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Algebra depth in tensor categories.

In memory of Daniel Kastler


Study of the quotient module of a finite-dimensional Hopf subalgebra pair in order to compute its depth yields a relative Maschke Theorem, in which semisimple extension is characterized as being separable, and is therefore an ordinary Frobenius extension. We study the core Hopf ideal of a Hopf subalgebra, noting that the length of the annihilator chain of tensor powers of the quotient module is linearly related to the depth, if the Hopf algebra is semisimple. A tensor categorical definition of depth is introduced, and a summary from this new point of view of previous results are included. It is shown in a last section that the depth, Bratteli diagram and relative cyclic homology of algebra extensions are Morita invariants.

1 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 modules, which have the same indecomposables but perhaps with different nonzero multiplicities, are said to be similar. Since an abelian category has direct sum [symmetry] that work as usual, similarity of two objects X, [UPSILON], denoted by X ~ [UPSILON], is defined by X [symmetry] * [congruent to] n [??] [UPSILON], i.e., "X divides a multiple of Y," and [UPSILON] [symmetry] * = m [??] X (or briefly [UPSILON] | m [??] X) for some multiplicities m, n [member of] N. In the presence of a uniqueness theorem for indecomposables that includes X, [UPSILON], they share isomorphic indecomposable summands. Also, the endomorphism rings of X and [UPSILON] are Morita equivalent in a particularly transparent way [1, 20]. For example, one may introduce the theory of basic algebras without complications using the regular representation and a similar direct sum of projective indecomposables with constant multiplicity one.

A special type of abelian category is a tensor category, which has a tensor product [cross product] satisfying the usual distributive, associative and unital laws up to natural isomorphism. An algebra A may then be defined in terms of multiplication A [cross product] A [right arrow] A as usual. Define the minimum depth of A to be the least 2n + 1 = 1, 3, 5, ... such that [A.sup.[symmetry](n)] = A [cross product]...[cross product] A (n times A) is similar to [A.sup.[cross product] (n+1)], which simplifies to [A.sup.[cross product] (n+1)] | q [??] [A.sup.[cross product](n)] for some q [member of] N, since [A.sup.[cross product](n)] | [A.sup.[cross product] (n+1)] follows from applying the multiplication and unit. This definition applied to an algebra A in the category of bimodules over a ring B with tensor [cross product] = [[cross product].sub.B], recovers the minimum odd depth of the ring extension B [right arrow] A [2], where it is applied to finite group algebra extensions to recover (together with minimum even depth) subgroup depth [7]. Interesting values of subgroup depth have been computed in [7, 2, 11, 13, 14, 18], where subgroup depth less than 3 are normal subgroups [3, 4, 25, 28, 26]. Several properties of subgroup depth extend to Hopf subalgebra (and left coideal subalgebra) pairs such as a characterization of normality [3] and unchanged minimum even depth when 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 minimum depth.

Problem 1.1. Is d(R, H) < [infinity]?

There are examples in subfactor theory by Haagerup of infinite depth, although not answering the problem. We bring up three other equivalent problems below.

In the opposite tensor category, algebra becomes a notion of coalgebra with the same definition of depth. In the tensor category of bimodules over B, a coal-gebra in this sense is a B-coring. Applying the definition of depth to the Sweedler coring of a ring extension, one recovers the minimum h-depth of the ring extension as defined in [29]. The minimum h-depth of a Hopf subalgebra pair R [??] H is shown in [31] to be precisely determined by the depth of their quotient module [Q.sub.H] = H /[R.sup.+] H in the finite tensor category of finite-dimensional H-modules [12]. In turn the depth of Q is determined precisely by the length of the descending chain of annihilator ideals of the tensor powers of Q, if the Hopf algebra is semisimple, as proven in Theorem 3.14. The quotient module Q has many uses, including the following equivalent reformulation of the problem above, either as an H- or R-module isoclass in the respective representation ring (see [31] or Section 3, the notion below is algebraic 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 a permutation module, which is algebraic [13, Ch. 9]. The question in general is only interesting for the projective-free summands of Q, since projectives form a finite rank ideal in the representation ring [15]. If either R or H has finite representation type (e.g., is semisimple, Nakayama serial), Q is similarly algebraic. Example 4.6 computes a finite depth where both Hopf algebras are of infinite representation type.

In Section 4, we study depth of a non-normal subalgebra in a factorisable Hopf algebra in terms of entwined subalgebras such as a matched pair of Hopf algebras. In Section 3, we prove a relative Maschke theorem characterizing semisimple extension of finite-dimensional Hopf algebras as a separable extension; as a corollary, these are ordinary (or untwisted) Frobenius extensions. We also define and study the core Hopf ideal of a Hopf subalgebra, which extends to Hopf algebras the usual notion of core of a subgroup pair of finite groups. We note that the length of the annihilator chain of tensor powers of the quotient module is linearly related to the depth if the Hopf algebra is semisimple, improving on some results in [15]. In Section 5, we make a categorical study of a Morita equivalence of noncommutative ring extensions. We show that depth and relative cyclic homology of a ring extension are Morita invariants, as is the inclusion matrix of a semisimple complex algebra extension.

1.1 Similar modules

Let A be a ring. Two left A-modules, [.sub.A]N and [.sub.A]M, are said to be similar ([1], or H-equivalent [20]) denoted by [.sub.A]M = [.sub.A]N if two conditions are met. First, for some positive integer r, N is isomorphic to a direct summand in the direct sum of r copies of M, denoted by [.sub.A]N [symmetry] * [congruent to] r [??] [.sub.A]M [??]

N | r [??] M [??] [there exists] [f.sub.i] [member of] Hom ([.sub.A]M, [.sub.A]N), [g.sub.i] [member of] Hom ([.sub.A]N, [.sub.A]M) : [[SIGMA].sub.i=1] [f.sub.i] [??] [g.sub.i] = [id.sub.N] (1)

Second, symmetrically there is s [member of] [Z.sub.+] 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 extend this definition of similarity to similarity of two objects in an abelian category, and to show that it is an equivalence relation.

Example 1.3. Suppose A is an artinian ring, with indecomposable A-modules {[P.sub.[alpha]] |[alpha] [member of] I} (representatives from each isomorphism class for some index set I). By Krull-Schmidt finitely generated modules [M.sub.A] and [N.sub.A] have a unique factorization into a direct sum of multiples of finitely many indecomposable module components. Denote the indecomposable constituents of [M.sub.A] by Indec (M) = {[P.sub.[alpha]] | [[P.sub.[alpha]], M] = 0} where [[P.sub.[alpha]], M] is the number of factors in M isomorphic to [P.sub.[alpha]]. 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.sub.A] = [n.sub.1] [P.sub.1] [symmetry] ... [symmetry] [n.sub.r][P.sub.r] is the decomposition of the regular module into its projective indecomposables. Let [P.sub.A] = [P.sub.1] [symmetry] ... [symmetry] [P.sub.r]. Then [P.sub.A] and [A.sub.A] are similar (and call P the basic A-module in the similarity class of A). Then A and End [P.sub.A] are Morita equivalent. The algebra End [P.sub.A] is of course the basic algebra of A.

Suppose A is a semisimple ring. Then [P.sub.i] = [S.sub.i] are simple modules. Note that the annihilator ideal Ann [S.sub.i] is a maximal ideal in A; denote it by [I.sub.i]. Note that Ann ([n.sub.i] [??] [S.sub.i]) = [I.sub.i], Ann ([n.sub.i] [??] [S.sub.i] [symmetry] [n.sub.j] [??] [S.sub.j]) = [I.sub.i] [intersection] [I.sub.j], and any ideal I is uniquely Ann ([S.sub.[i.sub.1]] [symmetry] ... [symmetry] [S.sub.[i.sub.s]] ) for the [2.sup.r] integer subsets, 1 [less than or equal to] [i.sub.1] < ... < [i.sub.s] [less than or equal to] r.

Proposition 1.4. If two modules are similar, then their annihilator ideals are equal. Conversely, if A is a semisimple ring, two finitely generated modules with equal annihilator ideals are similar.

Proof. Given modules M and N, if M [??] N, then Ann N [??] Ann M. It follows that 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.sub.[i.sub.1]] [intersection] ... [intersection] [I.sub.[i.sub.s]] for some integers 1 [less than or equal to] [i.sub.1] < ... < is [less than or equal to] r. It follows that [I.sub.[i.sub.1]] [THETA] ... [THETA] [I.sub.[i.sub.s]] is the basic module in the similarity class of both M and N; in particular, [MU] ~ N.

Example 1.5. Suppose R is an artinian ring that is not semisimple and with two additional indecomposable modules [I.sub.1], [I.sub.2] that are not projective and not isomorphic. Then the modules M = R [THETA] [I.sub.1] and N = R [THETA] [I.sub.2] are both faithful generators, but dissimilar by Krull-Schmidt. This contradicts the converse of the proposition for more general rings. (Without dissimilarity, one additional nonprojective indecomposable would suffice.)

1.2 Subring depth

Throughout this section, let A be a unital associative ring and B [??] A a subring where [1.sub.B] = [I.sub.A]; more generally, it suffices to assume B [right arrow] A is a unital ring homomorphism, called a ring extension, although we suppress this option notationally. Note the natural bimodules [.sub.B][A.sub.B] obtained by restriction of the natural A-A-bimodule (briefly A-bimodule) A, also to the natural bimodules [.sub.B][A.sub.A], [.sub.A][A.sub.B] or [.sub.B][A.sub.B], which are referred to with no further ado. Let [A.sup.[cross product]B(n)] denote A [[cross product].sub.B] ... [[cross product].sub.B] A (n times A, n [member of] N), where [A.sup.[[cross product].sub.B]0] = B. For n [greater than or equal to] 1, the [A.sup.[[cross product].sub.B](N)] has a natural A-bimodule structure which restricts to B-A-, A-B- and B-bimodule structures occurring in the next definition. Note that [A.sup.[[cross product].sub.B](N)] [A.sup.[[cross product].sub.B](n+1)] automatically occurs in any case for n [greater than or equal to] 2, since A [right arrow] A [[cross product].sub.B] A given by a [??] a [[cross product].sub.B] 1isa split monomorphism. For n = 1 and A-bimodules, this is the separability condition on A [??] B; otherwise, A | A [[cross product].sub.B] A as A-B- or B-A-bimodules (via the split epi a [[cross product].sub.B]a' [??] aa').

Definition 1.6. The subring B [??] A has depth 2n + 1 [greater than or equal to] 1 if as B-bimodules [A.sup.[[cross product].sub.B](N)] ~ [A.sup.[[cross product].sub.B](n+1)]. The subring B [??] A has left (respectively, right) depth 2n [greater than or equal to] 2 if [A.sup.[[cross product].sub.B](N)] ~ [A.sup.[[cross product].sub.B](N+1)] as B-A-bimodules (respectively, A-B-bimodules). Equivalently, A [??] B has depth 2n + 1 [greater than or equal to] 1, or left depth 2n [greater than or equal to] 2, if

[A.sup.[[cross product].sub.B](n+1)] [cross product]* [congruent to] q * [A.sup.[[cross product].sub.B](n)] (2)

as B-B-bimodules, or B-A-bimodules, respectively. Right depth 2n is defined similarly in terms of A-B-bimodules.

It is clear that if B [??] A has either left or right depth 2n, it has depth 2n + 1 by restricting the similarity condition to B-bimodules. If B [??] A has depth 2n + 1, it has depth 2n + 2 by tensoring the similarity by - [[cross product].sub.B] A or A [[cross product].sub.B] -. The minimum depth is denoted by d(B, A); if B [??] A has no finite depth, write d(B, A) = [infinity]. We similarly define minimum odd depth [d.sub.odd] (B, A) and minimum even depth [d.sub.even] (B, A).

A subring B [??] A has h-depth 2n - 1 if Eq. (2) is more strongly satisfied as A-A-bimodules (n = 1,2,3, ...). Note that B has h-depth 2n - 1 in A implies that it has h-depth 2n + 1 (also that it has depth 2n). Thus define the minimum h-depth [d.sub.h](B, A) (and set this equal to [infinity] if no such n [member of] N exists). Note that h-depth 1 is the Azumaya-like condition of Hirata in [20]. The notion of h-depth is studied in [29]; by elementary considerations the inequality |dh (B, A) - d(B, A)| [less than or equal to] 2 is satisfied if either the minimum depth or minimum h-depth is finite.

2 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 definition recovers minimum odd depth defined in [7] and h-depth defined in [30]. In particular, a coalgebra in bimodule tensor category is a coring, with depth defined in [16]. An algebra or coalgebra in a finite tensor category is an H-module algebra or H-module coalgebra with depth defined in [31].

2.1 Tensor Category

By a tensor category (M, [cross product], 1) we mean an abelian category M with unit object 1 [member of] Ob(M) and tensor product [cross product] : M x M [right arrow] M, an additive bifunctor (satisfying distributive laws w.r.t. [symmetry]) with associativity constraint, a natural isomorphism

[.sup.[alpha]]X,Y,Z : (X [cross product] Y) [cross product] Z [??] X [cross product] (Y [cross product] Z), X, Y, Z [member of] M

satisfying the pentagon axiom (a commutative pentagon with 4 arbitrary objects in a tensor product grouped together in different ways, see for example [41, (2.3)]), and unit constraints, natural isomorphisms i, r such that

[I.sub.X] : 1 [cross product] X [??] X, [r.sub.X] : X [cross product] 1 [??] X, X [member of]M

satisfy the triangle axiom (a commutative triangle with the unit object between two other arbitrary objects in a tensor product associated in two ways using [alpha], i, r, [41, (2.4)]). The Coherence Theorem of MacLane states that every diagram constructed from associativity and unit constraints commutes. (Here we are making no requirement of left and right duals satisfying rigidity axioms.)

A tensor functor between tensor categories (M, [cross product], 1) and (M', [cross product]', 1') is a functor F : M [right arrow] M' such that for every X, Y [pounds sterling] Ob(M), there are isomorphisms [J.sub.X,Y] : F(X) [cross product]' F(Y) [??] F(X [cross product] Y) defining a natural isomorphism, and [PHI] : 1' [??] F(1) is an isomorphism satisfying a commutative hexagon and two commutative rectangles, 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 [.sub.R][M.sub.R] denote the category of R-R-bimodules and their bimodule homomorphisms (denoted by Hom [.sub.R-R](X, Y) or Hom ([.sub.R][X.sub.R], [.sub.R][Y.sub.R])). Note that [.sub.R][M.sub.R] has a tensor product [[cross product].sub.R] and unit object [.su.R][R.sub.R], the natural bimodule structure on R itself. For example, [l.sub.X] : R [[cross product].sub.R] X [??] X is the well-known natural isomorphism. This makes ([.sub.R][M.sub.R], [[cross product].sub.R], [.sub.R][R.sub.R]) into a tensor category.

Let A, R are rings, [M.sub.A], [M.sub.R] their categories of right modules and homomorphisms. Recall that A and R are Morita equivalent rings if R [congruent to] End [P.sub.A] for some progenerator A-module P, if and only if the categories [M.sub.R] and [M.sub.A] are equivalent, via the additive functor - [[cross product].sub.R] P. The inverse bimodule of P is denoted without ambiguity by P* [congruent to] Hom ([P.sub.A], [A.sub.A]), since Hom ([P.sub.A], [A.sub.A]) = Hom ([.sub.R]P, [.sub.R]R) as A-R-bimodules (by a theorem of Morita [39]).

Lemma 2.2. Suppose T : [M.sub.R] [??] [M.sub.A] is an equivalence of categories given by T(X) X [[cross product].sub.R] [P.sub.A]. Then the categories [.sub.R][M.sub.R] and [.sub.A][M.sub.A] are tensor equivalent via F([.sub.R][Y.sub.R])= P* [[cross product].sub.R]Y [[cross product].sub.R]P.

Proof. The proof follows from F(X [[cross product].sub.R] Y) = P*[[cross product].sub.R] X [[cross product].sub.R] Y [[cross product].sub.R] P [congruent to] P*[[cross product].sub.R]X [[cross product].sub.R]R [[cross product].sub.R]Y [[cross product].sub.R]P [congruent to] P* [[cross product].sub.R]X [[cross product].sub.R]P [[cross product].sub.A] P* [[cross product].sub.R] Y [[cross product].sub.R]P [congruent to] F(X) [[cross product].sub.R]F(Y).

Also F([.sup.R][R.sub.R]) [congruent to] [.sub.A][A.sub.A]. The functor F is an equivalence with inverse functor [F.sup.-1]([.sub.A][Z.sub.A])=P [[cross product].sub.A] Z [[cross product].sub.A] P*.

In a tensor category (M, [cross product], 1M), one says (B, m, u) is an algebra in M if the multiplication m : B [cross product] B [right arrow] B, a morphism in M, satisfies a commutative pentagon [41, 3.9] w.r.t. associativity isomorphism [[alpha].sub.A,A,A] and "the unit" u : [1.sub.M] [right arrow] A, a morphism in M, satisfies two commutative rectangles [41, 3.10] w.r.t. the natural isomorphisms [l.sub.A], [r.sub.A] in the notation of Subsection 2.1. (Coalgebra (B, [DELTA], [epsilon]) is defined dually by coassociative comultiplication [DELTA] : B [right arrow] B [cross product] B and counit [epsilon] : B [right arrow] 1M satisfying the counit diagrams.) That [B.sup.[cross product] (n)]) | [B.sup.[cross product] (n+1)]) for n [greater than or equal to] 1 follows from using the multiplication epi, split by the unit (e.g., see commutative diagram [41, (3.10)]), or the counit splitting the comultiplication monomorphism.

Definition 2.3. Let B be an algebra (or coalgebra) in a tensor category M. Define B to have depth 1 if B ~ [1.sub.M]. Define B to have depth 2n + 1 (n [greater than or equal to] 1) if [B.sup.[cross product](n+1)] | q * [B.sup.[cross product] (n)]) for some q [member of] N ([??] [B.sup.[cross product] (n)]) ~ [B.sup.[cross product] (n+1))]; in this case, B also has depth 2n + 3,2n + 5, ... by tensoring repeatedly by - [cross product] B. If there is a finite n [member of] N like this, let d(B, M) denote the minimum depth (an odd number); otherwise, write d(B,M) = [infinity].

Example 2.4. Let A be a ring, with tensor category of bimodules [.sub.A][M.sub.A]. An algebra B (or monoid) in [.sub.A][M.sub.A] has unit mapping u : A [right arrow] B and multiplication B [[cross product].sub.A] B [right arrow] B satisfying associativity and unital axioms as usual. This is equivalently a ring extension. The depth just defined is the minimum odd depth; i.e., d(B, [.sub.A][M.sub.A]) = [d.sub.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 and bimodules over such algebras, as well as Morita equivalence between two such algebras. For example, a left module over algebra A in tensor category [.sub.B][M.sub.B] is an A-B-bimodule N as an exercise in applying these ideas. The category [.sub.A][M.sub.B] is equivalent to the category [.sub.A]M of left modules over A. If A' is another algebra in [.sub.B][M.sub.B] Morita equivalent in the sense of [41], then [.sub.A'][M.sub.B] is equivalent to [.sub.A][M.sub.b]. This is the case if the ring extensions B [right arrow] A and B [right arrow] A' are Morita equivalent in the sense of Section 5, cf. Diagram (34).

Example 2.6. Let B = C be an A-coring; i.e., a coalgebra (or comonoid) in the tensor category [.sub.A][M.sub.A]. Dual to algebra, there is a comultiplication [DELTA] : C [right arrow] C [[cross product].sub.A] C and counit [epsilon] : C [right arrow] A, both A-A-bimodule homomorphisms, satisfying coasso-ciativity and counit diagrams [5]. The definition of minimum depth d(C, [.sub.A][M.sub.A]) coincides with the depth d(C, A) of corings defined in [16, 2.1]: d(C, [.sub.A][M.sub.A]) = d(C, A).

Let A [??] B be a ring extension, and C = A [[cross product].sub.B] A its Sweedler A-coring, with comultiplication simplifying to [A.sup.[[cross product].sub.B](2)] [right arrow] [A.sup.[[cross product].sub.B](3)], [a.sub.1] [[cross product].sub.B] [a.sub.2] [??] [a.sub.1] [[cross product].sub.B] 1 [[cross product].sub.B] [a.sub.2], and counit [[epsilon].sub.C] : A [[cross product].sub.B] A [right arrow] A, [a.sub.1] [[cross product].sub.B] [a.sub.2] [??] [a.sub.1] [a.sub.2] ([a.sub.1], [a.sub.2] [member of] A). Comparing with Definition 1.6 and applying cancellations of the type X [[cross product].sub.A] A [congruent to] X, we see that coring depth of C recovers h-depth of the ring extension: d(C, [.sub.A][M.sub.A]) = [d.sub.h](B, A).

Suppose k is a field, the ground field below for all algebras, coalgebras, modules and unadorned tensor products in finite tensor categories (including the tensor category of finite-dimensional vector spaces, [Vect.sub.k]).

Example 2.7. Let H be a finite-dimensional Hopf k-algebra; its category of finite-dimensional modules [M.sub.H] is a finite tensor category [12]. The tensor [cross product] = [[cross product].sub.k] is defined by the diagonal action, where V [cross product] W: ([upsilon] [cross product] w) * h = [[upsilon]h.sub.(1)] [cross product] [wh.sub.(2)]. The unit module is [k.sub.[epsilon]] where [epsilon] : H [right arrow] k is the counit. An algebra A in [M.sub.H] is a right H-module algebra, which the reader may check satisfies the (measuring) axioms (ab).h = ([a.h.sub.(1)])([b.h.sub.(2)]) and [1.sub.A].h = [1.sub.A][epsilon](h) for all a, b [member of] A and h [member of] H. A coalgebra C in [M.sub.H] is a right H-module coalgebra (C, [DELTA], [[epsilon].sub.C]) satisfying

[DELTA](ch) = [c.sub.(1)][h.sub.(1)] [cross product] [c.sub.(2)][h.sub.(2)], [[epsilon].sub.C](ch) = [[epsilon].sub.C](c)[epsilon](h) (3)

for all c [member of] C, h [member of] H.

The depth d(A, [M.sub.H]) and d(C, [M.sub.H]) is a linear rescaling of the minimum depth of any object in [M.sub.H] defined in [31, 15, 16], not an important difference, though slightly more convenient in formulas given below.

Example 2.8. Continuing with H, the category of right H-comodules [M.sup.H] is a tensor category, where X, Y [member of] [M.sup.H] has tensor product X [cross product] Y as linear space with comultiplication x [cross product] y [??] [x.sub.(0)] [cross product] [y.sub.(0)] [cross product] [x.sub.(1)][y.sub.(1)]. The unit module is k with coaction [1.sub.k] [??] [1.sub.H]. An algebra A in [M.sub.H] has multiplication m : A [cross product] A [right arrow] A and unit k [right arrow] A right H-comodule morphisms. This condition is equivalent to the coaction of A, [RHO]A : A [right arrow] A [cross product] 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 summarize 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 coalgebra or algebra on the other hand (in a more manageable finite tensor category [12]). We study the quotient module Q of a finite-dimensional Hopf subal-gebra pair R [??] H in terms of core Hopf ideals, duals and Frobenius extensions, and under conditions of semisimplicity, relative or not.

Recall that an entwining structure of an algebra A and coalgebra C is given by a linear mapping [psi] : C [cross product] A [right arrow] A [cross product] C (called the entwining mapping) satisfying two commutative pentagons and two triangles (a bow-tie diagram on [5, p. 324]). Equivalently, (A [cross product] C, [id.sub.A] [cross product] [[DELTA].sub.C], [id.sub.A] [cross product] [[epsilon].sub.C]) is an A-coring with respect to the A-bimodule structure a(a' [cross product] c)a''= aa'[psi](c [cross product] a'') (or conversely defining [psi](c [cross product] a) = ([1.sub.A] [cross product] c)a) (details in [5, 32.6] or [9, Theorem 2.8.1]).

In more detail, an entwining structure mapping [psi] : C [cross product] A [right arrow] A [cross product] C takes values usually denoted by [psi](c [cross product] a) = [a.sub.[alpha]] [cross product] [a.sup.[alpha]] = [a.sub.[beta]] [cross product] [a.sup.[beta]], suppressing linear sums of rank one tensors, and satisfies the axioms: (for all a, b [member of] A, c [member of] C)

1. [psi](c [cross product] ab) = [a.sup.[alpha]][b.sub.[beta]] [cross product] [c.sup.[alpha][beta]];

2. [psi](c [cross product] [1.sub.A]) = [1.sub.A] [cross product] c;

3. [a.sub.[alpha]] [cross product] [[DELTA].sub.c]([c.sub.[alpha]]) = [a.sub.[alpha][beta]] [cross product] [c.sub.(1).sup.[beta]] [cross product] [c.sub.(2).sup.[alpha]]

4. [a.sub.[alpha]][[epsilon].sub.c]([c.sub.[alpha]]) = a[epsilon]c(c),

which is equivalent to two commutative pentagons (for axioms 1 and 3) and two commutative triangles (for axioms 2 and 4), in an exercise.

3.1 Doi-Koppinen entwinings [5, 9]

Let H be a finite-dimensional Hopf algebra. Suppose A is an algebra in the tensor category of right H-comodules, equivalently, A is a right H-comodule algebra. Moreover, let (C, [[DELTA].sub.C], [[epsilon].sub.C]) be a coalgebra in the tensor category [M.sub.H], right H-module coalgebra as noted in the example above 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 [rho] = [DELTA]. Given a Hopf subalgebra R [??] H the quotient module Q defined as Q = H/[R.sup.+] H. Note that Q is a right H-module coalgebra. So is (H, [DELTA], [epsilon]) trivially a right H-module coalgebra. The canonical epimorphism H [right arrow] Q denoted by h [??] [bar.h] is an epi of right H-module coalgebras. The module [Q.sub.H] is cyclic with generator [bar.1.sub.H].

The mapping [psi] : C [cross product] A [right arrow] A [cross product] C defined by [psi](c [cross product] a) = [a.sub.(0)] [cross product] [ca.sub.(1)] is an entwining (the Doi-Koppinen entwining [5, 33.4], [9, 2.1]). From the equivalence of corings with entwinings, it follows that A [cross product] C has A-coring structure

a(a' [cross product] c)a'' = [aa'a''.sub.(0)] [cross product] [ca''.sub.(1)] (4)

which defines the bimodule [.sub.A][(A [cross product] C).sub.A]. The coproduct is given by [id.sub.A] C [[DELTA].sub.C] and the counit by [id.sub.A] C [[epsilon].sub.C].

Note that Eq. (4) above, and Eq. (5) below, exhibit the category [M.sub.A] as a module category over [M.sub.H] [12].

Proposition 3.2. [16, Prop. 4.2] The depth of the A-coring A [cross product] C (of a Doi-Koppinen entwining) and the depth of the H-module coalgebra C are related by d(A [cross product] C, [.sub.A][M.sub.A]) [less than or equal to] d(C, [M.sub.H]).

Proof. One notes that [(A [cross product] C).sup.[cross product].sub.A(n)] [congruent to] A [cross product] [C.sup.[cross product](n)] as A-A-bimodules via cancellations of the type X [[cross product].sub.A] A [congruent to] X. Keeping track of the right A-module structure on A [cross product] [C.sup.[cross product](n)], one shows that it is given by

(a [cross product] [c.sub.1] [cross product] ... [cross product] [c.sub.n])b = [ab.sub.(0)] [cross product] [c.sub.1][b.sub.(1)] [cross product] ... [cross product] [c.sub.n][b.sub.(n)]. (5)

If d(C, [M.sub.H]) = n, then [C.sup.[cross product](n)] ~ [C.sup.[cross product](n+1)] in the finite tensor category [M.sub.H]. Applying an additive functor, it follows that A [cross product] [C.sup.[cross product](n)] ~ A [cross product] [C.sup.[cross product](n+1)] as A-bimodules. Then applying the isomorphism just above and Definition 2.3 obtains the inequality in the proposition.

For example, if A = H, and C a right H-module coalgebra, the Doi-Koppinen entwining mapping [psi] : C [cross product] H [right arrow] H [cross product] C is of course [psi](c [cross product] h) = [h.sub.(1)] [cross product] [ch.sub.(2)]. The associated H-coring H [cross product] C has coproduct [id.sub.H] [cross product] [[DELTA].sub.C] and counit [id.sub.H] [cross product] [[epsilon].sub.C] with H-bimodule structure: (x, y, h [member of] H, c [member of] C)

x(h [cross product] c)y = [xhy.sub.(1)] [cross product] [cy.sub.(2)] (6)

Corollary 3.3. [16, Prop. 3.2] The depth of the H-coring H [cross product] C and the depth of the H-module coalgebra C are related by d(H [cross product] C, H) = d(C, [M.sub.H]).

Proof. This follows immediately from the proposition, but the proof reverses as follows. If d(H [cross product] C, [.sub.H][M.sub.H]) = 2n + 1, so that H [cross product] [C.sup.[cross product](n)] ~ H [cross product] [C.sup.[cross product](n+1)] as H-H-bimodules, apply the additive functor k [[cross product].sub.H] - to the similarity and obtain the similarity of right H-modules, [C.sup.[cross product](n)] ~ [C.sup.[cross product](n+1)]. Thus d(C, [M.sub.H]) [less than or equal to] d(H [cross product] C, [.sub.H][M.sub.H]) as well.

The corollary applies as follows. Let K [??] H be a left coideal subalgebra of a finite-dimensional Hopf algebra; i.e., [DELTA](K) [??] H [cross product] K. Let [K.sup.+] denote the kernel of the counit restricted to K. Then [K.sup.+] H is a right H-submodule of H and a coideal by a short computation given in [5, 34.2]. Thus Q := H/[K.sup.+] H is a right H-module coalgebra (with a right H-module coalgebra epimorphism H [right arrow] Q given by h [??]) h + [K.sup.+] [ETA] := [bar.h]). The H-coring [ETA] [cross product] Q has grouplike element [1.sub.H] [cross product] [bar.1.sub.H]; in fact, [5, 34.2] together with [46] shows that this coring is Galois:

H [[cross product].sub.k] H [??] H [cross product] Q (7)

via x [[cross product].sub.y] [??] [xy.sub.(1)] [cross product] [bar.y.sub.(2)], an H-H-bimodule isomorphism. That [[ETA].sub.K] is faithfully flat follows from Skryabin's Theorem [46] that K is a Frobenius algebra and [[ETA].sub.K] is free. Note that an inverse to (7) is given by x [cross product] [bar.z] [??] xS([z.sub.(1)]) [[cross product].sub.K] [z.sub.(2)] for all x, z [member of] 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 is related to the depth of Q in [M.sub.H] by

[d.sub.h] (K, H) = d(Q, [M.sub.H]). (8)

If R is a Hopf subalgebra of H, the following holds:

[d.sub.even] (R, H) = d(Q, [M.sub.R]) + 1 (9)

The following is of use to computing depth graphically from a bicolored graph in case R and H are semisimple C -algebras. Let U denote the functor of restriction-induction, i.e., U = [Ind.sup.H.sub.R][Res.sup.H.sub.R] : [M.sub.H] [right arrow] [M.sub.H].

Proposition 3.5. The depth d(Q, [M.sub.H]) = 2n + 1 is the least n for which [U.sup.n](k) ~ [U.sup.n+1] (k).

Proof. Recall that Q [congruent to] k [[cross product].sub.R] H and for any module [M.sub.H], U(M) [congruent to] M [cross product] Q (tensor in [M.sub.H]) [31]. It follows by induction that [Q.sup.[cross product](n)] [congruent to] [U.sup.n](k).

Note that decomposing Q into its projective-free direct summand [Q.sub.0] and projective summand [Q.sub.1], such that Q = [Q.sub.0] [direct sum] [Q.sub.1], leads to the following from the fact that projectives form an ideal in the Green ring of H.

Proposition 3.6. The depth of the Hopf subalgebra, [d.sub.h] (R, H) < [infinity] if and only if the module depth d([Q.sub.0], [M.sub.H]) < [infinity].

Proof. For the statement and proof of this proposition, we apply the extended definition of module depth of any finitely generated module X [member of] [M.sub.H] in terms of the depth n condition, [T.sub.n](X) ~ [T.sub.n+1](X) where [T.sub.n](X) = X [direct sum] ... [direct sum] [X.sup.[cross product](n)] [31]. Since [T.sub.n] (X) | [T.sub.n+1] (X), any projective module [UPSILON] has finite depth, as there are a finite number of isoclasses of projective indecomposables. But Y [cross product] M is projective as well for any M [member of] [M.sub.H]. Then [Q.sup.[cross product](n)] = [Q.sup.[cross product](n).sub.0] [direct sum] [Q.sup.[cross product](n).sub.1] [direct sum] mixed terms of [Q.sub.0], [Q.sub.1], which are all projective. Thus [d.sub.h] (R, H) < [infinity] [??] [Q.sup.[cross product](n)] = [Q.sup.[cross product](n+1)] as H-modules for some n [member of] N, which implies that the summand [Q.sub.0] has finite depth by [31, Lemma 4.4]. Conversely, if [T.sub.n] ([Q.sub.0]) = [T.sub.n+1]([Q.sub.0]) as H-modules, from [T.sub.i](Q) |[T.sub.i+1](Q), we obtain that [T.sub.n+m](Q) ~ [T.sub.n+m+1](Q), equivalently [Q.sup.[cross product](n+m)] ~ [Q.sup.[cross product](n+m+1)], where m is the number of distinct isoclasses of projective indecom-posables.

3.2 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 is relative projective, i.e., N | N [[cross product].sub.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 multiplication epimorphism [[micro].sub.M] : M [[cross product].sub.B] A [right arrow] M splits [19], which also generalizes the straightforward notion of left semisimple extension. The following theorem is a relative Maschke theorem characterizing semisimple extensions of finite-dimensional Hopf algebras R [??] H. We freely use the notation Q = H/[R.sup.+] H and ground field k developed above.

Theorem 3.7. The Hopf subalgebra pair R [??] H is a right (or left) semisimple extension [??] [k.sub.H] | [Q.sub.H] [??] [k.sub.H] is R-relative projective [??] there is q [member of] Q such that [[epsilon].sub.Q] (q) [not equal to] 0 and qh = q[epsilon](h) for every h [member of] H [??] [there exists] s [member of] H : s[H.sup.+] [??] [R.sup.+] H and [epsilon](s) = 1 [??] H is a separable extension of R.

Proof. The counit of Q, given by [[epsilon].sub.Q]([bar.h]) = [epsilon](h) for h [member of] H, is always R-split by 1 [??] [bar.[1.sub.H]]. If all modules are relative projective, it follows that [[epsilon].sub.Q] [ETA]-splits, so [k.sub.H] is isomorphic to a direct summand of [Q.sub.H]. Conversely, if [Q.sub.H] [congruent to] [k.sub.H] [symmetry] [Q'.sub.H], then any H-module N satisfies by [31, Lemma 3.1]

N [[cross product].sub.R] H [congruent to] N. [cross product] Q. [congruent to] N [symmetry] (N. [cross product] Q'.)

since N. [cross product] k. [congruent to] [N.sub.H]. Thus, N and all H-modules are relative projective.

If [[epsilon].sub.Q] : Q [right arrow] k is split by an H-module mapping [k.sub.H] [right arrow] [Q.sub.H], where 1 [??] q under this mapping, then q satisfies the integral-like condition of the theorem as well as [[epsilon].sub.Q](q) = l. Moreover, q = [bar.s] [not equal to] [bar.0], satisfies [epsilon](s) = 1 and sh - s[epsilon](h) [member of] [R.sup.+] H for all h [member of] H, but all elements of [H.sup.+] are of the form h - [epsilon](h)[1.sub.H].

If an element s [member of] H exists satisfying the conditions of the theorem, for any H-module M, the epi [[micro].sub.M] : M [[cross product].sub.R] H [right arrow] M is split by m [??] mS([s.sub.(1)]) [[cross product].sub.R] [S.sub.2]. This is also seen from a commutative triangle using M [[cross product].sub.R] H [[right arrow].sup.[congruent to]] M. [cross product] Q. and the mappings in [31, Lemma 3.1]. Note that S([s.sub.(1)]) [[cross product].sub.R] [s.sub.2] is a separability element, for given any h [member of] H, sh = [epsilon](h)s - [[SIGMA].sub.i] [x.sub.i][h.sub.i] for some [x.sub.i] [member of] [R.sup.+], [h.sub.i] [member of] H. Applying [pi](S [cross product] id) [DELTA] (where [pi] : H [cross product] H [right arrow] H [[cross product].sub.R] H is the canonical epimorphism) to this equation: S([h.sub.(1)])S([s.sub.(1)]) [[cross product].sub.R][S.sub.(2)] [h.sub.(2)] =

[epsilon](h)S([s.sub.(1)]) [[cross product].sub.R] [s.sub.(2)] - [[summation].sub.i]S([h.sub.i(1)])S([x.sub.i(1)][[cross product].sub.R] [X.sub.i(2))[hi.sub.(2)] = [epsilon](h)S([s.sub.(1)]) [cross product][R.sup.s](2).

Then hS([s.sub.(1)]) [[cross product].sub.R] [s.sub.(2)] = S([s.sub.(1)]) [[cross product].sub.R] [s.sub.(2)]h for all h [member of] H follows from a standard application of [h.sub.(1)]S[h.sub.(2)]) [cross product] [h.sub.(3)]) =l [cross product] h.

Note that if R = [k1.sub.H], the theorem recovers the extended Maschke's theorem for Hopf algebras (e.g., [39, Ch. 2]), since [R.sup.+] = {0}, Q = H and q or s are integral elements of H with nonzero counit. For example, if [Q.sup.[cross product](n)] is projective as an H- or R-module for any n [member of] N, it follows from this theorem that R is semisimple, since [k.sub.R] | Q | ... | [Q.sup.[cross product](n)].

Let [t.sub.R], [t.sub.H] denote nonzero right integrals in R, H, respectively, for the proof of the corollary below.

Corollary 3.8. Suppose H [??] R is a semisimple extension of finite-dimensional Hopf algebras. Then

1. the modular functions of H and R satisfy [m.sub.H| R] = [m.sub.R];

2. the Nakayama automorphisms of H and R satisfy [[eta].sub.H|R] = [[eta].sub.R];

3. the extension H [??] R is an ordinary Frobenius extension.

Proof. Suppose s [member of] H satisfies the conditions of the theorem, [epsilon](s) = 1 and s[H.sup.+] [??] [R.sup.+] H. By [31, Lemma 3.2]. the quotient module

Q [[right arrow].sup.~] [t.sub.R]H,

which sends q = [bar.s] [??] [t.sub.R]s. Then [t.sub.R]s[H.sup.+] [member of] [t.sub.R][R.sup.+]H = {0}, i.e., [t.sub.R]s is a nonzero integral in H. Without loss of generality, set [t.sub.H] = [t.sub.R]s. Then for all r [member of] R,

[m.sub.H](r)[t.sub.H] = [rt.sub.H] = [rt.sub.R]s = [m.sub.R](r)[t.sub.H],

from which it follows that [m.sub.H] restricts on R to the modular function of R, [m.sub.R].

Recall that finite-dimensional Hopf subalgebra pairs such as H [??] R are [beta]-Frobenius extensions (Fischman-Montgomery-Schneider) with

[beta](r) = r [??] [m.sub.H] * [m.sup.-1.sub.R] = [[eta].sub.R]([[eta].sup.-1.sub.H] (r)).

See [24] or [45] for textbook coverages of the full details. Consequently, [[eta].sub.H] (r) = [[eta].sub.R](r), [m.sub.H](r) = [m.sub.R](r) and [beta](r) = r for all r [member of] R.

The hypothesis of semisimplicity that removes the twist in the Frobenius extension of Hopf algebra substantially uncomplicates the associated induction theory.

3.3 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.sup.+] = ker [epsilon] [intersection] R. The right quotient H-module Q := H/[R.sup.+] H controls induction of right H-module restricted to R-modules as follows: [for all] M [member of] [M.sub.H],

M [[cross product].sub.R] H [[right arrow].sup.[congruent to]] M. [cross product] Q., m [[cross product].sub.R] h [??] m[h.sub.(1)] [cross product][h.sub.(2)] (10)

with inverse mapping given by m [cross product] [bar.h]] [??] mS([h.sub.(1)]) [[cross product].sub.R] [h.sub.(2)] where S : [ETA] [right arrow] [ETA] denotes the 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-modules in a somewhat similar way: [for all]V M [member of] [M.sub.H],

M. [cross product] [Q.sup.*] [[right arrow].sup.[congruent to]] Hom ([H.sub.R],[M.sub.R]), m [cross product] [q.sup.*] [right arrow] (h [??] m[h.sub.(1)] [q.sup.*]([bar.([h.sub.(2)]])) (11)

Both Eqs. (10) and (11) are first recorded in [47, Ulbrich]; we use the notation for cosets 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 automorphism [beta] : R [right arrow] R given by a relative modular function or a relative Nakayama automorphism. If the twist is trivially the identity on R, the Hopf subalgebra is an 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* [congruent to] Q as right H-modules.

Proof. This follows from the characterization of Frobenius extension: for each right R-module N,

N [[cross product].sub.R] H [congruent to] Hom ([H.sub.R], [N.sub.R] ). (12)

Now apply this and the display equations above to [NU] = [MU] = [k.sub.[epsilon]].

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 via the correspondence I [??] l(I) := {a [member of] A: aI = 0} for every right ideal I of A, and inverse correspondence J [??] r(J) := {a [member of] A : Ja = 0} for every left ideal J of A. The following comes from the basic fact that l(I) [congruent to] Hom ([(A/I).sub.A], [A.sub.A]) and r(J) [congruent to] Hom ([.sub.A](A/J), [.sub.A]A). See [34, Lam II].

Proposition 3.10. Let [t.sub.R] denote a nonzero right integral in R, a Hopf subalgebra of H as above. Then l([R.sup.+] H) = H[t.sub.R],

Hom ([.sub.H](H/H[t.sub.R]), [.sub.H]H) [congruent to] [R.sup.+] H

and Hom ([Q.sub.H], [H.sub.H]) [congruent to] H[t.sub.R]. If H is a symmetric algebra, the k-duals Q* [congruent to] H[t.sub.R] and Q* [congruent to] [t.sub.R]H.

Proof. Note that H[t.sub.R][R.sup.+] H = 0. From [31, 3.2] Q [congruent to] [t.sub.R]H and dim Q = dim H/ dim R. By definition of Q, dim Q = dim H - dim [R.sup.+] H; similarly

dim H[t.sub.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 H[t.sub.R] = l([R.sup.+] H). The next two isomorphisms are applications of r(l(I) = I and l(r(J) = J. The last statement follows from

Hom ([M.sub.A], [A.sub.A]) [congruent to] M*

as left A-modules, for every A-module M, for a symmetric algebra A (and a similar statement for left A-modules, see [34]).

The equivalent problems in Section 1 have a third equivalent formulation based on elementary considerations using Eq. (1):

Problem 3.11. Is there an n [member of] N such that the composition

Hom ([Q.sup.[cross product](n)], [Q.sup.[cross product](n+1)]) [[cross product].sub.End [Q.sup.[cross product](n)]] Hom ([Q.sup.[cross product](n+1)], [Q.sup.[cross product](n)]) [right arrow] End [Q.sup.[cross product](n+1)]

is surjective?

Either R-modules or H-modules suffice above. If we assume that H [??] R is an ordinary Frobenius extension however, the following interesting isomorphisms of Hom-groups over H exist. Note that for any H-module M, there is a subring pair End [M.sub.H] [??] End [M.sub.R].

Proposition 3.12. There are End [Q.sup.[cross product](n).sub.H] := E-module isomorphisms,

Hom ([Q.sup.[cross product](n).sub.H], [Q.sup.[cross product](n+1).sub.H]) [congruent to] End [Q.sup.[cross product](n).sub.R] [congruent to] Hom ([Q.sup.[cross product](n+1).sub.H], [Q.sup.[cross product](n).sub.H])

(right and left E-modules respectively).

Proof. The second isomorphism follows from Eq. (10) and the hom-tensor adjoint isomorphism [1, 20.6]. The first isomorphism requires additionally the fact for any Frobenius extension H [??] R with modules [M.sub.H] and [N.sub.R]:

Hom ([M.sub.H], N [[cross product].sub.R] [H.sub.H]) = Hom ([M.sub.R], [N.sub.R]) (13)

which follows from a natural isomorphism Hom ([H.sub.R], [N.sub.R]) [congruent to] N [[cross product].sub.R] H as right [ETA]-modules, and the hom-tensor adjoint isomorphism.

It is worth remarking that the tensor powers of Q are also H-module coalgebra quotients, since they are pullbacks via [[DELTA].sup.n] : H [right arrow] [H.sup.[cross product](n)] of the quotient module of the Hopf subalgebra pair [R.sup.[cross product](n)] [??] [H.sup.[cross product](n)], which is isomorphic to [Q.sup.[cross product](n)] as [H.sup.[cross product](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] relating the depth of a quotient module Q to its descending chain of annihilator ideals of its tensor powers:

Ann Q [??] Ann (Q [cross product] Q) [??] ... [??] Ann [Q.sup.[cross product](n)] [??] .... (14)

The chain of ideals are either contained in [R.sup.+] or [H.sup.+] depending on whether Q is considered an R-module or H-module (as in Corollary 3.4). By classical theory recapitulated in [15, Section 4], for some n [member of] N we have Ann [Q.sup.[cross product](n)] = Ann [Q.sup.[cross product](n+m)] for all integers m [greater than or equal to] 1: this ideal I is a Hopf ideal, indeed the maximal Hopf ideal contained in Ann Q. Let [l.sub.Q] denote the least n for which this stabilization of the descending chain of annihilator ideals takes place; call [l.sub.Q] the length of the annihilator chain of tensor powers of the quotient module. This may be nuanced by [l.sub.[Q.sub.R]] or [l.sub.[Q.sub.H]] depending on which module Q is being considered: since for any module [M.sub.H] we have Ann [M.sub.R] = Ann [M.sub.H] [intersection] R, it follows that

[l.sub.[Q.sub.R]] [less than or equal to] [l.sub.[Q.sub.H]]. (15)

Let [S.sub.1], ..., [S.sub.t] be the simple composition factors of Q or one of its tensor powers; by elementary considerations with the composition series of [Q.sup.[cross product]i], we note that

I [??] [[intersection].sup.t.sub.j=1]Ann [S.sub.j], (16)

in particular, if some [Q.sup.[cross product]i] contains all simples (of R or H), I [??] [J.sub.[omega]], the (Chen-Hiss [8]) Hopf radical ideal, since [J.sub.[omega]] is the maximal Hopf ideal in the radical which is the intersection of the annihilator ideals of all simples. If one simple is projective, the corresponding [J.sub.[omega]] = 0 by a result in [8], whence Q is conditionally faithful, i.e., [Q.sup.[cross product](n)] is faithful for some n [member of] N [15].

Recall that the core of a subgroup U [less than or equal to] G is N := [[intersection].sub.g[member of]G]gU[g.sup.-1], and is the maximal 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 [less than or equal to] G is a subgroup pair. Then I is determined by the core N as follows: [I.sub.H] = [kN.sup.+] H and [I.sub.R] = [kN.sup.+] R.

Proof. Note that [kN.sup.+] H = [HkN.sup.+] is a Hopf ideal since N is normal in G. An arbitrary element in Q is the coset Ug annihilated by 1 - n for any n [member of] N, since N [??] U. Then [KN.sup.+] H [??] I, since I is maximal Hopf ideal in the annihilator of Q. Conversely, the Hopf ideal I = [k[??].sup.+] H for some normal subgroup [??] [??] G by a result in [43]. Since 1 - [??] annihilates each Ug, it follows that [??] [??] U, whence [??] = [NU] by maximality.

Due to the proposition, we propose calling the pair of Hopf ideals I = Ann [Q.sup.[cross product][l.sub.[Q.sub.H]]] and I [intersection] R = Ann [Q.sup.[cross product][l.sub.[Q.sub.R]]] the core Hopf ideals of the Hopf subal-gebra R [??] H.

Note that [15, Prop. 4.3] is equivalent to the inequality

2[l.sub.[Q.sub.R]] + 1 < [d.sub.even] (R, H), (17)

true without further conditions on H and R, since the even depth of Q, determined from similarity of tensor powers of Q as R-modules, results in equal annihilator ideals: see the first statement in Proposition 1.4. Similarly, considering the H-module Q and h-depth instead, we note that

2[l.sub.[Q.sub.H]] + 1 [less than or equal to] [d.sub.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.sub.even] (R, H) = 2[l.sub.[Q.sub.R]] + 2.

If moreover H is semisimple, then [d.sub.h] (R, H) = 2[l.sub.[Q.sub.H]] + 1.

Proof. Semisimple rings satisfy the equal-annihilator-similar-module condition in Proposition 1.4. The definition 2.3 of depth of Q depends on similarity of tensor powers of Q and involves a rescaling of 1 plus a factor of 2 with respect to [l.sub.Q]. The rest follows from the inequalities (17) and (18); see also [31, Theorem 5.1] for [d.sub.even](R, H) = d(Q, [M.sub.R]) + l.

For a semisimple Hopf subalgebra pair, also note the equalities that follow from Def. 2.3 and Prop. 1.4:

d(Q, [M.sub.H]) = 2[l.sub.[Q.sub.H]] + 1 (19)

d(Q, [M.sub.R]) = 2[l.sub.[Q.sub.R]] + 1. (20)

For semisimple Hopf algebra-subalgebra pairs, these formulas put the length [C.sub.Q] of the annihilator chain of tensor powers of Q in close relation to diameter of same colored points in the bicolored graph [7] as well as the base size or minimal number of "conjugates" of the Hopf subalgebra intersecting in the core, cf. [14, 7].

A general finite-dimensional Hopf subalgebra pair R [??] H may sometimes reduce to the hypothesis of the previous theorem via the following proposition, which extends [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 annihilator ideal of Q = H/[R.sup.+] H; let J = R [??] I denote the restricted Hopf ideal in R. Then h-depth [d.sub.h] (R, H) = [d.sub.h] (R/J, H/1). Similarly, minimum even depth satisfies [d.sub.even] (R, H) = [d.sub.even] (R/J, H/I).

Proof. Note that [d.sub.h](R, H) = d(Q, [M.sub.H]) by Corollary 3.4, and d(Q, [M.sub.H]) = d(Q, [M.sub.H/I]) by [16, Lemma 1.5]. Note that R/J [??] H/I is a Hopf subalgebra pair with quotient module isomorphic to Q by a Noether isomorphism theorem. Then [d.sub.h](R/J, H/I) = d(Q, [M.sub.H/I]).

3.5 Quotient module for the permutation group series

It is interesting at this point to compute the quotient module Q for the inclusion C [S.sub.n] [??] C [S.sub.n+1] of permutation group algebras. Notice that the proposition below implies that the character [[chi].sub.Q] = [[chi].sub.1] + [[chi].sub.t], where [[chi].sub.1] is the principal character and [[chi].sub.t] is the character of the standard irreducible representation (n, 1).

Proposition 3.16. The quotient module Q = C [[S.sub.n]/[S.sub.n+1]] is isomorphic to the standard representation of [S.sub.n+1] on [C.sub.n+1].

Proof. Recall the Artin presentation of [S.sub.n+1] with generators [[sigma].sub.i] = (i i + 1) for i = 1, ..., n and relations

[[sigma].sub.i][[sigma].sub.i]+1[[sigma].sub.i] = [[sigma].sub.i]+1 [[sigma].sub.i][[sigma].sub.i+1], [[sigma].sub.i] [[sigma].sub.j] = [[sigma].sub.j][[sigma].sub.i], [[sigma].sup.2.sub.i] = 1

for all |i-j| [greater than or equal to] 2. Note that [[sigma].sub.1], ..., [[sigma].sub.n-1] [member of] [S.sub.n]. An ordered basis for Q is given by

<[S.sub.n][S.sub.n][S.sub.n-1] *** [S.sub.n][[sigma].sub.n] *** [[sigma].sub.2] ***, [S.sub.n][[sigma].sub.n], [S.sub.n])

This ordered basis maps onto the ordered basis <[e.sub.1], ..., [e.sub.n+1]) of the [S.sub.n+1] representation space [C.sup.n+1] via the canonical order-preserving mapping. This mapping is an [S.sub.n+1]-module isomorphism, since [[sigma].sub.i] exchanges [e.sub.i] and [e.sub.i+1] as it does [S.sub.n][[sigma].sub.n] *** [[sigma].sub.i] and [S.sub.n][[sigma].sub.n] *** [[sigma].sub.i+1], respectively, (here we use [[sigma].sup.2.sub.i] = 1), and it leaves the other basis elements fixed, since [[sigma].sub.i] commutes with [[sigma].sub.i+2] and/or [[sigma].sub.i-2] (here we also use [[sigma].sub.i][[sigma].sub.i-1][[sigma].sub.i] = [[sigma].sub.i-1][[sigma].sub.i][[sigma].sub.i-1]) etc. while [[sigma].sub.i] [member of] [S.sub.n] for i < n. In more detail, note that

([S.sub.n][[sigma].sub.n] *** [[sigma].sub.i][[sigma].sub.i-1])[[sigma].sub.i] = [[sigma].sub.n][[sigma].sub.n] *** [[sigma].sub.i+1][[sigma].sub.i-1][[sigma].sub.i][[sigma].sub.i-1] = [S.sub.n][[sigma].sub.n] *** [[sigma].sub.i-1]

The rest of the proof is routine. (A second proof follows from Q [congruent to] U(1) [congruent to] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and Young diagram branching rule of adding a box.)

Since [S.sub.n] [??] [S.sub.n+1] is corefree, i.e., the core of the subgroup is trivial, it follows that the character [[chi].sub.Q] is faithful (equivalently, the annihilator idea of Q does not contain a nonzero Hopf ideal the representation of C G restricted to G has trivial kernel) [31,4.2]. The Burnside-Brauer Theorem [22, p. 49] implies for the character [[chi].sub.Q] that each irreducible character of [S.sub.n+1] is a constituent of its powers up to [[chi].sup.n.sub.Q], since dim Q = n + 1. This implies that d(Q, [M[.sub.S][.sub.n+1]]) [less than or equal to] n by reasoning along the lines of Example 1.3. Indeed d(Q, [M[.sub.S][.sub.n+1]]) = n follows from Corollary 3.4 and the graphical computation [d.sub.h]([S.sub.n], [S.sub.n+1]) = 2n + 1 in [31].

We mention the theorem in [37], that hooks generate the Green ring of a permutation group, as the full picture to the discussion above.

Theorem 3.17. [37, Marin] The representation ring A(C [S.sub.n+1]) is generated by the representations [[LAMBDA].sup.k][C.sup.n+1] for 0 [less than or equal to] k [less than or equal to] n.

Remark 3.18. Recall the notion of order of a module [V.sub.H] over a semisimple Hopf algebra H. The order ord(V) is the least natural number n such that [V.sup.[cross product](n)] has nonzero invariant subspace, i.e., dim[([V.sup.[cross product](n)]).sup.H] = 0. For example, ord([Q[.sub.S][.sub.n+1]) = 1 since [[chi].sub.Q] = [[chi].sub.1] + [[chi].sub.t]. For general semisimple Hopf subalgebra pairs H [??] R with quotient Q, one might conjecture that ord(Q) [less than or equal to] [C.sub.Q], since order of Q and [C.sub.Q] are both bounded above by the degree d of the minimal polynomial of [[chi].sub.Q] in the character ring of H (or H/J-modules where J = Ann [Q.sup.[[cross product]l].sub.Q]]] see [32, chs. 4,5, p. 37] and [7, 2.3], respectively). However, [32, p. 32] computes the order of a certain induced module V over the semidirect product group algebra H = C [[Z.sub.11]]#C [[Z.sub.5]] to be ord(V) = 3: with R = C [[Z.sub.q], in fact V [congruent to] [Q.sub.H]. We deduce that d(Q, [M.sub.H]) = 3, since [d.sub.h](R, H) = 1 forces R = H by [31, Cor. 3.3]), and [C[.sub.Q][.sub.H]]] = 1, since R is a normal Hopf subalgebra in H: so in general ord(Q) [??] [C.sub.Q].

4 Factorisable algebras

An algebra factorisation of a unital (associative) algebra C into two unital sub-lgebras A and B occurs when the multiplication mapping B [cross product] A [??] C is a B-A-bimodule isomorphism [5, 9]. Conversely, the algebra C may be constructed from B and A as a twisted tensor product (denoted by B [[cross product].sup.R] A) as follows: linearly C = B [cross product] A with multiplication given by the structure mapping R : A [cross product] B [right arrow] B [cross product] A, values denoted by R(a [cross product] b) = [b.sup.r] [cross product] [a.sup.r] or [b.sup.R] [cross product] [a.sup.R], where summation over more than one simple tensor is suppressed. In this case, the multiplication in B [cross product] A is given by

([b.sub.1] [cross product] [a.sub.1]) ([b.sub.2] [cross product] [a.sub.2]) = [b.sub.1] [b.sup.r.sub.2] [cross product] [a.sup.r.sub.1] [a.sub.2] (21)

In order for C to be associative, R must satisfy two pentagonal commutative diagrams, equationally given by

R([[micro].sub.A] [cross product] B) = (B [cross product] [[micro].sub.A]) (R [cross product] A) (A [cross product] R) (22)

(where [mu]A denotes multiplication in A), and

R(A [cross product] [[micro].sub.B]) = ([[micro].sub.B] [cross product] A) (B [cross product] R) (R [cross product] B) (23)

in Hom (A [cross product] B [cross product] B, B [cross product] A). These equations are satisfied if and only if C is associative. Additionally, the structure map R satisfies two commutative triangles given equationally by R(A [cross product] [1.sub.B]) = [1.sub.B] [cross product] A and R([1.sub.A] [cross product] B) = B [cross product] [1.sub.A]. It follows that A [right arrow] C, a [??] [1.sub.B] [cross product] a and B [right arrow] C, b [??] b [cross product] [1.sub.A] are algebra monomor-phisms.

Example 4.1. Let B be an algebra in [.sub.H]M, where A = H is a Hopf algebra as before. Let R : B [cross product] H [right arrow] H [cross product] B be given by R(b [cross product] h) = [h.sub.(1)].b [cross product] [h.sub.(2)]. Then B [[cross product].sub.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 ofH embedded canonically in the smash product B#H satisfies

[d.sub.odd] (H, B#H) = d(B, [.sub.H] M) (24)

Proof. Via cancellations of the type X [[cross product].sub.H] H [congruent to] X, one establishes an H-H-bimodule isomorphism,

[(B#H)[.sup.[cross product][.sub.H[.sup.n]]]] = [B.sup.[cross product](n)] [cross product] H, (25)

where the left H-module structure on [B.sup.[cross product](n)] [cross product] H is given by the diagonal action:

x.([b.sub.1] [cross product] *** [cross product] [b.sub.n] [cross product] h) = [x.sub.(1)].[b.sub.1] [cross product] *** [cross product] [x.sub.(n)].[b.sub.n] [cross product] [x.sub.(n+i)].h

If [B.sup.[cross product](n+1)] | q * [B.sup.[cross product](n)] in [.sub.H]M for some q [member of] N, then tensoring this by - [cross product] H yields [(B#H).sup.[cross product]H(n+1)] | q * [(B#H).sup.[cross product]Hn] as H-H-bimodules. Thus the minimum odd depth [d.sup.odd](H, B#H) [less than or equal to] d(B, [.sup.H]M) by Definition 1.6.

Conversely, if [(B#H).sup.[cross product]H(n+1)] | q * [(B#H)[.sup.[cross product][.sub.H[.sup.n]]]] as H-H-bimodules, then [B.sup.[cross product](n+1)] [cross product] H | q * [B.sup.[cross product](n)] [cross product] H, to which one applies - [cross product] [.sub.H]k, obtaining [B.sup.[cross product](n+1)] | q * [B.sup.[cross product](n)] in [H.sub.]M. Therefore d(B, [.sub.H]M) [less than or equal to] [d.sub.odd](H, B#H).

Using the notation developed in Section 3 for a finite-dimensional Hopf sub-algebra pair R [??] H with quotient right H-module coalgebra Q, we note that its k-dual Q* becomes a left H-module algebra via <hq*, q> = (q*, qh). Yet another equivalent formulation of the fundamental problem in Section 1 follows easily from the proposition since d([Q*.sub.H]M) = d(Q, [M.sub.H]) [31, 15].

Problem 4.3. Is d(H, Q*#H) < [infinity] or d(R, Q*#R) < [infinity]?

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.sub.B] with action denoted by h [??] b, and B is coalgebra in [.sub.H]M with action denoted by h [??] b satisfying compatibility conditions given in [36, (7.7)-(7.9)]. A twisting R : H [cross product] B [right arrow] B [cross product] H is given by

R(h [cross product] b) = [h.sub.(1)] [??] [b.sub.(1)] [cross product] [h.sub.(2)] [??] [b.sub.(2)], (26)

which defines an algebra structure on B [[cross product].sup.R] H = B [??] H; moreover, this is a Hopf algebra, called the double cross product, where H and B are canonically Hopf subalgebras [36].

For example, H and its dual Hopf algebra (with opposite multiplication) B = [H.sup.op*] are a matched pair via [??], the left coadjoint action of H on [H.sup.*],

h [??] b = [b.sub.(2)] <([Sb.sub.(1)])[b.sub.(3)], h>, (27)

and [??] the analogous left coadjoint action of [H.sup.*] on H. This defines the Drinfeld double D(H) as a special case of double cross product, D(H) = [H.sup.op*] [??] H.

Proposition 4.5. Let B and H be a matched pair of finite-dimensional Hopf algebras with A = B [??] H their double cross product. Then the minimum h-depth and even depth of the Hopf subalgebra B in A is given by the depth of H in the finite tensor category [M.sub.B] (w.r.t. [??] in Example 4.4): [d.sub.h](B, A) = d(H,[M.sub.B]) and [d.sub.even](B, A) = d(H,[M.sub.B]) + 1. Similarly, the Hopf subalgebra H has depth in A given by [d.sub.h] (H, A) = d(B, [.sub.H]M) (w.r.t. [??]) and [d.sub.even] (H, A) = d(B, [.sub.H]M)+1.

Proof. This follows from Cor. 3.4 if we show that the quotient module [Q.sub.B] [congruent to] ([H.sub.B],[??]). Note that Q = [BETA] [??] H/[B.sup.+](B [??] H) [congruent to] [ETA] via [bar.b[??]h] [??] [.sup.[epsilon].sub.B](b)h, and

[bar.h]b = [bar.([1.sub.B] [??] h)(b [??] [1.sub.h])] = [bar.[h.sub.(1)]] [??] [b.sub.(1)] [??] [h.sub.(1)] [??] [b.sub.(2)] = [.sup.[epsilon].sub.b]([h.sub.(1)] [??] [b.sub.(1))[bar.[h.sub.(2)] [??] [b.sub.(2)]] = [bar.h[??]b], where we use axiom (3) for B, a left H-module coalgebra.

For example, if B = [H.sup.op*] and B [??] H = D(H), suppose H is cocommutative. From the formula for coadjoint action, it is apparent that [H.sub.B] [congruent to] (dim H)*k, so d(H,[M.sub.b]) = 1 and d([H.sup.*], D(H)) [less than or equal to] 2. Indeed, it is known that D(H) [congruent to] [H.sup.*]#H in case 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.sub.8] (see for example [31, Example 4.9] for its Hopf algebra structure) and its quantum double D([H.sub.8]) indicates that minimum depth satisfies 3 [less than or equal to] d([H.sub.8], D([H.sub.8])) [less than or equal to] 4. The method is to compute D([H.sub.8]) in terms of generators and relations, compute the quotient Q as an 8-dimensional [H.sub.8]-module, then decompose it into its indecomposable summands (twice each simple, and two 2-dimensional indecomposables), compute the tensor products between these indecomposables, noting that Q ~ Q [cross product] Q as [H.sub.8]-modules, and using Eq. (9). Since both algebras have infinite representation type, we cannot otherwise predict a finite depth from known results [31, 17].

Let []H denote the adjoint action of H on itself, given by h.x = [h.sub.(1)]xS([h.sub.(2)]) for all h, x [member of] H.

Corollary 4.7. [15, Cor. 5.4] Let G be a finite group and D(G) its Drinfeld double as a complex group algebra. Then d(C G, D(G)) = d([]C G, [.sub.C G]M).

Proof. From the remark about cocommutativity just above, the double D(G) = [H.sup.*]#H (with H = C G) is a smash product to which Proposition 4.2 applies: thus [d.sub.odd](C G,D(G)) = d([H.sup.*],[.sub.CG]M). The smash product multiplication formula for g, h [member of] G, [p.sub.g], [p.sub.h] [member of] [H.sup.*] one-point projections, is given by

([p.sub.x] #g)([p.sub.y] #h)= [p.sub.x][p.sub.gyg.sup.-1]#gh (28)

which visibly demonstrates that [.sub.H][H.sup.*] [congruent to] [][H.sup.*] [congruent to] []C G.

It remains to show that [d.sub.even](C G, D(G)) = 1 + [d.sub.odd](C G, D(G)). Note that (S([p.sub.x]) = [p.sub.x.sup-1],


whence using Eq. (27)


the adjoint action of h on [p.sub.x]. Use Proposition 4.5 to conclude the proof.

5 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 provide the classical background theory, and prove that depth, relative cyclic homology as well as the bipartite graphs of a semisimple complex subalgebra pair are all Morita invariant properties of a ring or algebra extension. In addition, we note a natural example of Morita equivalence in towers of Frobenius extensions.

Define two ring extensions A | B and R | S to be Morita equivalent if there are additive equivalences P : [.sub.R]M [right arrow] [.sub.A]M and Q : [.sub.s]M [right arrow] [.sub.B]M satisfying a commutative rectangle (up to a natural isomorphism) with respect to the functors of restriction from R-modules into S-modules, and from A-modules into B-modules.


The requirement then is that there be a natural isomorphism QRes[??] [??] Res[??] P. One shows in an exercise that this is an equivalence relation on ring extensions by using operations on natural transformation by functors.

From ordinary Morita theory we know that P([.sub.r]R) = [.sub.a]P, a progenerator such that End [.sub.A]P [congruent to] R, so that P is in fact an A-R-bimodule with P([.sub.R]X) = P [cross product][.sub.R]X for all [.sub.R]X. The dual of P is unequivocally [P.sup.*] = Hom ([P.sub.R], [R.sub.R]), an R-A-bimodule, since Hom ([.sub.A]P, [.sub.A]A) [cross product] [P.sup.*] as R-A-bimodules by [39, Theorem 1.1]. Then [P.sup.*] [cross product][.sub.A] - : [.sub.A]M [right arrow] [.sub.R]M is an inverse equivalence to P: one has bimodule isomorphisms [P.sup.*] [cross product][.sub.a]P [equivalent to] [.sub.r][R.sub.r] and P[cross product][.sub.r][P.sup.*] [congruent to] [.sub.a][A.sub.a].

Similarly there is an invertible Morita bimodule [.sub.B][Q.sub.S], a left and right pro-generator module, such that Q([.sub.s]Y) = [.sub.B]Q [cross product][.sub.s]Y. The condition that the rectangle above commutes applied to R [member of] [.sub.R]M becomes [.sub.B]Q [cross product][.sub.s] R [congruent to] [.sub.B]P, also valid as B-R-bimodules due to naturality, noted as an equivalent condition in the proposition below.

Example 5.1. Given a ring extension R [[contains].bar] S, let A = [M.sub.n] (R) [[contains].bar] B = [M.sub.n] (S). Of course, A and R are Morita equivalent via P = n * R, also B and S are Morita equivalent via Q = n * S. Note that

[.sub.b]Q[cross product][.sub.s][R.sub.r] [congruent to] n * R = [.sub.b][P.sub.r].

Thus, as one would expect, the ring extensions R [[contains].bar] S and A [[contains].bar] B are Morita equivalent.

Example 5.2. Suppose B [[subset].bar] A and S [[subset].bar] R are ring extensions with ring isomorphism [PSI] : A [??] R restricting to a ring isomorphism [eta] : B [??] S. Defining bimodules [.sub.a][P.sub.r] := [.sub.[PSI]][R.sub.r] and [.sub.b][Q.sub.s] := [.sub.[eta]][S.sub.s], one shows in an exercise that the two ring extensions are Morita equivalent.

The proposition below characterizes Morita equivalence of ring extensions in many equivalent ways, condition (2) being the definition in [38, 21, 48].

Proposition 5.3. The following conditions on ring extensions A [[contains].bar] B and R [[contains].bar] S are equivalent:

1. A [[contains].bar] B and R [[contains].bar] S are Morita equivalent;

2. there are Morita bimodules [.sub.A][P.sub.R] and [.sub.B][Q.sub.S] satisfying [.sub.B]Q [cross product][.sub.S] [R.sub.R] [congruent to] [.sub.B][P.sub.R] [38];

3. there are Morita bimodules [.sub.A][P.sub.R] and [.sub.B][Q.sub.S] satisfying [.sub.R]R [cross product][.sub.S] [Q.sup.*][.sub.B] [congruent to] [.sub.R][P.sup.*][.sub.B];

4. there are Morita bimodules [.sub.A][P.sub.R] and [.sub.B][Q.sub.S] satisfying [.sub.A]A [cross product][.sub.B] [Q.sub.S] [congruent to] [.sub.A][P.sub.s];

5. there are Morita bimodules [.sub.A][P.sub.R] and [.sub.B][Q.sub.S] satisfying [.sub.S][Q.sup.*] [cross product][.sub.B] [A.sub.A] = [.sub.S][P.sup.*][.sub.A];

6. the following rectangle, with sides representing the induction functors, commutes up to a natural isomorphism,


7. the following rectangle, with sides representing the coinduction functors, commutes up to a natural isomorphism,


8. any of the conditions above stated identically with right module categories [M.sub.R], [M.sub.A], [M.sub.S], and [M.sub.B] replacing the corresponding left module categories.

Proof. (1) [??] (2) is sketched above. (2) [??] (3) follows from the computation [.sub.R][P*.sub.B] [congruent to] [.sub.R]Hom [([P.sub.R], [R.sub.R]).sub.B] [congruent to] [.sub.R]Hom [(Q [[cross product].sub.S] [R.sub.R], [R.sub.R]).sub.B] [congruent to] [.sub.R]Hom [([Q.sub.S], [R.sub.S]).sub.B]

[congruent to] [.sub.R]R [[cross product].sub.S] [Q*.sub.B] using adjoint theorems in [1, pp. 240, 243]. This shows (2) [??] (3). This argument reverses by using the reflexive property of progenerators ([.sub.A]Hom [([.sub.R]P*, [.sub.R]R).sub.R] [congruent to] [.sub.A][P.sub.R]).

(3) [??] (4) and (8). The following rectangle is commutative up to a natural isomorphism:


since for any module [X.sub.R] one has

X [[cross product].sub.R] [P*.sub.B] [congruent to] X [[cross product].sub.R]R [[cross product].sub.S] [Q*.sub.B] [congruent to] X [[cross product].sub.S] [Q*.sub.B].

To the natural isomorphism identifying the sides of this rectangle, apply the functor--[[cross product].sub.B] Q from the left and the functor--[[cross product].sub.A] P from the right to obtain the following commutative rectangle up to natural isomorphism:


(4) now follows from applying the rectangle to [A.sub.A]. (4) [??] (5). The same type of argument as in (2) [??] (3) above shows that

[.sub.S][P*.sub.A] [congruent to] [.sub.S]Hom [([.sub.A]P, [.sub.A]A).sub.A] [congruent to] [.sub.S]Hom ([.sub.B]Q, [.sub.B]B) [[cross product].sub.B] [A.sub.A] [congruent to] [.sub.S]Q* [[cross product].sub.B] [A.sub.a].

(4) [??] (6). By using (4), compute for any module [.sub.S][UPSILON],

[.sub.A]A [[cross product].sub.B]Q [[cross product].sub.S][UPSILON] [congruent to] [.sub.A]P [[cross product].sub.S][UPSILON] [congruent to] [.sub.A]P [[cross product].sub.R] R [[cross product].sub.S] [UPSILON],

which shows the rectangle (6) is commutative up to a natural isomorphism. The converse (6) [??] (4) follows from applying the rectangle to [.sub.S]S [member of] [.sub.S]M as well as naturality.

(5) [??] (7) For any module [.sub.S]W, it suffices to show that P [[cross product].sub.R] Hom ([.sub.S]R, [.sub.S]W) [congruent to] Hom ([.sub.B]A, [.sub.B]Q [[cross product].sub.S] W) using natural isomorphisms in [1,20.6,20.11, exercise 20.12] and (5):

[.sub.A]P [[cross product].sub.R] Hom ([.sub.S]R, [.sub.S]W) [congruent to] [.sub.A] Hom ([.sub.S]P*, [.sub.S]W) [congruent to] [.sub.A]Hom ([.sub.S]Q* [[cross product].sub.B] A, [.sub.S]W)

[congruent to] [.sub.A]Hom ([.sub.B]A, [.sub.B]Hom ([.sub.S]Q*, [.sub.S]W)) [congruent to] [.sub.A]Hom ([.sub.b]A, [.sub.b]Q [[cross product].sub.S] W)

The rest of the proof is similar and left as an exercise. *

In the following proposition, we note some different, quick proofs for certain results in [21], while building up results which show that depth and bipartite graphs are 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 with identical inclusion matrix and bipartite graph;

5. the following diagram oftensor categories and functors commutes up to natural isomorphism:


where F([.sub.r][X.sub.r]) := [.sub.A]P [[cross product].sub.r] X [[cross product].sub.R] [P*.sub.A] and G([.sub.s][Y.sub.s]) := [.sub.b]Q [[cross product].sub.S] [UPSILON] [[cross product].sub.S] [Q*.sub.B] define tensor equivalences;


7. the centralizers are isomorphic: [A.sup.B] [congruent to] [R.sup.S] [21];

8. the ring extensions A | B and R | S have the same minimum depth and h-depth.

Proof. 1. Let 0 [right arrow] V [right arrow] W [right arrow] U [right arrow] 0 be a short exact sequence in [.sub.A]M that is split exact when restricted to [.sub.B]M. By Rafael's characterization [44] of separability, the short exact sequence splits in [.sub.A]M. The rest of the proof follows from applying the commutative rectangle (29).

2. Suppose [.sub.A]V is (A, B)-projective (or "relative projective"), i.e., [.sub.A]V | [.sub.A]A [[cross product].sub.B] V (or the multiplication epi A [[cross product].sub.B] V [right arrow] V splits as an A-module map). By the relative Faith-Walker theorem for QF extensions [40], V is also (A, B)-injective: i.e., the canonical A-module monomorphism V [??] Hom ([.sub.B]A, [.sub.b]V) splits. In fact the class of relative projectives coincides with the class of relative injectives for QF extensions. It is clear from the commutative diagram (30) that the equivalence P sends relative projectives into relative projective; similarly, it is clear from the commutative rectangle (31) that relative injectives are sent by an equivalence into relative injectives. The rest of the proof is then an application of the relative Faith-Walker characterization of QF extension.

3. The proof is an application of the commutative rectangles (30) and (31) and the characterization of Frobenius extensions as having naturally isomorphic induction and coinduction functors. Suppose R [??] S is Frobenius. Then

[Ind.sup.A.sub.B] Q [CONGRUENT TO] P [Ind.sup.R.sub.S] [congruent to] P [CoInd.sup.R.sub.S] [CONGRUENT TO] [CoInd.sup.A.sub.B] Q.

Since Q is an equivalence, it follows that [Ind.sup.A.sub.B] and [CoInd.sup.A.sub.B] are naturally isomorphic functors, whence A [??] B is Frobenius.

4. Let [V.sub.1],..., [V.sub.s] be the simples of S (up to isomorphism). Then [U.sub.i] := Q [[cross product].sub.S] [V.sub.i] are representatives of the simple isoclasses of B by Morita theory. Induce each [V.sub.i] to an R-module, expressing this uniquely up to isomorphism as a sum of nonnegative multiples of the simples of R, [W.sub.1],..., [W.sub.r]:

R [[cross product].sub.S] [V.sub.i] [congruent to] [[cross product].sup.r.sub.j+1] [r.sub.ij][W.sub.j].

The s x r matrix is the inclusion matrix [K.sub.0] (S) [right arrow] [K.sub.0] (R) of the semisimple complex subalgebra pair S [??] R. This matrix determines the bipartite graph of the inclusion, an edge connecting black dot i with white dot j in case the (i, j)-entry is nonzero.

Since A and R Morita equivalent rings, both are semisimple complex algebras; the same is true of B and S. Moreover, their centers are isomorphic, 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.sub.1],..., [X.sub.r] where [X.sub.i] [congruent to] P [[cross product].sub.R] [W.sub.i] for each i. Suppose the inclusion matrix of B [??] A is given by A [[cross product].sub.B] [U.sub.i] [congruent to] [[cross product].sup.r.sub.j+1][b.sub.ij][X.sub.j]. Since

A [[cross product].sub.B] [U.sub.i] [congruent to] A [[cross product].sub.B]Q [[cross product].sub.S] [V.sub.i] [congruent to] P [[cross product].sub.R]R [[cross product].sub.S] [V.sub.I] [congruent to] [[cross product].sup.r.sub.j+1][r.sub.ij][X.sub.j]

this implies by Krull-Schmidt that the inclusion matrices ([b.sub.ij]) and ([r.sub.ij]) are equal. Thus the bipartite graphs are equal.

5. The functors F and G are tensor equivalences according to Lemma 2.2. Let [.sub.R][X.sub.R] be a bimodule. Note that [Res.sup.[A.sup.e].sub.[B.sup.e]](F(X)) = [.sub.B]P [[cross product].sub.R] X [[cross product].sub.R] [P*.sub.B] [congruent to] [.sub.B]Q [[cross product].sub.S] R [[cross product].sub.R] X [[cross product].sub.R] R [[cross product].sub.S] [Q*.sub.B] [CONGRUENT TO] G([Res.sup.[R.sup.e].sub.[S.sup.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([.sub.S][R.sub.S]) [CONGRUENT TO] [.sub.B][A.sub.B] and from the tensor functor property of G that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A computation similar to the one in (4) of this proof shows that the following rectangle is commutative:


where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([.sub.S][Z.sup.S]) := [.sub.R]R [[cross product].sub.S] Z [[cross product].sub.S] [R.sub.R]. Since F preserves tensor category unit objects, F([.sub.R][R.sub.R]) [congruent to] [.sub.A][A.sub.A]. Starting with [.sub.S][S.sub.S] [member of] [.sub.S][M.sub.S], the rectangle shows that F([.sub.R]R [[cross product].sub.s] [R.sub.R]) [congruent to] [.sub.A]A [[cross product].sub.B] [A.sub.A]. Starting with [R.sup.[cross product].sub.S.sup.(n)] [member of] [.sub.S][M.sub.S] in the rectangle, we note that for n [greater than or equal to] 1,

F([.sub.R][[R.sup.[cross product].sub.S.sup.(n+2)].sub.R]) [congruent to] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([A.sup.[cross product].sub.B.sup.(n)]) = [.sub.A][[A.sup.[cross product].sub.B.sup.(n+2)].sub.A].

7. Note the equivalence of bimodule categories H : [.sub.S][M.sub.R] [right arrow] [.sub.B][M.sub.A] given by H([.sub.S][W.sub.R]) := [.sub.B]Q [[cross product].sub.S]W [[cross product].sub.R] [P*.sub.a]. We claim that H([.sub.S][R.sub.R]) [congruent to] [.sub.B][A.sub.A]; moreover,

H([.sub.S][[R.sup.[cross product].sub.S.sup.(n)].sub.R]) [congruent to] [.sub.B][[A.sup.[cross product].sub.B.sup.(n)].sub.a] (33)

for all n [greater than or equal to] 1. This follows from the diagram below, commutative up to natural isomorphism.


which is established by a short computation using (2) in Prop. 5.3. Applied to [R.sup.[cross product].sub.S.sup.(n)] [member of] [.sub.R][M.sub.R], we obtain Eq. (33).

Note that the centralizer [R.sub.S] = {r [member of] R : [[for all].sub.s] [member of] S, rs = sr} is isomorphic to End ([.sub.S][R.sub.R]) [congruent to] [R.sub.S] via f [??] f (1). Recall that an equivalence H satisfies

End ([.sub.S][R.sub.R]) [congruent to] End (H([.sub.S][R.sub.R])) [congruent to] End ([.sub.B][A.sub.A]) [congruent to] [A.sub.B].

8. Similarly to Eq. (33), we establish that the equivalence of bimodule categories given by H' : [.sub.R][M.sub.S] [right arrow] [.sub.A][M.sub.B], [.sub.R][V.sub.S] [??] P [[cross product].sub.R] V [[cross product].sub.S] [Q.sup.*] satisfies

H' ([R.sup.[cross product].sub.S.sup.(n)]) [congruent to] [.sub.A][[A.sup.[cross product].sub.B.sup.(n)].sub.B] (35)

Of course, equivalences preserve similarity of modules since they are additive. Suppose [R.sup.[cross product].sub.S.sup.(n)] ~ [R.sup.[cross product].sub.S.sup.(n+1)] as R-S-bimodules, i.e., R | S has right depth 2n. Applying H', one obtains [A.sup.[cross product].sub.B.sup.(n)] ~ [A.sup.[cross product].sub.B.sup.(n+1)] as A-B-bimodules, i.e., A | B has right depth 2n. Similarly for left depth 2n using the equivalence H. Similarly, if R | S has depth 2n + 1, applying G we obtain that A | B has depth 2n + 1. Going in the reverse direction using [G.sup.-1], [H.sup.-1] , we obtain d(S, R) = d(B, A). Using F we likewise show that [d.sub.h](S, R) = [d.sub.h](B, A). *

5.1 Example: tower above Frobenius extension

A Frobenius extension A [??] B is characterized by any of the following four conditions [24]. First, that [A.sub.B] is finite projective and [.sub.B][A.sub.A] [congruent to] Hom ([A.sub.B], [B.sub.B]). Secondly, that [.sub.B]A is finite projective and [.sub.A][A.sub.B] [congruent to] Hom ([.sub.B]A, [.sub.B]B). Thirdly, that coinduction and induction of right (or left) B-modules into A-modules are naturally isomorphic functors. Fourth, there is a Frobenius coordinate system (E : A [right arrow] B; [x.sub.1], ..., [x.sub.m], [y.sub.1], ..., [y.sub.m] [member of] A), which satisfies ([[for all].sub.a] [member of] A)

E [member of] Hom ([.sub.B][A.sub.B], [.sub.B][B.sub.B]), [m.summation over (i=1)] E([ax.sub.i])[y.sub.i] = a = [m.summation over (i=1)] [x.sub.i]E([y.sub.i]a). (36)

These equations may be used to show that [[summation].sub.i] [x.sub.i] [cross product] [y.sub.i] .sub. [(A [[cross product].sub.B] A)].sup.A].

By [30, Lemma 4.1], a Frobenius extension A [??] B has both [A.sub.B] and [.sub.B]A generator modules if and only if the Frobenius homomorphism E : A [right arrow] B is surjective: although most Frobenius extensions in the literature are generator extensions, there is a somewhat pathological example in [24, 2.7] of a matrix algebra Frobenius extension with a non-surjective Frobenius homomorphism.

A Frobenius extension A [??] B enjoys an endomorphism ring theorem, which states that [A.sub.2] := End [A.sub.b] [??] A is itself a Frobenius extension, where the ring monomorphism A [right arrow] [A.sub.2] is the left multiplication mapping [lambda] : a [??] [[lambda].sub.a], [[lambda].sub.a] (x) = ax. It is worth noting that [lambda] is a left split A-monomorphism (by evaluation at [1.sub.A]) so [.sub.A][A.sub.2] is a generator. It is an exercise to check that [A.sub.2] [congruent to] A [[cross product].sub.B] A via f [??] [[summation].sub.i] f ([x.sub.i]) [[cross product].sub.B] [y.sub.i]; the induced ring structure on A [[cross product].sub.B] A is the "E-multiplication," given by

(a [[cross product].sub.B] c)(d [.sub.[cross product]B] e)= aE(cd) [[cross product].sub.B] e. (37)

The identity is given 1 = [[summation].sub.i] [x.sub.i] [[cross product].sub.B] [y.sub.i]. The Frobenius coordinate system for [A.sub.2] [??] [A.sub.1] is given by [E.sub.2](a [[cross product].sub.B] c) = ac (always surjective!) with dual bases {[x.sub.i] [[cross product].sub.B] 1} and {1 [[cross product].sub.B] [y.sub.i]}.

The tower of a Frobenius extension is obtained by iteration of the endomorphism ring and [lambda], obtaining a tower of Frobenius extensions; with the notation B := [A.sub.0], A := [A.sub.1] and defining [A.sub.n+1] = End [A.sub.nA.sub.n-1], we obtain the tower,

[A.sub.0] [??] [A.sub.1] [??] [A.sub.2] [??] ... [??] [A.sub.n] [??] [A.sub.n+1] [??] ... (38)

By transitivity of Frobenius extension or QF extension [42], all sub-extensions [A.sub.m] [??] [A.sub.m+n] in the tower are also Frobenius extensions. Note that [A.sub.n] [congruent to] [A.sup.[cross product].sub.B.sup.(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 data notation given above. Then [A.sub.n-1] [??] [A.sub.n-2] is Morita equivalent to [A.sub.n+1] [??] [A.sub.n] for all integers n > 1. Also A [??] B is Morita equivalent to [A.sub.3] [??] [A.sub.2] if the Frobenius homomorphism is epi.

Proof. It suffices to assume E : A [right arrow] B is surjective, let S = [A.sub.2] = End [A.sub.B], R = [A.sub.3], and show that B [??] A is Morita equivalent to [A.sub.2] [??] [A.sub.3]. Since A is a Frobenius extension of B with surjective Frobenius homomorphism, it follows that the module [A.sub.B] is a progenerator; since [A.sub.2] = End [A.sub.B], it follows that B and [A.sub.2] are Morita equivalent rings. Similarly, A and [A.sub.3] [congruent to] End A [[cross product].sub.B] [A.sub.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 [[cross product].sub.B] A. Thus [.sub.S]Q [[cross product].sub.B] [A.sub.A] [congruent to] [.sub.S][P.sub.A], the condition in the proposition for Morita equivalent ring extensions.

The theorem states in other words that the tower above a Frobenius extension has up to Morita equivalence period two. Note that consecutive ring extensions in the tower are almost never Morita equivalent: in [30, Example 1.12], the depth is d([S.sub.3], [S.sub.4]) = 5, but of its reflected graph, the depth is d(A, [A.sub.2]) = 6 (where A = C [S.sub.4], using the graph-theoretic depth calculation in [7, Section 3]).

5.2 Relative cyclic homology of ring extensions is Morita invariant

We extend a result in [23] that relative cyclic homology of a ring extension R [??] S and of its n x n-matrix ring extension [M.sub.n](R) [??] [M.sub.n](S) are isomorphic via a Dennis trace map adapted to this set-up. The relative cyclic homology (or any of its several variant homologies) is computed from cyclic modules

[Z.sub.n](R, S) := R [[cross product].sub.S.sup.e] [R.sup.[cross product].sub.s.sup.(n)],

which has the effect of considering tensor products of the natural bimodule [.sub.S][R.sub.S] with itself over S n + 1 times arranged in a circle (in place of a line). For each n [greater than or equal to] 0, there are n + 1 face maps are given by [d.sub.i] : [Z.sub.n] (R, S) [right arrow] [Z.sub.n-1] (R, S) defined from tensoring n - 1 copies of the [id.sub.S.sup.R.sub.S] with one copy of the multiplication [micro] [member of] Hom ([.sub.S]R [[cross product].sub.S] [R.sub.S], [.sub.S][R.sub.S]) at the ith position, there are n + 1 degeneracy mappings [s.sub.j] : [Z.sub.n] (R, S) [right arrow] [Z.sub.n+1] (R, S) by tensoring n copies of [id.sub.S.sup.R.sub.S] with one copy of the unit mapping [eta] [member of] Hom ([.sub.S][S.sub.S], [.sub.S][R.sub.S]) in the ith position, and a cyclic permutation [t.sub.n] : [Z.sub.n](R, S) [right arrow] [Z.sub.n](R, S) of order n + 1 (see [23] for the Connes cyclic object relations [10] and the textbook [35] for further details).

Suppose ring extensions R [??] S and A [??] B are Morita equivalent, and assume the same structural bimodules and module equivalences with notation as in this section. Now recall from the diagram (32) that the tensor equivalence G : [.sub.S][M.sub.S] [right arrow] [.sub.B][M.sub.B], defined by G(X) = Q [[cross product].sub.S] X [[cross product].sub.S] Q*, sends [.sub.S][R.sub.S] into [.sub.B][A.sub.B]. We note the following commutative diagram,


where A[b.sub.B] denotes [.sub.B][M.sub.B] [cross product][.sub.B.sup.E] [.sub.B][M.sub.B], a subcategory of abelian groups (and similarly for A[b.sub.S]), from a computation with X,Y [member of] [.sub.S][M.sub.S]:

G(X) [cross product][.sub.B.sup.E] G(Y) [congruent to] X [cross product][.sub.S] [Q.sup.*] [cross product][.sub.b] Q [cross product][.sub.S.sup.E] [Q.sup.*] [cross product][.sub.b]Q [cross product][.sub.S] Y

[congruent to] X [cross product][.sub.S] S [cross product][.sub.S.sup.E] S [cross product][.sub.S] Y [congruent to] X [cross product][.sub.S.sup.E] Y.

It follows that [Z.sub.n](R, S) [??] [Z.sub.n](A, B) via [??] (restricted to the cyclic modules) as abelian groups for each n [greater than or equal to] 0. Now [??] commutes with face maps since the functor G sends the multiplication of R [??] S,

[micro] [member of] Hom ([.sub.S]R [cross product][.sub.S] [R.sub.S], [.sub.S][R.sub.S]) [??] [micro] [member of] Hom ([.sub.B]A [cross product][.sub.B] [A.sub.B], [.sub.B][A.sub.B]),

the multiplication of the ring extension A [??] B. That [??] : [Z.sub.n] (R, S) [right arrow] [Z.sub.n] (A, B) commutes with the degeneracy maps follows from the functor G sending the unit [eta] [member of] Hom ([.sub.S][S.sub.S],[.sub.S][R.sub.S]) into the unit [eta] [member of] Hom ([.sub.B][B.sub.B], [.sub.B][A.sub.B]). That [??] : [Z.sub.n](R, S) [right arrow] [Z.sub.n] (A, B) commutes with the cyclic group action generator [t.sub.n] follows from G x G commuting with simple exchange X x Y [??] Y x X. We have sketched the proof of the 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 are isomorphic: H[C.sub.N](R, S) [congruent to] H[C.sub.N](A, B), all n [member of] N.

The isomorphism is given by a generalized Dennis trace mapping as follows. Suppose the S-bimodule isomorphism [Q.sub.*] [cross product][.sub.B] Q [??] S sends [[SIGMA].sup.R.sub.I=1] [q.sup.*.sub.i] [cross product] [q.sub.i] [??] [1.sub.S]. Then an isomorphism of cyclic modules [Z.sub.n] (A, B) [??] [Z.sub.n] (R, S) is given by


In the matrix example 5.1 of Morita equivalent ring extensions, where each [a.sub.i] denotes an n x n-matrix, this expression simplifies to the classical Dennis trace isomorphism of cyclic modules noted in [23],


5.3 Acknowledgements

The author thanks the organizers of the "New Trends in Hopf algebras and Tensor Categories" in Brussels, June 2-5, 2015, for a nice conference including Joost Vercruysse and Mio Iovanov for discussions about Proposition 5.4, item (3), Alberto Hernandez for interesting mathematical conversations about several subjects in this paper, and CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020, as well as Professor Manuel Delgado, and the Invited Scientist program of CMUP for financial support.


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Lars Kadison

Departamento de Matematica Faculdade de Ciencias da Universidade do Porto Rua Campo Alegre, 687 4169-007 Porto, Portugal


Received by the editors in November 2015.

Communicated by J. Vercruysse.

2010 Mathematics Subject Classification : 16D20, 16D90, 16T05, 18D10, 20C05.

Key words and phrases : subgroup depth, Morita equivalent ring extensions, Frobenius extension, semisimple extension, tensor category, core Hopf ideals, relative Maschke theorem.
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