n +1 is corefree, i.e., the core of the subgroup is trivial, it followsthat the character χ Q is faithful (equivalently, the annihilator idea of Q does notcontain a nonzero Hopf ideal ⇔ the representation of C G restricted to G has trivialkernel) [31, 4.2]. The Burnside-Brauer Theorem [22, p. 49] implies for the character χ Q that each irreducible character of S n +1 is a constituent of its powers up to χ nQ ,since dim Q = n + 1. This implies that d ( Q, M S n +1 ) ≤ n by reasoning along the LGEBRA DEPTH IN TENSOR CATEGORIES 15 lines of Example 1.3. Indeed d ( Q, M S n +1 ) = n follows from Corollary 3.4 and thegraphical computation d h ( S n , S n +1 ) = 2 n + 1 in [31].We mention the theorem in [37], that hooks generate the Green ring of a per-mutation group, as the full picture to the discussion above. Theorem 3.17. [37, Marin]
The representation ring A ( C S n +1 ) is generated by therepresentations Λ k C n +1 for ≤ k ≤ n . Remark 3.18.
Recall the notion of order of a module V H over a semisimple Hopfalgebra H . The order ord ( V ) is the least natural number n such that V ⊗ ( n ) hasnonzero invariant subspace, i.e., dim( V ⊗ ( n ) ) H = 0. For example, ord ( Q S n +1 ) = 1since χ Q = χ + χ t . For general semisimple Hopf subalgebra pairs H ⊇ R withquotient Q , one might conjecture that ord ( Q ) ≤ ℓ Q , since order of Q and ℓ Q are bothbounded above by the degree d of the minimal polynomial of χ Q in the characterring of H (or H/J -modules where J = Ann Q ⊗ ℓ Q see [32, chs. 4,5, p. 37] and [7,2.3], respectively). However, [32, p. 32] computes the order of a certain inducedmodule V over the semidirect product group algebra H = C [ Z ] C [ Z ] to be ord ( V ) = 3: with R = C [ Z q ], in fact V ∼ = Q H . We deduce that d ( Q, M H ) = 3,since d h ( R, H ) = 1 forces R = H by [31, Cor. 3.3]), and ℓ Q H = 1, since R is anormal Hopf subalgebra in H : so in general ord ( Q ) ℓ Q .4. Factorisable algebras An algebra factorisation of a unital (associative) algebra C into two unital sub-algebras A and B occurs when the multiplication mapping B ⊗ A ∼ −→ C is a B - A -bimodule isomorphism [5, 9]. Conversely, the algebra C may be constructedfrom B and A as a twisted tensor product (denoted by B ⊗ R A ) as follows: linearly C = B ⊗ A with multiplication given by the structure mapping R : A ⊗ B → B ⊗ A ,values denoted by R ( a ⊗ b ) = b r ⊗ a r or b R ⊗ a R , where summation over more thanone simple tensor is suppressed. In this case, the multiplication in B ⊗ A is givenby ( b ⊗ a )( b ⊗ a ) = b b r ⊗ a r a (21)In order for C to be associative, R must satisfy two pentagonal commutative dia-grams, equationally given by R ( µ A ⊗ B ) = ( B ⊗ µ A )( R ⊗ A )( A ⊗ R ) (22)(where µ A denotes multiplication in A ), and R ( A ⊗ µ B ) = ( µ B ⊗ A )( B ⊗ R )( R ⊗ B ) (23)in Hom ( A ⊗ B ⊗ B, B ⊗ A ). These equations are satisfied if and only if C isassociative. Additionally, the structure map R satisfies two commutative trianglesgiven equationally by R ( A ⊗ B ) = 1 B ⊗ A and R (1 A ⊗ B ) = B ⊗ A . It followsthat A → C, a B ⊗ a and B → C, b b ⊗ A are algebra monomorphisms. Example 4.1.
Let B be an algebra in H M , where A = H is a Hopf algebra asbefore. Let R : B ⊗ H → H ⊗ B be given by R ( b ⊗ h ) = h (1) .b ⊗ h (2) . Then B ⊗ R H = B H , the smash product of H with a left H -module algebra B . Proposition 4.2. [15, Theorem 5.2]
The minimum odd depth of H embeddedcanonically in the smash product B H satisfies d odd ( H, B H ) = d ( B, H M ) (24) IN MEMORY OF DANIEL KASTLER
Proof.
Via cancellations of the type X ⊗ H H ∼ = X , one establishes an H - H -bimoduleisomorphism, ( B H ) ⊗ H n ∼ = B ⊗ ( n ) ⊗ H, (25)where the left H -module structure on B ⊗ ( n ) ⊗ H is given by the diagonal action: x. ( b ⊗ · · · ⊗ b n ⊗ h ) = x (1) .b ⊗ · · · ⊗ x ( n ) .b n ⊗ x ( n +1) .h If B ⊗ ( n +1) | q · B ⊗ ( n ) in H M for some q ∈ N , then tensoring this by − ⊗ H yields( B H ) ⊗ H ( n +1) | q · ( B H ) ⊗ H n as H - H -bimodules. Thus the minimum odd depth d odd ( H, B H ) ≤ d ( B, H M ) by Definition 1.6.Conversely, if ( B H ) ⊗ H ( n +1) | q · ( B H ) ⊗ H n as H - H -bimodules, then B ⊗ ( n +1) ⊗ H | q · B ⊗ ( n ) ⊗ H , to which one applies − ⊗ H k , obtaining B ⊗ ( n +1) | q · B ⊗ ( n ) in H M . Therefore d ( B, H M ) ≤ d odd ( H, B H ). (cid:3) Using the notation developed in Section 3 for a finite-dimensional Hopf subal-gebra pair R ⊆ H with quotient right H -module coalgebra Q , we note that its k -dual Q ∗ becomes a left H -module algebra via h hq ∗ , q i = h q ∗ , qh i . Yet anotherequivalent formulation of the fundamental problem in Section 1 follows easily fromthe proposition since d ( Q ∗ , H M ) = d ( Q, M H ) [31, 15]. Problem 4.3. Is d ( H, Q ∗ H ) < ∞ or d ( R, Q ∗ R ) < ∞ ? Example 4.4.
Suppose B and H are a matched pair of Hopf algebras (see [36,7.2.1] or [33, IX.2.2]). I.e., H is a coalgebra in M B with action denoted by h ⊳ b ,and B is coalgebra in H M with action denoted by h ⊲ b satisfying compatibilityconditions given in [36, (7.7)-(7.9)]. A twisting R : H ⊗ B → B ⊗ H is given by R ( h ⊗ b ) = h (1) ⊲ b (1) ⊗ h (2) ⊳ b (2) , (26)which defines an algebra structure on B ⊗ R H = B ⊲⊳ H ; moreover, this is aHopf algebra, called the double cross product, where H and B are canonically Hopfsubalgebras [36].For example, H and its dual Hopf algebra (with opposite multiplication) B = H op ∗ are a matched pair via ⊲ , the left coadjoint action of H on H ∗ , h ⊲ b = b (2) h ( Sb (1) ) b (3) , h i , (27)and ⊳ the analogous left coadjoint action of H ∗ on H . This defines the Drinfelddouble D ( H ) as a special case of double cross product, D ( H ) = H op ∗ ⊲⊳ H . Proposition 4.5.
Let B and H be a matched pair of finite-dimensional Hopf al-gebras with A = B ⊲⊳ H their double cross product. Then the minimum h-depthand even depth of the Hopf subalgebra B in A is given by the depth of H in thefinite tensor category M B (w.r.t. ⊳ in Example 4.4): d h ( B, A ) = d ( H, M B ) and d even ( B, A ) = d ( H, M B ) + 1 . Similarly, the Hopf subalgebra H has depth in A given by d h ( H, A ) = d ( B, H M ) (w.r.t. ⊲ ) and d even ( H, A ) = d ( B, H M ) + 1 .Proof. This follows from Cor. 3.4 if we show that the quotient module Q B ∼ =( H B , ⊳ ). Note that Q = B ⊲⊳ H/B + ( B ⊲⊳ H ) ∼ = H via b ⊲⊳ h ε B ( b ) h , and hb = (1 B ⊲⊳ h )( b ⊲⊳ H ) = h (1) ⊲ b (1) ⊲⊳ h (1) ⊳ b (2) = ε B ( h (1) ⊲ b (1) ) h (2) ⊳ b (2) = h ⊳ b , where we use axiom (3) for B , a left H -module coalgebra. (cid:3) LGEBRA DEPTH IN TENSOR CATEGORIES 17
For example, if B = H op ∗ and B ⊲⊳ H = D ( H ), suppose H is cocommutative.From the formula for coadjoint action, it is apparent that H B ∼ = (dim H ) · k , so d ( H, M B ) = 1 and d ( H ∗ , D ( H )) ≤
2. Indeed, it is known that D ( H ) ∼ = H ∗ H incase H is quasitriangular [36, Majid, 1991, 7.4]), but a smash product is a Hopf-Galois extension of its left H -module algebra (which has depth 2). Example 4.6.
A study of the 8-dimensional small quantum group H (see forexample [31, Example 4.9] for its Hopf algebra structure) and its quantum double D ( H ) indicates that minimum depth satisfies 3 ≤ d ( H , D ( H )) ≤
4. The methodis to compute D ( H ) in terms of generators and relations, compute the quotient Q asan 8-dimensional H -module, then decompose it into its indecomposable summands(twice each simple, and two 2-dimensional indecomposables), compute the tensorproducts between these indecomposables, noting that Q ∼ Q ⊗ Q as H -modules,and using Eq. (9). Since both algebras have infinite representation type, we cannototherwise predict a finite depth from known results [31, 17].Let ad H denote the adjoint action of H on itself, given by h.x = h (1) xS ( h (2) )for all h, x ∈ H . Corollary 4.7. [15, Cor. 5.4]
Let G be a finite group and D ( G ) its Drinfeld doubleas a complex group algebra. Then d ( C G, D ( G )) = d ( ad C G, C G M ) .Proof. From the remark about cocommutativity just above, the double D ( G ) = H ∗ H (with H = C G ) is a smash product to which Proposition 4.2 applies: thus d odd ( C G, D ( G )) = d ( H ∗ , C G M ). The smash product multiplication formula for g, h ∈ G , p g , p h ∈ H ∗ one-point projections, is given by( p x g )( p y h ) = p x p gyg − gh (28)which visibly demonstrates that H H ∗ ∼ = ad H ∗ ∼ = ad C G .It remains to show that d even ( C G, D ( G )) = 1 + d odd ( C G, D ( G )). Note that S ( p x ) = p x − , ∆ ( p x ) = X z,y ∈ G p z ⊗ p z − y ⊗ p y − x whence using Eq. (27) h ⊲ p x = X z,y ∈ G p z − y h p z − p y − x , h i = X z,y ∈ G h p z − , h ih p y − x , h i p z − y = p hxh − , the adjoint action of h on p x . Use Proposition 4.5 to conclude the proof. (cid:3) Morita equivalent ring extensions
In this section we continue a study of Morita equivalence of ring extensions in [38,21, 48], though with an emphasis on functors and categories. We will briefly providethe classical background theory, and prove that depth, relative cyclic homology aswell as the bipartite graphs of a semisimple complex subalgebra pair are all Moritainvariant properties of a ring or algebra extension. In addition, we note a naturalexample of Morita equivalence in towers of Frobenius extensions.Define two ring extensions A | B and R | S to be Morita equivalent if there are ad-ditive equivalences P : R M → A M and Q : S M → B M satisfying a commutative IN MEMORY OF DANIEL KASTLER rectangle (up to a natural isomorphism) with respect to the functors of restrictionfrom R -modules into S -modules, and from A -modules into B -modules. R M ∼P ✲ A M S M Res RS ❄ ∼Q ✲ B M Res AB ❄ (29)The requirement then is that there be a natural isomorphism Q Res RS ∼ → Res AB P .One shows in an exercise that this is an equivalence relation on ring extensions byusing operations on natural transformation by functors.From ordinary Morita theory we know that P ( R R ) = A P , a progenerator suchthat End A P ∼ = R , so that P is in fact an A - R -bimodule with P ( R X ) = P ⊗ R X for all R X . The dual of P is unequivocally P ∗ = Hom ( P R , R R ), an R - A -bimodule,since Hom ( A P, A A ) ∼ = P ∗ as R - A -bimodules by [39, Theorem 1.1]. Then P ∗ ⊗ A − : A M → R M is an inverse equivalence to P : one has bimodule isomorphisms P ∗ ⊗ A P ∼ = R R R and P ⊗ R P ∗ ∼ = A A A .Similarly there is an invertible Morita bimodule B Q S , a left and right progen-erator module, such that Q ( S Y ) = B Q ⊗ S Y . The condition that the rectangleabove commutes applied to R ∈ R M becomes B Q ⊗ S R ∼ = B P , also valid as B - R -bimodules due to naturality, noted as an equivalent condition in the propositionbelow. Example 5.1.
Given a ring extension R ⊇ S , let A = M n ( R ) ⊇ B = M n ( S ). Ofcourse, A and R are Morita equivalent via P = n · R , also B and S are Moritaequivalent via Q = n · S . Note that B Q ⊗ S R R ∼ = n · R = B P R . Thus, as one would expect, the ring extensions R ⊇ S and A ⊇ B are Moritaequivalent. Example 5.2.
Suppose B ⊆ A and S ⊆ R are ring extensions with ring isomor-phism ψ : A ∼ → R restricting to a ring isomorphism η : B ∼ → S . Defining bimodules A P R := ψ R R and B Q S := η S S , one shows in an exercise that the two ring extensionsare Morita equivalent.The proposition below characterizes Morita equivalence of ring extensions inmany equivalent ways, condition (2) being the definition in [38, 21, 48]. Proposition 5.3.
The following conditions on ring extensions A ⊇ B and R ⊇ S are equivalent: (1) A ⊇ B and R ⊇ S are Morita equivalent; (2) there are Morita bimodules A P R and B Q S satisfying B Q ⊗ S R R ∼ = B P R [38] ; (3) there are Morita bimodules A P R and B Q S satisfying R R ⊗ S Q ∗ B ∼ = R P ∗ B ; (4) there are Morita bimodules A P R and B Q S satisfying A A ⊗ B Q S ∼ = A P S ; (5) there are Morita bimodules A P R and B Q S satisfying S Q ∗ ⊗ B A A ∼ = S P ∗ A ; LGEBRA DEPTH IN TENSOR CATEGORIES 19 (6) the following rectangle, with sides representing the induction functors, com-mutes up to a natural isomorphism, R M ∼P ✲ A M S M Ind RS ✻ ∼Q ✲ B M Ind AB ✻ (30)(7) the following rectangle, with sides representing the coinduction functors,commutes up to a natural isomorphism, R M ∼P ✲ A M S M CoInd RS ✻ ∼Q ✲ B M CoInd AB ✻ (31)(8) any of the conditions above stated identically with right module categories M R , M A , M S , and M B replacing the corresponding left module cate-gories.Proof. (1) ⇒ (2) is sketched above. (2) ⇔ (3) follows from the computation R P ∗ B ∼ = R Hom ( P R , R R ) B ∼ = R Hom ( Q ⊗ S R R , R R ) B ∼ = R Hom ( Q S , R S ) B ∼ = R R ⊗ S Q ∗ B using adjoint theorems in [1, pp. 240, 243]. This shows (2) ⇒ (3). This argument reverses by using the reflexive property of progenerators( A Hom ( R P ∗ , R R ) R ∼ = A P R ).(3) ⇒ (4) and (8). The following rectangle is commutative up to a naturalisomorphism: M R ∼− ⊗ R P ∗ ✲ M A M S Res RS ❄ ∼− ⊗ S Q ∗ ✲ M B Res AB ❄ since for any module X R one has X ⊗ R P ∗ B ∼ = X ⊗ R R ⊗ S Q ∗ B ∼ = X ⊗ S Q ∗ B . To the natural isomorphism identifying the sides of this rectangle, apply the functor − ⊗ B Q from the left and the functor − ⊗ A P from the right to obtain the following IN MEMORY OF DANIEL KASTLER commutative rectangle up to natural isomorphism: M A ∼− ⊗ A P ✲ M R M B Res AB ❄ ∼− ⊗ B Q ✲ M S Res RS ❄ (4) now follows from applying the rectangle to A A . (4) ⇒ (5). The same type ofargument as in (2) ⇒ (3) above shows that S P ∗ A ∼ = S Hom ( A P, A A ) A ∼ = S Hom ( B Q, B B ) ⊗ B A A ∼ = S Q ∗ ⊗ B A A . (4) ⇒ (6). By using (4), compute for any module S Y , A A ⊗ B Q ⊗ S Y ∼ = A P ⊗ S Y ∼ = A P ⊗ R R ⊗ S Y, which shows the rectangle (6) is commutative up to a natural isomorphism. Theconverse (6) ⇒ (4) follows from applying the rectangle to S S ∈ S M as well asnaturality.(5) ⇒ (7) For any module S W , it suffices to show that P ⊗ R Hom ( S R, S W ) ∼ =Hom ( B A, B Q ⊗ S W ) using natural isomorphisms in [1, 20.6, 20.11, exercise 20.12]and (5): A P ⊗ R Hom ( S R, S W ) ∼ = A Hom ( S P ∗ , S W ) ∼ = A Hom ( S Q ∗ ⊗ B A, S W ) ∼ = A Hom ( B A, B Hom ( S Q ∗ , S W )) ∼ = A Hom ( B A, B Q ⊗ S W )The rest of the proof is similar and left as an exercise. (cid:3) In the following proposition, we note some different, quick proofs for certainresults in [21], while building up results which show that depth and bipartite graphsare Morita invariants of ring extensions.
Proposition 5.4.
Suppose A | B and R | S are Morita equivalent ring extensions.In the notation of the previous proposition, it follows that (1) if the extension A ⊇ B is a separable, then R ⊇ S is a separable extension [21] ; (2) if the extension A ⊇ B is QF, then R ⊇ S is a QF extension [21] ; (3) if the extension A ⊇ B is Frobenius, then R ⊇ S is a Frobenius extension [21] ; (4) if B ⊆ A is a semisimple complex subalgebra pair, then so is S ⊆ R withidentical inclusion matrix and bipartite graph; (5) the following diagram of tensor categories and functors commutes up tonatural isomorphism: R M R ∼ F ✲ A M AS M S Res R e S e ❄ ∼ G ✲ B M B Res A e B e ❄ (32) LGEBRA DEPTH IN TENSOR CATEGORIES 21 where F ( R X R ) := A P ⊗ R X ⊗ R P ∗ A and G ( S Y S ) := B Q ⊗ S Y ⊗ S Q ∗ B define tensor equivalences; (6) G ( R ⊗ S ( n ) ) ∼ = A ⊗ B ( n ) as B - B -bimodules and F ( R ⊗ S ( n ) ) ∼ = A ⊗ B ( n ) as A - A -bimodules for each n ∈ N ; (7) the centralizers are isomorphic: A B ∼ = R S [21] ; (8) the ring extensions A | B and R | S have the same minimum depth and h-depth.Proof. (1) Let 0 → V → W → U → A M thatis split exact when restricted to B M . By Rafael’s characterization [44] ofseparability, the short exact sequence splits in A M . The rest of the prooffollows from applying the commutative rectangle (29).(2) Suppose A V is ( A, B )-projective (or “relative projective”), i.e., A V | A A ⊗ B V (or the multiplication epi A ⊗ B V → V splits as an A -module map). Bythe relative Faith-Walker theorem for QF extensions [40], V is also ( A, B )-injective: i.e., the canonical A -module monomorphism V ֒ → Hom ( B A, B V )splits. In fact the class of relative projectives coincides with the class ofrelative injectives for QF extensions. It is clear from the commutative di-agram (30) that the equivalence P sends relative projectives into relativeprojective; similarly, it is clear from the commutative rectangle (31) thatrelative injectives are sent by an equivalence into relative injectives. Therest of the proof is then an application of the relative Faith-Walker charac-terization of QF extension.(3) The proof is an application of the commutative rectangles (30) and (31) andthe characterization of Frobenius extensions as having naturally isomorphicinduction and coinduction functors. Suppose R ⊇ S is Frobenius. ThenInd AB Q ∼ = P Ind RS ∼ = P CoInd RS ∼ = CoInd AB Q . Since Q is an equivalence, it follows that Ind AB and CoInd AB are naturallyisomorphic functors, whence A ⊇ B is Frobenius.(4) Let V , . . . , V s be the simples of S (up to isomorphism). Then U i := Q ⊗ S V i are representatives of the simple isoclasses of B by Morita theory. Induceeach V i to an R -module, expressing this uniquely up to isomorphism as asum of nonnegative multiples of the simples of R , W , . . . , W r : R ⊗ S V i ∼ = ⊕ rj =1 r ij W j . The s × r matrix is the inclusion matrix K ( S ) → K ( R ) of the semisimplecomplex subalgebra pair S ⊆ R . This matrix determines the bipartitegraph of the inclusion, an edge connecting black dot i with white dot j incase the ( i, j )-entry is nonzero.Since A and R Morita equivalent rings, both are semisimple complexalgebras; the same is true of B and S . Moreover, their centers are isomor-phic, thus A and R each have r distinct simples, and B , S each have s pairwise nonisomorphic simples. Denote the simples of A by X , . . . , X r where X i ∼ = P ⊗ R W i for each i . Suppose the inclusion matrix of B ⊆ A isgiven by A ⊗ B U i ∼ = ⊕ rj =1 b ij X j . Since A ⊗ B U i ∼ = A ⊗ B Q ⊗ S V i ∼ = P ⊗ R R ⊗ S V i ∼ = ⊕ rj =1 r ij X j this implies by Krull-Schmidt that the inclusion matrices ( b ij ) and ( r ij ) areequal. Thus the bipartite graphs are equal. IN MEMORY OF DANIEL KASTLER (5) The functors F and G are tensor equivalences according to Lemma 2.2.Let R X R be a bimodule. Note that Res A e B e ( F ( X )) = B P ⊗ R X ⊗ R P ∗ B ∼ = B Q ⊗ S R ⊗ R X ⊗ R R ⊗ S Q ∗ B ∼ = G (Res R e S e ( X )) by applying (2) and (3)in Proposition 5.3. Whence the rectangle is commutative.(6) From the commutative rectangle just established it follows that G ( S R S ) ∼ = B A B and from the tensor functor property of G that G ( R ⊗ S ( n ) ) ∼ = B A ⊗ B ( n ) B .A computation similar to the one in (4) of this proof shows that thefollowing rectangle is commutative: R M R ∼ F ✲ A M AS M S Ind R e S e ✻ ∼ G ✲ B M B Ind A e B e ✻ where Ind R e S e ( S Z S ) := R R ⊗ S Z ⊗ S R R . Since F preserves tensor categoryunit objects, F ( R R R ) ∼ = A A A . Starting with S S S ∈ S M S , the rectangleshows that F ( R R ⊗ S R R ) ∼ = A A ⊗ B A A . Starting with R ⊗ S ( n ) ∈ S M S inthe rectangle, we note that for n ≥ F ( R R ⊗ S ( n +2) R ) ∼ = Ind A e B e ( A ⊗ B ( n ) ) = A A ⊗ B ( n +2) A . (7) Note the equivalence of bimodule categories H : S M R → B M A given by H ( S W R ) := B Q ⊗ S W ⊗ R P ∗ A . We claim that H ( S R R ) ∼ = B A A ; moreover, H ( S R ⊗ S ( n ) R ) ∼ = B A ⊗ B ( n ) A (33)for all n ≥
1. This follows from the diagram below, commutative up tonatural isomorphism. R M R ∼ F ✲ A M AS M R Res RS ❄ ∼ H ✲ B M A Res AB ❄ (34)which is established by a short computation using (2) in Prop. 5.3. Appliedto R ⊗ S ( n ) ∈ R M R , we obtain Eq. (33).Note that the centralizer R S = { r ∈ R : ∀ s ∈ S, rs = sr } is isomorphicto End ( S R R ) ∼ = R S via f f (1). Recall that an equivalence H satisfiesEnd ( S R R ) ∼ = End ( H ( S R R )) ∼ = End ( B A A ) ∼ = A B . (8) Similarly to Eq. (33), we establish that the equivalence of bimodule cate-gories given by H ′ : R M S → A M B , R V S P ⊗ R V ⊗ S Q ∗ satisfies H ′ ( R ⊗ S ( n ) ) ∼ = A A ⊗ B ( n ) B (35)Of course, equivalences preserve similarity of modules since they are addi-tive. Suppose R ⊗ S ( n ) ∼ R ⊗ S ( n +1) as R - S -bimodules, i.e., R | S has rightdepth 2 n . Applying H ′ , one obtains A ⊗ B ( n ) ∼ A ⊗ B ( n +1) as A - B -bimodules, LGEBRA DEPTH IN TENSOR CATEGORIES 23 i.e., A | B has right depth 2 n . Similarly for left depth 2 n using the equiva-lence H . Similarly, if R | S has depth 2 n +1, applying G we obtain that A | B has depth 2 n + 1. Going in the reverse direction using G − , H − , we obtain d ( S, R ) = d ( B, A ). Using F we likewise show that d h ( S, R ) = d h ( B, A ). (cid:3) Example: tower above Frobenius extension.
A Frobenius extension A ⊇ B is characterized by any of the following four conditions [24]. First, that A B isfinite projective and B A A ∼ = Hom ( A B , B B ). Secondly, that B A is finite projectiveand A A B ∼ = Hom ( B A, B B ). Thirdly, that coinduction and induction of right (orleft) B -modules into A -modules are naturally isomorphic functors. Fourth, thereis a Frobenius coordinate system ( E : A → B ; x , . . . , x m , y , . . . , y m ∈ A ), whichsatisfies ( ∀ a ∈ A ) E ∈ Hom ( B A B , B B B ) , m X i =1 E ( ax i ) y i = a = m X i =1 x i E ( y i a ) . (36)These equations may be used to show that P i x i ⊗ y i ∈ ( A ⊗ B A ) A .By [30, Lemma 4.1], a Frobenius extension A ⊇ B has both A B and B A generatormodules if and only if the Frobenius homomorphism E : A → B is surjective:although most Frobenius extensions in the literature are generator extensions, thereis a somewhat pathological example in [24, 2.7] of a matrix algebra Frobeniusextension with a non-surjective Frobenius homomorphism.A Frobenius extension A ⊇ B enjoys an endomorphism ring theorem, whichstates that A := End A B ⊇ A is itself a Frobenius extension, where the ringmonomorphism A → A is the left multiplication mapping λ : a λ a , λ a ( x ) = ax . It is worth noting that λ is a left split A -monomorphism (by evaluation at1 A ) so A A is a generator. It is an exercise to check that A ∼ = A ⊗ B A via f P i f ( x i ) ⊗ B y i ; the induced ring structure on A ⊗ B A is the “E-multiplication,”given by ( a ⊗ B c )( d ⊗ B e ) = aE ( cd ) ⊗ B e. (37)The identity is given 1 = P i x i ⊗ B y i . The Frobenius coordinate system for A ⊇ A is given by E ( a ⊗ B c ) = ac (always surjective!) with dual bases { x i ⊗ B } and { ⊗ B y i } .The tower of a Frobenius extension is obtained by iteration of the endomorphismring and λ , obtaining a tower of Frobenius extensions; with the notation B := A , A := A and defining A n +1 = End A nA n − , we obtain the tower, A ֒ → A ֒ → A ֒ → · · · ֒ → A n ֒ → A n +1 ֒ → · · · (38)By transitivity of Frobenius extension or QF extension [42], all sub-extensions A m ֒ → A m + n in the tower are also Frobenius extensions. Note that A n ∼ = A ⊗ B ( n ) :the ring, module and Frobenius structures in the tower are worked out in [30]. Theorem 5.5.
Suppose A ⊇ B is a Frobenius extension with the tower and datanotation given above. Then A n − ⊇ A n − is Morita equivalent to A n +1 ⊇ A n forall integers n > . Also A ⊇ B is Morita equivalent to A ⊇ A if the Frobeniushomomorphism is epi.Proof. It suffices to assume E : A → B is surjective, let S = A = End A B , R = A ,and show that B ֒ → A is Morita equivalent to A ֒ → A . Since A is a Frobeniusextension of B with surjective Frobenius homorphism, it follows that the module IN MEMORY OF DANIEL KASTLER A B is a progenerator; since A = End A B , it follows that B and A are Moritaequivalent rings. Similarly, A and A ∼ = End A ⊗ B A A are Morita equivalent rings.In the notation of Proposition 5.3 (exchanging R with A , B with S ), note that Q = A and P = A ⊗ B A . Thus S Q ⊗ B A A ∼ = S P A , the condition in the propositionfor Morita equivalent ring extensions. (cid:3) The theorem states in other words that the tower above a Frobenius extensionhas up to Morita equivalence period two. Note that consecutive ring extensions inthe tower are almost never Morita equivalent: in [30, Example 1.12], the depth is d ( S , S ) = 5, but of its reflected graph, the depth is d ( A, A ) = 6 (where A = C S ,using the graph-theoretic depth calculation in [7, Section 3]).5.2. Relative cyclic homology of ring extensions is Morita invariant.
Weextend a result in [23] that relative cyclic homology of a ring extension R ⊇ S andof its n × n -matrix ring extension M n ( R ) ⊇ M n ( S ) are isomorphic via a Dennistrace map adapted to this set-up. The relative cyclic homology (or any of its severalvariant homologies) is computed from cyclic modules Z n ( R, S ) := R ⊗ S e R ⊗ S ( n ) , which has the effect of considering tensor products of the natural bimodule S R S with itself over S n + 1 times arranged in a circle (in place of a line). For each n ≥
0, there are n + 1 face maps are given by d i : Z n ( R, S ) → Z n − ( R, S ) definedfrom tensoring n − S R S with one copy of the multiplication µ ∈ Hom ( S R ⊗ S R S , S R S ) at the i th position, there are n + 1 degeneracy mappings s j : Z n ( R, S ) → Z n +1 ( R, S ) by tensoring n copies of id S R S with one copy of theunit mapping η ∈ Hom ( S S S , S R S ) in the i th position, and a cyclic permutation t n : Z n ( R, S ) → Z n ( R, S ) of order n + 1 (see [23] for the Connes cyclic objectrelations [10] and the textbook [35] for further details).Suppose ring extensions R ⊇ S and A ⊇ B are Morita equivalent, and assume thesame structural bimodules and module equivalences with notation as in this section.Now recall from the diagram (32) that the tensor equivalence G : S M S → B M B ,defined by G ( X ) = Q ⊗ S X ⊗ S Q ∗ , sends S R S into B A B . We note the followingcommutative diagram, S M S × S M S ∼ G × G ✲ B M B × B M B Ab S − ⊗ S e − ❄ ∼ ˆ G ✲ Ab B − ⊗ B e − ❄ (39)where Ab B denotes B M B ⊗ B e B M B , a subcategory of abelian groups (and similarlyfor Ab S ), from a computation with X, Y ∈ S M S : G ( X ) ⊗ B e G ( Y ) ∼ = X ⊗ S Q ∗ ⊗ B Q ⊗ S e Q ∗ ⊗ B Q ⊗ S Y ∼ = X ⊗ S S ⊗ S e S ⊗ S Y ∼ = X ⊗ S e Y. It follows that Z n ( R, S ) ∼ = −→ Z n ( A, B ) via ˆ G (restricted to the cyclic modules) asabelian groups for each n ≥
0. Now ˆ G commutes with face maps since the functor LGEBRA DEPTH IN TENSOR CATEGORIES 25 G sends the multiplication of R ⊇ S , µ ∈ Hom ( S R ⊗ S R S , S R S ) µ ∈ Hom ( B A ⊗ B A B , B A B ) , the multiplication of the ring extension A ⊇ B . That ˆ G : Z n ( R, S ) → Z n ( A, B )commutes with the degeneracy maps follows from the functor G sending the unit η ∈ Hom ( S S S , S R S ) into the unit η ∈ Hom ( B B B , B A B ). That ˆ G : Z n ( R, S ) → Z n ( A, B ) commutes with the cyclic group action generator t n follows from G × G commuting with simple exchange X × Y Y × X . We have sketched the proof ofthe next proposition. Proposition 5.6. If R ⊇ S and A ⊇ B are Morita equivalent ring extensions,then their cyclic modules, cyclic chain complexes and cyclic homology groups areisomorphic: HC n ( R, S ) ∼ = HC n ( A, B ) , all n ∈ N . The isomorphism is given by a generalized Dennis trace mapping as follows.Suppose the S -bimodule isomorphism Q ∗ ⊗ B Q ∼ = −→ S sends P ri =1 q ∗ i ⊗ q i S .Then an isomorphism of cyclic modules Z n ( A, B ) ∼ = −→ Z n ( R, S ) is given by a ⊗ · · · ⊗ a n r X i ,...,i n =1 q i ⊗ a ⊗ q ∗ i ⊗ q i ⊗ · · · ⊗ q i n ⊗ a n ⊗ q ∗ i (40)In the matrix example 5.1 of Morita equivalent ring extensions, where each a i denotes an n × n -matrix, this expression simplifies to the classical Dennis traceisomorphism of cyclic modules noted in [23], a ⊗ B e a ⊗ B · · · ⊗ B a n r X i ,...,i n =1 a i i ⊗ S e a i i ⊗ S · · · ⊗ S a i n i n . Acknowledgements.
The author thanks the organizers of the “New Trendsin Hopf algebras and Tensor Categories” in Brussels, June 2-5, 2015, for a nice con-ference including Joost Vercruysse and Mio Iovanov for discussions about Propo-sition 5.4, item (3), Alberto Hernandez for interesting mathematical conversationsabout several subjects in this paper, and CMUP (UID/MAT/00144/2013), whichis funded by FCT (Portugal) with national (MEC) and European structural fundsthrough the programs FEDER, under the partnership agreement PT2020, as well asProfessor Manuel Delgado, and the Invited Scientist program of CMUP for financialsupport.
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