# Subring Depth, Frobenius Extensions, and Towers

Authors: Lars Kadison
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

The minimum depthd(B,A) of a subring B⊆A introduced in the work of Boltje, Danz and Külshammer (2011) is studied and compared with the tower depth of a Frobenius extension. We show that d(B,A) < ∞ if A is a finite-dimensional algebra and Be has finite representation type. Some conditions in terms of depth and QF property are given that ensure that the modular function of a Hopf algebra restricts to the modular function of a Hopf subalgebra. If A⊇B is a QF extension, minimum left and right even subring depths are shown to coincide. If A⊇B is a Frobenius extension with surjective Frobenius, homomorphism, its subring depth is shown to coincide with its tower depth. Formulas for the ring, module, Frobenius and Temperley-Lieb structures are noted for the tower over a Frobenius extension in its realization as tensor powers. A depth 3 QF extension is embedded in a depth 2 QF extension; in turn certain depth n extensions embed in depth 3 extensions if they are Frobenius extensions or other special ring extensions with ring structures on their relative Hochschild bar resolution groups.


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## Body

## 1. Introduction and Preliminaries

A basic lemma in representation theory states that if a subalgebraB of a finite-dimensional algebra A has μ:A⊗BA→A,a⊗a′↦aa′ a split epimorphism of A-A-bimodules, then A has finite representation type if B has. Weakening the condition on μ to a split epimorphism of A-B-bimodules does not place any restriction on B⊆A, but the opposite hypothesis that a split monomorphism exists from A⊗BA into a multiple nA=A⊕⋯⊕A captures the notion of normality of a subalgebra in the context of group algebras [1], Hopf algebras [2], and semisimple algebras [3]. If A is a Frobenius extension of B, where AB is a progenerator module (but A and B may be infinite-dimensional algebras), the “depth two” condition as the opposite hypothesis is known as, implies that A is a Galois extension of B, where the bimodule endomorphism ring of the extension may be given the structure of a Hopf algebroid (which acts naturally on A with invariant subalgebra B) [4, 5]. Such theorems first appeared in [6, 7] for certain finite index subfactors of depth two. The left bialgebroid aspect of the definition of Hopf algebroid was influenced by a study of Lie groupoids in Poisson geometry [8]. The publication of [9] clarified the role played by Galois theory in depth two theory.After the focus on depth two, the study of how to generalize depth three and more from subfactor theory to algebra occurred in three stages after [10]. At first the depth two condition was generalized from a subalgebra pair B⊆A to a tower of three rings C⊆B⊆A [11]. This was applied to the tower of iterated right endomorphism rings above a Frobenius extension B⊆A⊆A1↪A2↪⋯, so that B⊆A has (tower) depth n if B↪An-3↪An-2 has the generalized depth two property (called a depth 3 tower in [11]). This yields a compact matrix inequality condition
(1.1)M[n+1]≤qM[n-1]
(some q∈ℕ) for when a subalgebra pair of semisimple complex algebras has depth n in terms of the inclusion matrix M, equivalently the incidence matrix of the Bratteli diagram of the inclusion B↪A [3, 18]. Since M[2]=MMt, M[3]=MMtM,…, already in this matrix condition the odd and even depth become distinguished from one another in terms of square and rectangular matrices. From [3], Boltje et al. [12] have extended the definition to a subring B⊆A, which has (right) depth 2n if the relative Hochschild n+1 bar resolution group Cn+1(A,B) maps as a split monomorphism into a multiple of a smaller group, qCm(A,B) as A-B-bimodules, and depth 2n+1 if this condition only holds as natural B-B-bimodules. Since subring B⊆A having depth m implies that it has depth m+1, the minimum depth d(B,A) is the more interesting positive integer.The algebraic definition of depth of subring pairs of Artin algebras is closely related to induced and restricted modules or characters in the case of group algebras. The depths of several class subgroups are recently computed, both as induced complex representations [3] and as induced representations of group algebras over an arbitrary ground ring [12]. For example, the minimum depth of the permutation groups Sn⊂Sn+1 is 2n-1 over any ground ring k and depends only on a combinatorial depth of a subgroup H<G defined in terms of G×H-sets and diagonal action in the same way as depth is defined for a subring [12]. The main theorem in [12] is that an extension k[G]⊇k[H] of finite group algebras over any ground ring k has finite depth, in fact bounded by twice the index [G:NG(H)] of the normalizer subgroup.The notion of subring depthd(B,A) in [12] is defined in equivalent terms in (1.7). In case B and A are semisimple complex algebras, it is shown in an appendix of [12] how subring depth equals the notion of depth based on induction-restriction table, equivalently inclusion matrix M in [3] and given in (1.1). Such a pair A⊇B is a special case of a split, separable Frobenius extension; in Theorem 5.2 we show that subring depth is equal to the tower depth of Frobenius extensions [11] satisfying only a generator module condition. The authors of [12] define left and right even depth and show these are the same on group algebra extensions; Theorem 3.2 shows this equality holds for any quasi-Frobenius (QF) extension.It is intriguing that the definition of subring depth makes use of the bar resolution groups of relative homological algebra, although in a fundamentally different way. The tower of iterated endomorphism rings above a ring extension becomes in the case of Frobenius extensions a tower of rings on the bar resolution groupsCn(A,B) (n=0,1,2,…) with Frobenius and Temperley-Lieb structures explicitly calculated from their more usual iterative definition in Section 4.1. At the same time Frobenius extensions of depth more than 2 are known to have depth 2 further out in the tower: we extend this observation in [11] with different proofs to include other ring extensions satisfying the hypotheses of Proposition 4.3. In Section 1 it is noted that a subalgebra B of a finite-dimensional algebra A has finite depth if its enveloping algebra Be has finite representation type.


### 1.1.H-Equivalent Modules

LetA be a ring. Two left A-modules, AN and AM, are said to be h-equivalent, denoted by AM~hAN, 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 AN⊕*≅AMr⇔(1.2)N∣rM⟺∃fi∈Hom(AM,AN),gi∈Hom(AN,AM):∑i=1rfi∘gi=idN.
Second, symmetrically there is s∈ℤ+ such that M∣sN. It is easy to extend this definition of h-equivalence (sometimes referred to as similarity) to h-equivalence of two objects in an abelian category and to show that it is an equivalence relation.If two modules areh-equivalent, AN~hAM, then they have Morita equivalent endomorphism rings, ℰN:=EndAN and ℰM:=EndAM, since a Morita context of bimodules is given by H(M,N):=Hom(AM,AN), which is an ℰN-ℰM-bimodule via composition, and the bimodule ℰMH(N,M)ℰN; these are progenerator modules, by applying to (1.2) or its reverse, M∣sN, any of the four Hom-functors such as Hom(A-,AM) from the category of left A-modules into the category of left EM-modules. Then, the explicit conditions on mappings for h-equivalence show that H(M,N)⊗ℰMH(N,M)→ℰN and the reverse mapping given by composition are surjections.The theory ofh-equivalent modules applies to bimodules TMB~hTNB by letting A=T⊗ℤBop, which sets up an equivalence of abelian categories between T-B-bimodules and left A-modules. Two additive functors F,G:𝒞↪𝒟 are h-equivalent if there are natural split epis F(X)n↪G(X) and G(X)m↪F(X) for all X in 𝒞. We leave the proof of the lemma below as an elementary exercise.Lemma 1.1.
Suppose twoA-modules are h-equivalent, M~hN, and two additive functors from A-modules to an abelian category are h-equivalent, F~hG. Then, F(M)~hG(N).For example, the following substitution in equations involving the~h-equivalence relation follows from the lemma:
(1.3)APT~hAQT,TUB~hTVB⟹AP⊗TUB~hAQ⊗TVB.Example 1.2.
SupposeA is a finite-dimensional algebra with indecomposable A-modules {Pα∣α∈I} (representatives from each isomorphism class for some index set I). By Krull-Schmidt finitely generated modules MA and NA have a unique factorization into a direct sum of multiples of finitely many indecomposable module components. Denote the indecomposable constituents of MA by Indec(M)={Pα∣[Pα,M]≠0}, where [Pα,M] is the number of factors in M isomorphic to Pα. Note that M∣qN for some positive q if and only if Indec(M)⊆Indec(N). It follows that M~hN if and only if Indec(M)=Indec(N).
SupposeAA=n1P1⊕⋯⊕nrPr is the decomposition of the regular module into its projective indecomposables. Let PA=P1⊕⋯⊕Pr. Then, PA and AA are h-equivalent, so that A and EndPA are Morita equivalent. The algebra EndPA is the basic algebra of A.


### 1.2. Depth Two

A subring pairB⊆A is said to have left depth 2 (or be a left depth two extension [4]) if A⊗BA~hA as natural B-A-bimodules. Right depth 2 is defined similarly in terms of h-equivalence of natural A-B-bimodules. In [4] it was noted that the left condition implies the right and conversely if A is a Frobenius extension of B. Also in [4] a Galois theory of Hopf algebroids was defined on the endomorphism ring H:=EndBAB as total ring and the centralizer R:=AB as base ring. The antipode is the natural anti-isomorphism stemming from following the arrows:
(1.4)EndAB⟶≅A⊗BA⟶≅(EndBA)op
restricted to the intersection EndBAB=EndAB∩EndBA.The Galois extension properties of a depth two extensionA⊇B are as follows. If AB is faithfully flat, balanced or B equals its double centralizer in A, the natural action of H on A has invariant subalgebra AH satisfying the Galois property of AH=B. Also the well-known Galois property of the endomorphism ring as a cross-product holds: the right endomorphism ring EndAB≅A#H, where the latter has smash product ring structure on A⊗RH [4]. There is also a duality structure by going a step further along in the tower above B⊆A↪EndAB↪EndA⊗BAA, where the Hopf algebroid H′:=(A⊗BA)B is the R-dual of H and acts naturally on EndAB in such a way that End(A⊗BA)A has a smash product ring structure [4].Conversely, Galois extensions have depth2. For example, an H-comodule algebra A with invariant subalgebra B and finite-dimensional Hopf algebra H over a base field k, which has a Galois isomorphism from A⊗BA→≅A⊗kH given by a′⊗a↦a′a(0)⊗a(1), satisfies (strongly) the depth two condition A⊗BA≅AdimH as A-B-bimodules. The Hopf subalgebras within a finite-dimensional Hopf algebra, which have depth 2, are precisely the normal Hopf subalgebras; if normal, it has depth 2 by applying the observation about Hopf-Galois extension just made. The converse follows from an argument noted in Boltje-Külshammer [2], which divides the normality notion into right and left (like the notion of depth 2), where left normal is invariance under the left adjoint action. In the context of an augmented algebra A their results extend to the following proposition. Let ε:A→k be an algebra homomorphism into the ground field k. Let A+ denote kerε, and, for a subalgebra B⊆A, let B+ denote kerε∩B.Proposition 1.3.
SupposeB⊆A is a subalgebra of an augmented algebra. If B⊆A has right depth 2, then AB+⊆B+A.The proof of this proposition is an exercise in tensoring both sides ofA⊗BA⊕*≅qA by the unit A-module k, then passing to the annihilator ideal of a module and a direct summand. The opposite inclusion is of course satisfied by a left depth 2 extension of augmented algebras.Example 1.4.
LetA=Tn(k) be the algebra of n by n upper triangular matrices where n>1, and B=Dn(k) the subalgebra of diagonal matrices. Note that there are n augmentations εi:A→k given by εi(X)=Xii, and each of the Bi+ satisfies the inclusions above if left or right depth two. This is a clear contradiction, thus d(B,A)>2. We will see below that d(B,A)=3.Also subalgebra pairs of semisimple complex algebras have depth2 exactly when they are normal in a classical sense of Rieffel. The theorem in [3] is given below and one may prove the forward direction in the manner indicated for the previous proposition.Theorem 1.5 see ([3] Theorem 4.6).
SupposeB⊆A is a subalgebra pair of semisimple complex algebras. Then, B⊆A has depth 2 if and only if, for every maximal ideal I in A, one has A(I∩B)=(I∩B)A.For example, subalgebra pairs of semisimple complex algebras that satisfy this normality condition are then by our sketch above examples of weak Hopf-Galois extensions, since the centralizerR mentioned above is semisimple (see Kaplansky's Fields and Rings for a C*-theoretic reason), the extension is Frobenius [18], and weak Hopf algebras are equivalently Hopf algebroids over a separable base algebra [4].


### 1.3. Subring Depth

Throughout this paper, letA be a unital associative ring and B⊆A a subring where 1B=1A. Note the natural bimodules BAB obtained by restriction of the natural A-A-bimodule (briefly A-bimodule) A, also to the natural bimodules BAA, AAB or BAB, which are referred to with no further ado.LetC0(A,B)=B, and, for n≥1,
(1.5)Cn(A,B)=A⊗B⋯⊗BA(ntimesA),
For n≥1, Cn(A,B) has a natural A-bimodule structure, which restricts to B-A-, A-B-, and B-bimodule structures occurring in the next definition.Definition 1.6.
The subringB⊆A has depth 2n+1≥1 if as B-bimodules Cn(A,B)~hCn+1(A,B). The subring B⊆A has left (resp., right) depth 2n≥2 if Cn(A,B)~hCn+1(A,B) as B-A-bimodules (resp., A-B-bimodules).It is clear that ifB⊆A has either left or right depth 2n, it has depth 2n+1 by restricting the h-equivalence condition to B-bimodules. If it has depth 2n+1, it has depth 2n+2 by tensoring the h-equivalence by -⊗BA or A⊗B-. The minimum depth is denoted by d(B,A); if B⊆A has no finite depth, write d(B,A)=∞.Note that the minimum left and right minimumeven depths may differ by 2 (in which case d(B,A) is the lesser of the two). In the next section we provide a general condition, which includes a Hopf subalgebra pair B⊆A of symmetric (Frobenius) algebras, where the left and right minimum even depths coincide.Also note that a subalgebra pair of Artin algebrasB⊆A have depth 2n+1 if and only if the indecomposable module constituents of Cn+m(A,B) remain the same for all m≥0 as those already found in Cn(A,B) (see Example 1.2). This corresponds well with the classical notion of finite depth in subfactor theory.Example 1.7.
Again letA=Tn(k) and B=Dn(k)≅kn, where n>1. Let eij denote the matrix units, ki the n simple B-modules, and kij for 1≤i≤j≤n the n(n+1)/2 simple components of BAB. Note that A⊗BA as a B-bimodule has components keis⊗Besj≅kij where i≤s≤j, so A⊗BA∣nA as B-bimodules. Thus, d(B,A)≤3. But d(B,A)≠2 by the remark following Proposition 1.3; then d(B,A)=3.


### 1.4.ℋ-Depth

A subringB⊆A has ℋ-depth 2n-1 if Cn+1(A,B)~hCn(A,B) 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 dℋ(B,A) if it exists. Note that the definition of ℋ-depth 2n-1 is equivalent to the condition on a subring B⊆A that Cn+1(A,B)∣qCn(A,B) for some q∈ℕ. This is clear for n≠2 since Cn(A,B)∣Cn+1(A,B). For n=1, the H-separability condition
(1.6)AA⊗BAA⊕*≅AAAq
implies the separability condition AAA⊕*≅AA⊗BAA as argued in the paper [13] by Hirata. The notion of ℋ-depth is studied in [14] where it is noted that |dℋ(B,A)-d(B,A)|≤2 if one or the other minimum depth is finite. See Section 2 for which Hopf subalgebras satisfy the dℋ(B,A)=1 condition in (1.6).Remark 1.8.
SupposeB is a subring of A. The minimum depth of the subring B⊆A as defined in Boltje-Danz-Külshammer [12] coincides with d(B,A). In fact, for n>0, the depth 2n+1 condition in [12] is that for some q∈ℤ+(1.7)Cn+1(A,B)∣qCn(A,B)
as B-bimodules. The left depth 2n condition in [12] is (1.7) more strongly as natural B-A-bimodules (and as A-B-bimodules for the right depth 2n condition). But (using a pair of classical face and degeneracy maps of homological algebra) we always have Cn(A,B)∣Cn+1(A,B) as A-B-, B-A-, or B-bimodules, so that the depth 2n as well as 2n+1 conditions coincide in the case of subring having depth 2n and 2n+1 conditions above.
Note that depth1 in this paper is equivalent to the subring depth 1 notion in, for example, [4, 12, 15] since A is h-equivalent to B as B-bimodules if and only if A is centrally projective over B (i.e., A∣qB as B-bimodules). This follows from the lemma below.Lemma 1.9.
SupposeB is a subring of ring A such that BAB∣mBBB for some integer m≥1. Then, BBB∣BAB.Proof.
From the central projectivity condition onA, one obtains m maps hi∈Hom(BAB,BBB) and m maps gi∈Hom(BBB,BAB)↦≅vi∈AB such that ∑i=1m‍vihi(a)=a for every a∈A. It follows that A≅B⊗Z(B)AB since hi(AB)⊆Z(B). Note that restricting the equation to the centralizer AB shows that AB is a finitely generated projective Z(B)-module. But Z(B)⊆AB is a commutative subring, whence AB is a generator Z(B)-module. From Z(B)⊕*≅nAB for some positive integer n, it follows from the tensor algebra decomposition of A that BBB∣nBAB. Whence there are n maps fi∈Hom(BAB,BBB) and n elements ri∈AB such that ∑i=1n‍fi(ri)=1A. Define a (condition expectation or) bimodule projection E(a):=∑i=1n‍fi(ari) of A onto B.Example 1.10.
The paper [12] asks in its introduction about the depth d(B,A) of invariant subrings in classical invariant theory, where K is a field, A=K[X1,…,Xn], B=k[X1,…,Xn]G and G is a finite group in GLn(K) acting by linear substitution of the variables. In any case AB is finitely generated and B is a finitely generated affine K-algebra. We note here that if G is generated by pseudoreflections (such as G=Sn, the symmetric group) and the characteristic of K is coprime to |G|, B is itself an n-variable polynomial algebra and A is a free B-module; consequences of the Shephard-Todd Theorem [16, 17]. Since A is a commutative algebra, it follows that d(B,A)=1.Example 1.11.
LetB⊆A be a subring pair of semisimple complex algebras. Then, the minimum depth d(B,A) may be computed from the inclusion matrix M, alternatively an r-by-s induction-restriction table of rB-simples induced to nonnegative integer linear combination of sA-simples along rows, and by Frobenius reciprocity, columns show restriction of A-simples in terms of B-simples. The procedure to obtain d(B,A) given in the paper [3] is to compute the bracketed powers of M given in Section 1, and check for which nth power of M satisfies the matrix inequality in (1.1): d(B,A) is the least such n by results in [12, appendix] (or Theorem 5.2 below combined with [3, 18]). One may note that d(B,A)≤2d-1 where MMt has degree d minimal polynomial [3]. A GAP subprogram exists to compute d(B,A) for a complex group algebra extension by converting character tables to an induction-restriction table M, then counting the number of zero entries in the bracketed powers of M, which decreases nonstrictly with increasing even and odd powers of M, d(B,A) being the least point of no decrease.
In terms of the bipartite graph of the inclusionB⊆A, d(B,A) is the lesser of the minimum odd depth and the minimum even depth [3]. The matrix M is an incidence matrix of this bipartite graph if all entries greater than 1 are changed to 1, while zero entries are retained as 0: let the B-simples be represented by r black dots in a bottom row of the graph and A-simples by s white dots in a top row, connected by edges joining black and white dots (or not) according to the 0-1-matrix entries obtained from M. The minimum odd depth of the bipartite graph is 1 plus the diameter in edges of the row of black dots (indeed an odd number), while the minimum even depth is 2 plus the largest of the diameters of the bottom row where a subset of black dots under one white dot is identified with one another.
For example, letA=ℂS4, the complex group algebra of the permutation group on four letters, and B=ℂS3. The inclusion diagram pictured in Figure with the degrees of the irreducible representations is determined from the character tables of S3 and S4 or the branching rule (for the Young diagrams labelled by the partitions of n and representing the irreducibles of Sn).
This graph has minimum odd depth 5 and minimum even depth 6, whenced(B,A)=5.Example 1.12.
The induction-restriction tableM of the inclusion of permutation groups Sn×Sm<Sn+m via
(1.8)(σ,τ)↦(1⋯nn+1⋯n+mσ(1)⋯σ(n)n+τ(1)⋯n+τ(m))
may be computed combinatorially from the Littlewood-Richardson coefficients cμνγ∈ℕ, where μ is partition of n, ν=(ν1,…,νm) a partition of m, and λ a partition of n+m. Briefly, the coefficient number cμνγ is zero if γ does not contain μ or is the number of Littlewood-Richardson fillings with content ν of γ with μ removed. A Littlewood-Richardson filling of a skew Young tableau is with integers i=1,2,…,m occuring νi times in rows that are weakly increasing from left to right, columns are strictly increasing from top to bottom, and the entries when listed from right to left in rows, top to bottom row, form a lattice word [19].
For example, computing the matrixM for the subgroup S2×S3<S5 with respect to the ordered bases of irreducible characters of the subgroup λ(12)×μ(13), λ(12)×μ(2,1), λ(12)×μ(3), λ(2)×μ(13), λ(2)×μ(2,1), λ(2)×μ(3) and of the group γ(15),γ(2,13), γ(22,1), γ(3,2), γ(3,12), γ(4,1), γ(5) yields
(1.9)M=(111000001111000000110010010000111100001011).
The bracketed powers of M satisfy a minimum depth 5 inequality (1.1) so that d(S2×S3,S5)=5. We mentioned before that d(Sn×S1,Sn+1)=2n-1 [3, 12]; however, a formula for d(Sn×Sm,Sn+m) is not known.


### 1.5. Finite Depth and Finite Representation Type

For the next proposition we adopt the notationBe for the (enveloping) algebra B⊗kBop and recall that a finite-dimensional algebra has finite representation type if it only has finitely many isomorphism classes of indecomposable modules.For example, a group algebra over a base field of characteristicp has finite representation type if and only if its Sylow p-subgroup is cyclic. Thus, B having finite representation type does not imply that Be has finite representation type.Proposition 1.13.
SupposeB⊆A is a subalgebra pair of finite-dimensional algebras where Be has in all r indecomposable Be-module isomorphism classes. Then, d(B,A)≤2r+1.Proof.
This follows from the observation in Example1.2 that since Cn(A,B) is the image of Cn+1(A,B) under an obvious split epimorphism of Be-modules (equivalently, B-bimodules), there is an increasing chain of subset inclusions
(1.10)Indec(A)⊆Indec(A⊗BA)⊆Indec(A⊗BA⊗BA)⊆⋯,
which stops strictly increasing in at most r steps. When Indec(Cn(A,B))=Indec(Cn+1(A,B)), then Cn(A,B)~hCn+1(A,B) as Be-modules, whence A⊇B has depth 2n+1≤2r+1.Remarkably, the result in [12] is that all finite group algebra pairs have finite depth. The proposition says something about finite depth of interesting classes of finite-dimensional Hopf algebra pairs B⊆A, where research on which Hopf algebras have finite representation type is a current topic (although the paper [20] studies how tensor algebras seldom have finite representation type when the component algebras are not semisimple). (Note that Be is a Hopf algebra and semisimple if B is so.) For example, we have the following corollary.Corollary 1.14.
SupposeB is a semisimple Hopf subalgebra in a finite dimensional Hopf algebra A. Suppose that B has n nonisomorphic simple modules. Then, d(B,A)≤2n2+1.


## 1.1.H-Equivalent Modules

LetA be a ring. Two left A-modules, AN and AM, are said to be h-equivalent, denoted by AM~hAN, 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 AN⊕*≅AMr⇔(1.2)N∣rM⟺∃fi∈Hom(AM,AN),gi∈Hom(AN,AM):∑i=1rfi∘gi=idN.
Second, symmetrically there is s∈ℤ+ such that M∣sN. It is easy to extend this definition of h-equivalence (sometimes referred to as similarity) to h-equivalence of two objects in an abelian category and to show that it is an equivalence relation.If two modules areh-equivalent, AN~hAM, then they have Morita equivalent endomorphism rings, ℰN:=EndAN and ℰM:=EndAM, since a Morita context of bimodules is given by H(M,N):=Hom(AM,AN), which is an ℰN-ℰM-bimodule via composition, and the bimodule ℰMH(N,M)ℰN; these are progenerator modules, by applying to (1.2) or its reverse, M∣sN, any of the four Hom-functors such as Hom(A-,AM) from the category of left A-modules into the category of left EM-modules. Then, the explicit conditions on mappings for h-equivalence show that H(M,N)⊗ℰMH(N,M)→ℰN and the reverse mapping given by composition are surjections.The theory ofh-equivalent modules applies to bimodules TMB~hTNB by letting A=T⊗ℤBop, which sets up an equivalence of abelian categories between T-B-bimodules and left A-modules. Two additive functors F,G:𝒞↪𝒟 are h-equivalent if there are natural split epis F(X)n↪G(X) and G(X)m↪F(X) for all X in 𝒞. We leave the proof of the lemma below as an elementary exercise.Lemma 1.1.
Suppose twoA-modules are h-equivalent, M~hN, and two additive functors from A-modules to an abelian category are h-equivalent, F~hG. Then, F(M)~hG(N).For example, the following substitution in equations involving the~h-equivalence relation follows from the lemma:
(1.3)APT~hAQT,TUB~hTVB⟹AP⊗TUB~hAQ⊗TVB.Example 1.2.
SupposeA is a finite-dimensional algebra with indecomposable A-modules {Pα∣α∈I} (representatives from each isomorphism class for some index set I). By Krull-Schmidt finitely generated modules MA and NA have a unique factorization into a direct sum of multiples of finitely many indecomposable module components. Denote the indecomposable constituents of MA by Indec(M)={Pα∣[Pα,M]≠0}, where [Pα,M] is the number of factors in M isomorphic to Pα. Note that M∣qN for some positive q if and only if Indec(M)⊆Indec(N). It follows that M~hN if and only if Indec(M)=Indec(N).
SupposeAA=n1P1⊕⋯⊕nrPr is the decomposition of the regular module into its projective indecomposables. Let PA=P1⊕⋯⊕Pr. Then, PA and AA are h-equivalent, so that A and EndPA are Morita equivalent. The algebra EndPA is the basic algebra of A.


## 1.2. Depth Two

A subring pairB⊆A is said to have left depth 2 (or be a left depth two extension [4]) if A⊗BA~hA as natural B-A-bimodules. Right depth 2 is defined similarly in terms of h-equivalence of natural A-B-bimodules. In [4] it was noted that the left condition implies the right and conversely if A is a Frobenius extension of B. Also in [4] a Galois theory of Hopf algebroids was defined on the endomorphism ring H:=EndBAB as total ring and the centralizer R:=AB as base ring. The antipode is the natural anti-isomorphism stemming from following the arrows:
(1.4)EndAB⟶≅A⊗BA⟶≅(EndBA)op
restricted to the intersection EndBAB=EndAB∩EndBA.The Galois extension properties of a depth two extensionA⊇B are as follows. If AB is faithfully flat, balanced or B equals its double centralizer in A, the natural action of H on A has invariant subalgebra AH satisfying the Galois property of AH=B. Also the well-known Galois property of the endomorphism ring as a cross-product holds: the right endomorphism ring EndAB≅A#H, where the latter has smash product ring structure on A⊗RH [4]. There is also a duality structure by going a step further along in the tower above B⊆A↪EndAB↪EndA⊗BAA, where the Hopf algebroid H′:=(A⊗BA)B is the R-dual of H and acts naturally on EndAB in such a way that End(A⊗BA)A has a smash product ring structure [4].Conversely, Galois extensions have depth2. For example, an H-comodule algebra A with invariant subalgebra B and finite-dimensional Hopf algebra H over a base field k, which has a Galois isomorphism from A⊗BA→≅A⊗kH given by a′⊗a↦a′a(0)⊗a(1), satisfies (strongly) the depth two condition A⊗BA≅AdimH as A-B-bimodules. The Hopf subalgebras within a finite-dimensional Hopf algebra, which have depth 2, are precisely the normal Hopf subalgebras; if normal, it has depth 2 by applying the observation about Hopf-Galois extension just made. The converse follows from an argument noted in Boltje-Külshammer [2], which divides the normality notion into right and left (like the notion of depth 2), where left normal is invariance under the left adjoint action. In the context of an augmented algebra A their results extend to the following proposition. Let ε:A→k be an algebra homomorphism into the ground field k. Let A+ denote kerε, and, for a subalgebra B⊆A, let B+ denote kerε∩B.Proposition 1.3.
SupposeB⊆A is a subalgebra of an augmented algebra. If B⊆A has right depth 2, then AB+⊆B+A.The proof of this proposition is an exercise in tensoring both sides ofA⊗BA⊕*≅qA by the unit A-module k, then passing to the annihilator ideal of a module and a direct summand. The opposite inclusion is of course satisfied by a left depth 2 extension of augmented algebras.Example 1.4.
LetA=Tn(k) be the algebra of n by n upper triangular matrices where n>1, and B=Dn(k) the subalgebra of diagonal matrices. Note that there are n augmentations εi:A→k given by εi(X)=Xii, and each of the Bi+ satisfies the inclusions above if left or right depth two. This is a clear contradiction, thus d(B,A)>2. We will see below that d(B,A)=3.Also subalgebra pairs of semisimple complex algebras have depth2 exactly when they are normal in a classical sense of Rieffel. The theorem in [3] is given below and one may prove the forward direction in the manner indicated for the previous proposition.Theorem 1.5 see ([3] Theorem 4.6).
SupposeB⊆A is a subalgebra pair of semisimple complex algebras. Then, B⊆A has depth 2 if and only if, for every maximal ideal I in A, one has A(I∩B)=(I∩B)A.For example, subalgebra pairs of semisimple complex algebras that satisfy this normality condition are then by our sketch above examples of weak Hopf-Galois extensions, since the centralizerR mentioned above is semisimple (see Kaplansky's Fields and Rings for a C*-theoretic reason), the extension is Frobenius [18], and weak Hopf algebras are equivalently Hopf algebroids over a separable base algebra [4].


## 1.3. Subring Depth

Throughout this paper, letA be a unital associative ring and B⊆A a subring where 1B=1A. Note the natural bimodules BAB obtained by restriction of the natural A-A-bimodule (briefly A-bimodule) A, also to the natural bimodules BAA, AAB or BAB, which are referred to with no further ado.LetC0(A,B)=B, and, for n≥1,
(1.5)Cn(A,B)=A⊗B⋯⊗BA(ntimesA),
For n≥1, Cn(A,B) has a natural A-bimodule structure, which restricts to B-A-, A-B-, and B-bimodule structures occurring in the next definition.Definition 1.6.
The subringB⊆A has depth 2n+1≥1 if as B-bimodules Cn(A,B)~hCn+1(A,B). The subring B⊆A has left (resp., right) depth 2n≥2 if Cn(A,B)~hCn+1(A,B) as B-A-bimodules (resp., A-B-bimodules).It is clear that ifB⊆A has either left or right depth 2n, it has depth 2n+1 by restricting the h-equivalence condition to B-bimodules. If it has depth 2n+1, it has depth 2n+2 by tensoring the h-equivalence by -⊗BA or A⊗B-. The minimum depth is denoted by d(B,A); if B⊆A has no finite depth, write d(B,A)=∞.Note that the minimum left and right minimumeven depths may differ by 2 (in which case d(B,A) is the lesser of the two). In the next section we provide a general condition, which includes a Hopf subalgebra pair B⊆A of symmetric (Frobenius) algebras, where the left and right minimum even depths coincide.Also note that a subalgebra pair of Artin algebrasB⊆A have depth 2n+1 if and only if the indecomposable module constituents of Cn+m(A,B) remain the same for all m≥0 as those already found in Cn(A,B) (see Example 1.2). This corresponds well with the classical notion of finite depth in subfactor theory.Example 1.7.
Again letA=Tn(k) and B=Dn(k)≅kn, where n>1. Let eij denote the matrix units, ki the n simple B-modules, and kij for 1≤i≤j≤n the n(n+1)/2 simple components of BAB. Note that A⊗BA as a B-bimodule has components keis⊗Besj≅kij where i≤s≤j, so A⊗BA∣nA as B-bimodules. Thus, d(B,A)≤3. But d(B,A)≠2 by the remark following Proposition 1.3; then d(B,A)=3.


## 1.4.ℋ-Depth

A subringB⊆A has ℋ-depth 2n-1 if Cn+1(A,B)~hCn(A,B) 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 dℋ(B,A) if it exists. Note that the definition of ℋ-depth 2n-1 is equivalent to the condition on a subring B⊆A that Cn+1(A,B)∣qCn(A,B) for some q∈ℕ. This is clear for n≠2 since Cn(A,B)∣Cn+1(A,B). For n=1, the H-separability condition
(1.6)AA⊗BAA⊕*≅AAAq
implies the separability condition AAA⊕*≅AA⊗BAA as argued in the paper [13] by Hirata. The notion of ℋ-depth is studied in [14] where it is noted that |dℋ(B,A)-d(B,A)|≤2 if one or the other minimum depth is finite. See Section 2 for which Hopf subalgebras satisfy the dℋ(B,A)=1 condition in (1.6).Remark 1.8.
SupposeB is a subring of A. The minimum depth of the subring B⊆A as defined in Boltje-Danz-Külshammer [12] coincides with d(B,A). In fact, for n>0, the depth 2n+1 condition in [12] is that for some q∈ℤ+(1.7)Cn+1(A,B)∣qCn(A,B)
as B-bimodules. The left depth 2n condition in [12] is (1.7) more strongly as natural B-A-bimodules (and as A-B-bimodules for the right depth 2n condition). But (using a pair of classical face and degeneracy maps of homological algebra) we always have Cn(A,B)∣Cn+1(A,B) as A-B-, B-A-, or B-bimodules, so that the depth 2n as well as 2n+1 conditions coincide in the case of subring having depth 2n and 2n+1 conditions above.
Note that depth1 in this paper is equivalent to the subring depth 1 notion in, for example, [4, 12, 15] since A is h-equivalent to B as B-bimodules if and only if A is centrally projective over B (i.e., A∣qB as B-bimodules). This follows from the lemma below.Lemma 1.9.
SupposeB is a subring of ring A such that BAB∣mBBB for some integer m≥1. Then, BBB∣BAB.Proof.
From the central projectivity condition onA, one obtains m maps hi∈Hom(BAB,BBB) and m maps gi∈Hom(BBB,BAB)↦≅vi∈AB such that ∑i=1m‍vihi(a)=a for every a∈A. It follows that A≅B⊗Z(B)AB since hi(AB)⊆Z(B). Note that restricting the equation to the centralizer AB shows that AB is a finitely generated projective Z(B)-module. But Z(B)⊆AB is a commutative subring, whence AB is a generator Z(B)-module. From Z(B)⊕*≅nAB for some positive integer n, it follows from the tensor algebra decomposition of A that BBB∣nBAB. Whence there are n maps fi∈Hom(BAB,BBB) and n elements ri∈AB such that ∑i=1n‍fi(ri)=1A. Define a (condition expectation or) bimodule projection E(a):=∑i=1n‍fi(ari) of A onto B.Example 1.10.
The paper [12] asks in its introduction about the depth d(B,A) of invariant subrings in classical invariant theory, where K is a field, A=K[X1,…,Xn], B=k[X1,…,Xn]G and G is a finite group in GLn(K) acting by linear substitution of the variables. In any case AB is finitely generated and B is a finitely generated affine K-algebra. We note here that if G is generated by pseudoreflections (such as G=Sn, the symmetric group) and the characteristic of K is coprime to |G|, B is itself an n-variable polynomial algebra and A is a free B-module; consequences of the Shephard-Todd Theorem [16, 17]. Since A is a commutative algebra, it follows that d(B,A)=1.Example 1.11.
LetB⊆A be a subring pair of semisimple complex algebras. Then, the minimum depth d(B,A) may be computed from the inclusion matrix M, alternatively an r-by-s induction-restriction table of rB-simples induced to nonnegative integer linear combination of sA-simples along rows, and by Frobenius reciprocity, columns show restriction of A-simples in terms of B-simples. The procedure to obtain d(B,A) given in the paper [3] is to compute the bracketed powers of M given in Section 1, and check for which nth power of M satisfies the matrix inequality in (1.1): d(B,A) is the least such n by results in [12, appendix] (or Theorem 5.2 below combined with [3, 18]). One may note that d(B,A)≤2d-1 where MMt has degree d minimal polynomial [3]. A GAP subprogram exists to compute d(B,A) for a complex group algebra extension by converting character tables to an induction-restriction table M, then counting the number of zero entries in the bracketed powers of M, which decreases nonstrictly with increasing even and odd powers of M, d(B,A) being the least point of no decrease.
In terms of the bipartite graph of the inclusionB⊆A, d(B,A) is the lesser of the minimum odd depth and the minimum even depth [3]. The matrix M is an incidence matrix of this bipartite graph if all entries greater than 1 are changed to 1, while zero entries are retained as 0: let the B-simples be represented by r black dots in a bottom row of the graph and A-simples by s white dots in a top row, connected by edges joining black and white dots (or not) according to the 0-1-matrix entries obtained from M. The minimum odd depth of the bipartite graph is 1 plus the diameter in edges of the row of black dots (indeed an odd number), while the minimum even depth is 2 plus the largest of the diameters of the bottom row where a subset of black dots under one white dot is identified with one another.
For example, letA=ℂS4, the complex group algebra of the permutation group on four letters, and B=ℂS3. The inclusion diagram pictured in Figure with the degrees of the irreducible representations is determined from the character tables of S3 and S4 or the branching rule (for the Young diagrams labelled by the partitions of n and representing the irreducibles of Sn).
This graph has minimum odd depth 5 and minimum even depth 6, whenced(B,A)=5.Example 1.12.
The induction-restriction tableM of the inclusion of permutation groups Sn×Sm<Sn+m via
(1.8)(σ,τ)↦(1⋯nn+1⋯n+mσ(1)⋯σ(n)n+τ(1)⋯n+τ(m))
may be computed combinatorially from the Littlewood-Richardson coefficients cμνγ∈ℕ, where μ is partition of n, ν=(ν1,…,νm) a partition of m, and λ a partition of n+m. Briefly, the coefficient number cμνγ is zero if γ does not contain μ or is the number of Littlewood-Richardson fillings with content ν of γ with μ removed. A Littlewood-Richardson filling of a skew Young tableau is with integers i=1,2,…,m occuring νi times in rows that are weakly increasing from left to right, columns are strictly increasing from top to bottom, and the entries when listed from right to left in rows, top to bottom row, form a lattice word [19].
For example, computing the matrixM for the subgroup S2×S3<S5 with respect to the ordered bases of irreducible characters of the subgroup λ(12)×μ(13), λ(12)×μ(2,1), λ(12)×μ(3), λ(2)×μ(13), λ(2)×μ(2,1), λ(2)×μ(3) and of the group γ(15),γ(2,13), γ(22,1), γ(3,2), γ(3,12), γ(4,1), γ(5) yields
(1.9)M=(111000001111000000110010010000111100001011).
The bracketed powers of M satisfy a minimum depth 5 inequality (1.1) so that d(S2×S3,S5)=5. We mentioned before that d(Sn×S1,Sn+1)=2n-1 [3, 12]; however, a formula for d(Sn×Sm,Sn+m) is not known.


## 1.5. Finite Depth and Finite Representation Type

For the next proposition we adopt the notationBe for the (enveloping) algebra B⊗kBop and recall that a finite-dimensional algebra has finite representation type if it only has finitely many isomorphism classes of indecomposable modules.For example, a group algebra over a base field of characteristicp has finite representation type if and only if its Sylow p-subgroup is cyclic. Thus, B having finite representation type does not imply that Be has finite representation type.Proposition 1.13.
SupposeB⊆A is a subalgebra pair of finite-dimensional algebras where Be has in all r indecomposable Be-module isomorphism classes. Then, d(B,A)≤2r+1.Proof.
This follows from the observation in Example1.2 that since Cn(A,B) is the image of Cn+1(A,B) under an obvious split epimorphism of Be-modules (equivalently, B-bimodules), there is an increasing chain of subset inclusions
(1.10)Indec(A)⊆Indec(A⊗BA)⊆Indec(A⊗BA⊗BA)⊆⋯,
which stops strictly increasing in at most r steps. When Indec(Cn(A,B))=Indec(Cn+1(A,B)), then Cn(A,B)~hCn+1(A,B) as Be-modules, whence A⊇B has depth 2n+1≤2r+1.Remarkably, the result in [12] is that all finite group algebra pairs have finite depth. The proposition says something about finite depth of interesting classes of finite-dimensional Hopf algebra pairs B⊆A, where research on which Hopf algebras have finite representation type is a current topic (although the paper [20] studies how tensor algebras seldom have finite representation type when the component algebras are not semisimple). (Note that Be is a Hopf algebra and semisimple if B is so.) For example, we have the following corollary.Corollary 1.14.
SupposeB is a semisimple Hopf subalgebra in a finite dimensional Hopf algebra A. Suppose that B has n nonisomorphic simple modules. Then, d(B,A)≤2n2+1.


## 2.   When Frobenius Extensions of the Second Kind Are Ordinary

A (proper) ring extensionA⊇B is a subring or more generally a monomorphism ι:B↪A, which is equivalent to a subring ι(B)⊆A. Restricted modules such as Aι(B) and pullback modules AB are identified, and these are the type of modules we refer to below unless otherwise stated. (Almost all that we have to say holds for a ring homomorphism B→A and its pullback modules such as AB; however, certain conditions needed below such as AB is a generator imply that B→A is monic.)A ring extensionA⊇B is a left QF extension if the module BA is finitely generated projective and the natural A-B-bimodules satisfy A∣qHom(BA,BB) for some positive integer q. A right QF extension is oppositely defined. A QF extensionA⊇B is both a left and right QF extension and may be characterized by both AB and BA being finite projective, and two h-equivalences of bimodules given by AAB~hAHom(BA,BB)B and (BAA~hHomB(AB,BB)A [21]. For example, a Frobenius extension A⊇B is a QF extension since it is left and right finite projective and satisfies the stronger conditions that A is isomorphic to its right B-dual A* and its left B-dual *A as natural B-A-bimodules, respectively A-B-bimodules; the more precise definition is given in the next section.


### 2.1.β-Frobenius Extensions

In Hopf algebras and quantum algebras, examples of Frobenius extensions often occur with a twist foreseen by Nakayama and Tzuzuku, their so-called beta-Frobenius extension or Frobenius extensions of the second kind. Letβ be an automorphism of the ring B and B⊆A a subring pair. Denote the pullback module of a module BM along β:B→B by βM, the so-called twisted module. A ring extension A⊇B is a β-Frobenius extension if AB is finite projective and there is a bimodule isomorphism BAA≅βHom(AB,BB). One shows that A⊇B is a Frobenius extension if and only if  β  is an inner automorphism. A subring pair of Frobenius algebras B⊆A is β-Frobenius extension so long as AB is finite projective and the Nakayama automorphism ηA of A stabilizes B, in which case β=ηB∘ηA-1 [22]. For instance a finite-dimensional Hopf algebra A=H and B=K a Hopf subalgebra of H are a pair of Frobenius algebras satisfying the conditions just given: the formula for β reduces to the following given in terms of the modular functions of H and K and the antipode S [23, 7.8]: for x∈K,
(2.1)β(x)=∑(x)mH(x(1))mK(S(x(2)))x(3),Given the bimodule isomorphism aboveBAA→≅βHom(AB,BB), apply it to 1A and let its value be E:A→B, which then is a cyclic generator of βHom(AB,BB)A satisfying E(b1ab2)=β(b1)E(a)b2 for all b1,b2∈B,a∈A. If x1,…,xm∈A and ϕ1,…,ϕm∈Hom(AB,BB) are projective bases of AB, and Eyj:=E(yj-)=ϕj the equations
(2.2)∑j=1mxjE(yja)=a,∑j=1mβ-1(E(axj))yj=a
hold for all a∈A. Call (E,xj,yj) a β-Frobeniuscoordinatesystem of A⊇B. Note that also BA is finite projective, that a β-Frobenius coordinate system is equivalent to the ring extension A∣B being β-Frobenius and that β=idB if B is in the center of A. Additionally, one notes that there is an automorphism η of the centralizer subring AB such that E(ac)=E(η(c)a) for all a∈A and c∈AB. Also an isomorphism Aβ⊗BA≅EndAB is easily defined from the data and equations above, where ∑j‍xj⊗yj↦idA, so that if (E,zi,wi) is another β-Frobenius coordinate system (sharing the same E:A→B), then ∑i‍zi⊗Bwi=∑j‍xj⊗Byj in (Aβ⊗BA)A.When aβ-Frobenius extension is a QF extension is addressed in the next proposition.Proposition 2.1.
Aβ-Frobenius extension A⊇B is a left QF extension if and only if there are ui,vi∈A(i=1,…,n) such that sui=uiβ(s) and vis=β(s)vi for all i and s∈B, and
(2.3)β-1(s)=∑i=1nuisvi.Proof.
SupposeA⊇B is β-Frobenius extension with β-Frobenius system satisfying the equations above. Given the elements ui,vi∈A satisfying the equations above, let Ei=E(ui-), which defines n mappings in (the untwisted) Hom(BAB,BBB). Also define n mappings ψi∈Hom(A(*A)B,AAB) by ψi(g)=∑j=1m‍xjg(viyj) where it is not hard to show using the β-Frobenius coordinate equations that ∑j‍xj⊗Bviyj∈(A⊗BA)A for each i (a Casimir element). It follows that ∑i=1n‍ψi(Ei)=1A and that A∣n(*A) as natural A-B-bimodules, whence A is a left QF extension of B.
Conversely, assume the left QF conditionBA*A∣An, equivalent to AAB∣n(*A) by applying the right B-dual functor and noting (*A)*≅A as well *(A*)≅A. Also assume the slightly rewritten β-Frobenius condition β-1AA≅B(A*)A, which then implies β-1AA∣nA. So there are n mappings gi∈Hom(β-1AA,BAA) and n mappings fi∈Hom(BAA,β-1AA) such that ∑i=1n‍fi∘gi=idA. Equivalently, with ui:=f(1A) and vi:=g(1A), ∑i=1n‍uivi=1A, and the equations in the proposition are satisfied.The following corollary weakens one of the equivalent conditions in [24, 25]. It implies that a finite dimensional Hopf algebra that is QF over a Hopf subalgebra is necessarily Frobenius over it. (Nontrivial examples of QF extensions occur for weak Hopf algebras over their separable base algebra [26].)Corollary 2.2.
LetH be a finite dimensional Hopf algebra and K a Hopf subalgebra. In the notation of (2.1) the following are equivalent.  (1)
The automorphismβ=idK and H⊇K is a Frobenius extension.(2)
The algebra extensionH⊇K is a QF extension.(3)
The modular functionsmH(x)=mK(x) for all x∈K.Proof.
(1⇒2) A Frobenius extension is a QF extension. (2⇒3) Set s=1 in (2.3), and apply the counit ε to see that ε(∑i‍uivi)=1. Reapply ε to (2.3) to obtain ε∘β=ε. Apply ε to (2.1), and use uniqueness of inverse in convolution algebra Hom(K,k), where mK∘S=mK-1 and 1=ε, to show that mH=mK on K. (3⇒1) This follows from (2.1).The following observation for a normal Hopf subalgebraK⊆H has not been explicitly noted before in the literature.Corollary 2.3.
The modular function of a finite dimensional Hopf algebraH restricts to the modular function of a Hopf subalgebra K⊆H if K has depth d(K,H)≤2.Proof.
If the Hopf subalgebraK has depth 1 in H, it has depth 2. If it has depth 2, it is equivalently a normal Hopf subalgebra by the result of [2]. But a normal Hopf subalgebra K⊆H is an H¯-Galois extension: here H¯:=H/HK+ denotes the quotient Hopf algebra, H→H¯, h↦h¯ denotes the quotient map, and the Galois isomorphism can:H⊗KH→H⊗H¯ is given by can(h⊗h′)=hh′(1)⊗h′¯(2) [27]. In the same paper [27] it is shown that a Hopf-Galois extension of a finite dimensional Hopf algebra is a Frobenius extension. Then, β=id in the corollary above, so mK=mH∣K.The corollary extends to some extent to quasi-Hopf algebras [23] and Hopf algebras over commutative rings [28], since the following identity may be established along the lines of [29] for the modular functions of subalgebra pairs of augmented Frobenius algebras B⊆A.Lemma 2.4.
Let(A,ε) be an augmented Frobenius algebra with Nakayama automorphism ηA, B a subalgebra and Frobenius algebra where ηA(B)=B, and AB finitely generated projective. It follows that A⊇B is a β-Frobenius extension where β=ηB∘ηA-1, a relative Nakayama automorphism [22, Satz 7], [29, Paragraph 5.1]. Then the modular automorphisms of A and B satisfy
(2.4)mA∣B=mB∘β.Proof.
Let(ϕ,xi,yi,ηA) be a Frobenius coordinate system for A, tA∈A a right norm satisfying ϕtA=ε, then tA is a right integral, satisfying tAx=tAε(x) for all x∈A, spanning the one-dimensional space of integrals in A. Let mA be the augmentation on A defined by xtA=mA(x)tA for x∈A. It follows that ε=mA∘ηA by expressing tA in terms of dual bases, ε and mA=tAϕ (and note that (ϕ,yi,ηA(xi)) are also dual bases) [29, Paragraph 3.2]. Similarly let (ψ,uj,vj,ηB) be a Frobenius coordinate system for B and tB a right norm satisfying ψtB=ε∣B, then tB is a right integral in B and xtB=mB(x)tB defines the k-valued algebra homomorphism mB, which satisfies ε∣B=mB∘ηB. It follows that mB∘β=mB∘ηB∘ηA-1=ε∘ηA-1∣B=mA∣B.Note that (2.4) for Hopf subalgebras also follows from (2.1). Corollary 2.3 does not extend to depth 3 Hopf subalgebras by the next example.Example 2.5.
The Taft-Hopf algebraH over its cyclic group subalgebra K is a nontrivial β-Frobenius extension [23]. The algebra H is generated over ℂ by a grouplike g of order n≥2, a nilpotent x of index n, and (g,1)-primitive element where xg=ψgx for ψ∈ℂ a primitive nth root of unity. This is a Hopf algebra having right integral tH=xn-1∑j=0n-1‍gj with modular function mH(g)=ψ [23]. The Hopf subalgebra K is generated by g. Then the twist automorphism of K is given by β(gj)=ψjgj. Of course, mH restricted to K is not equal to mK=ε∣K. The depth d(K,H)=3 is computed in [30].Finally we note that unimodular Hopf algebra extensions are trivial if theℋ-depth condition dℋ(B,A)=1 is imposed.Proposition 2.6.
SupposeH is a finite-dimensional Hopf algebra and K is a Hopf subalgebra of H. If dℋ(K,H)=1, then K satisfies a double centralizer result; in particular, if H is unimodular, then K=H.Proof.
SinceH is a finite-dimensional Hopf algebra, it is a free extension of the Hopf subalgebra K, therefore faithfully flat. If dℋ(K,H)=1, then the ring extension satisfies the generalized Azumaya condition H⊗KH≅HomZ(H)(CH(K),H) via x⊗Ky↦λx∘ρy, left and right multiplication [23, 31], where CH(K) denotes the centralizer subalgebra of K in H. If d∈CH(CH(K)), then it is obvious from this that d⊗K1H=1H⊗Kd, so that d∈K: it follows that
(2.5)K=CH(CH(K)).
SinceH is unimodular, it has a two-sided nonzero integral t. Note that t∈Z(H)⊆CH(CH(K)), whence t∈K. Let λ:H→k (where k is the arbitrary ground field) be the left integral in the dual Hopf algebra H* such that λ↼t=ε. The bijective antipode S:H→H satisfies S(a)=∑(t)‍t(1)λ(at(2)) since ∑(a)‍a(1)S(a(2))=1Hε(a) and λ⇀x=λ(x)1H for all x,a∈H. Since Δ(t)=∑(t)‍t(1)⊗t(2)∈K⊗K, it follows that S(a)∈K for all a∈H. Thus H=K.


## 2.1.β-Frobenius Extensions

In Hopf algebras and quantum algebras, examples of Frobenius extensions often occur with a twist foreseen by Nakayama and Tzuzuku, their so-called beta-Frobenius extension or Frobenius extensions of the second kind. Letβ be an automorphism of the ring B and B⊆A a subring pair. Denote the pullback module of a module BM along β:B→B by βM, the so-called twisted module. A ring extension A⊇B is a β-Frobenius extension if AB is finite projective and there is a bimodule isomorphism BAA≅βHom(AB,BB). One shows that A⊇B is a Frobenius extension if and only if  β  is an inner automorphism. A subring pair of Frobenius algebras B⊆A is β-Frobenius extension so long as AB is finite projective and the Nakayama automorphism ηA of A stabilizes B, in which case β=ηB∘ηA-1 [22]. For instance a finite-dimensional Hopf algebra A=H and B=K a Hopf subalgebra of H are a pair of Frobenius algebras satisfying the conditions just given: the formula for β reduces to the following given in terms of the modular functions of H and K and the antipode S [23, 7.8]: for x∈K,
(2.1)β(x)=∑(x)mH(x(1))mK(S(x(2)))x(3),Given the bimodule isomorphism aboveBAA→≅βHom(AB,BB), apply it to 1A and let its value be E:A→B, which then is a cyclic generator of βHom(AB,BB)A satisfying E(b1ab2)=β(b1)E(a)b2 for all b1,b2∈B,a∈A. If x1,…,xm∈A and ϕ1,…,ϕm∈Hom(AB,BB) are projective bases of AB, and Eyj:=E(yj-)=ϕj the equations
(2.2)∑j=1mxjE(yja)=a,∑j=1mβ-1(E(axj))yj=a
hold for all a∈A. Call (E,xj,yj) a β-Frobeniuscoordinatesystem of A⊇B. Note that also BA is finite projective, that a β-Frobenius coordinate system is equivalent to the ring extension A∣B being β-Frobenius and that β=idB if B is in the center of A. Additionally, one notes that there is an automorphism η of the centralizer subring AB such that E(ac)=E(η(c)a) for all a∈A and c∈AB. Also an isomorphism Aβ⊗BA≅EndAB is easily defined from the data and equations above, where ∑j‍xj⊗yj↦idA, so that if (E,zi,wi) is another β-Frobenius coordinate system (sharing the same E:A→B), then ∑i‍zi⊗Bwi=∑j‍xj⊗Byj in (Aβ⊗BA)A.When aβ-Frobenius extension is a QF extension is addressed in the next proposition.Proposition 2.1.
Aβ-Frobenius extension A⊇B is a left QF extension if and only if there are ui,vi∈A(i=1,…,n) such that sui=uiβ(s) and vis=β(s)vi for all i and s∈B, and
(2.3)β-1(s)=∑i=1nuisvi.Proof.
SupposeA⊇B is β-Frobenius extension with β-Frobenius system satisfying the equations above. Given the elements ui,vi∈A satisfying the equations above, let Ei=E(ui-), which defines n mappings in (the untwisted) Hom(BAB,BBB). Also define n mappings ψi∈Hom(A(*A)B,AAB) by ψi(g)=∑j=1m‍xjg(viyj) where it is not hard to show using the β-Frobenius coordinate equations that ∑j‍xj⊗Bviyj∈(A⊗BA)A for each i (a Casimir element). It follows that ∑i=1n‍ψi(Ei)=1A and that A∣n(*A) as natural A-B-bimodules, whence A is a left QF extension of B.
Conversely, assume the left QF conditionBA*A∣An, equivalent to AAB∣n(*A) by applying the right B-dual functor and noting (*A)*≅A as well *(A*)≅A. Also assume the slightly rewritten β-Frobenius condition β-1AA≅B(A*)A, which then implies β-1AA∣nA. So there are n mappings gi∈Hom(β-1AA,BAA) and n mappings fi∈Hom(BAA,β-1AA) such that ∑i=1n‍fi∘gi=idA. Equivalently, with ui:=f(1A) and vi:=g(1A), ∑i=1n‍uivi=1A, and the equations in the proposition are satisfied.The following corollary weakens one of the equivalent conditions in [24, 25]. It implies that a finite dimensional Hopf algebra that is QF over a Hopf subalgebra is necessarily Frobenius over it. (Nontrivial examples of QF extensions occur for weak Hopf algebras over their separable base algebra [26].)Corollary 2.2.
LetH be a finite dimensional Hopf algebra and K a Hopf subalgebra. In the notation of (2.1) the following are equivalent.  (1)
The automorphismβ=idK and H⊇K is a Frobenius extension.(2)
The algebra extensionH⊇K is a QF extension.(3)
The modular functionsmH(x)=mK(x) for all x∈K.Proof.
(1⇒2) A Frobenius extension is a QF extension. (2⇒3) Set s=1 in (2.3), and apply the counit ε to see that ε(∑i‍uivi)=1. Reapply ε to (2.3) to obtain ε∘β=ε. Apply ε to (2.1), and use uniqueness of inverse in convolution algebra Hom(K,k), where mK∘S=mK-1 and 1=ε, to show that mH=mK on K. (3⇒1) This follows from (2.1).The following observation for a normal Hopf subalgebraK⊆H has not been explicitly noted before in the literature.Corollary 2.3.
The modular function of a finite dimensional Hopf algebraH restricts to the modular function of a Hopf subalgebra K⊆H if K has depth d(K,H)≤2.Proof.
If the Hopf subalgebraK has depth 1 in H, it has depth 2. If it has depth 2, it is equivalently a normal Hopf subalgebra by the result of [2]. But a normal Hopf subalgebra K⊆H is an H¯-Galois extension: here H¯:=H/HK+ denotes the quotient Hopf algebra, H→H¯, h↦h¯ denotes the quotient map, and the Galois isomorphism can:H⊗KH→H⊗H¯ is given by can(h⊗h′)=hh′(1)⊗h′¯(2) [27]. In the same paper [27] it is shown that a Hopf-Galois extension of a finite dimensional Hopf algebra is a Frobenius extension. Then, β=id in the corollary above, so mK=mH∣K.The corollary extends to some extent to quasi-Hopf algebras [23] and Hopf algebras over commutative rings [28], since the following identity may be established along the lines of [29] for the modular functions of subalgebra pairs of augmented Frobenius algebras B⊆A.Lemma 2.4.
Let(A,ε) be an augmented Frobenius algebra with Nakayama automorphism ηA, B a subalgebra and Frobenius algebra where ηA(B)=B, and AB finitely generated projective. It follows that A⊇B is a β-Frobenius extension where β=ηB∘ηA-1, a relative Nakayama automorphism [22, Satz 7], [29, Paragraph 5.1]. Then the modular automorphisms of A and B satisfy
(2.4)mA∣B=mB∘β.Proof.
Let(ϕ,xi,yi,ηA) be a Frobenius coordinate system for A, tA∈A a right norm satisfying ϕtA=ε, then tA is a right integral, satisfying tAx=tAε(x) for all x∈A, spanning the one-dimensional space of integrals in A. Let mA be the augmentation on A defined by xtA=mA(x)tA for x∈A. It follows that ε=mA∘ηA by expressing tA in terms of dual bases, ε and mA=tAϕ (and note that (ϕ,yi,ηA(xi)) are also dual bases) [29, Paragraph 3.2]. Similarly let (ψ,uj,vj,ηB) be a Frobenius coordinate system for B and tB a right norm satisfying ψtB=ε∣B, then tB is a right integral in B and xtB=mB(x)tB defines the k-valued algebra homomorphism mB, which satisfies ε∣B=mB∘ηB. It follows that mB∘β=mB∘ηB∘ηA-1=ε∘ηA-1∣B=mA∣B.Note that (2.4) for Hopf subalgebras also follows from (2.1). Corollary 2.3 does not extend to depth 3 Hopf subalgebras by the next example.Example 2.5.
The Taft-Hopf algebraH over its cyclic group subalgebra K is a nontrivial β-Frobenius extension [23]. The algebra H is generated over ℂ by a grouplike g of order n≥2, a nilpotent x of index n, and (g,1)-primitive element where xg=ψgx for ψ∈ℂ a primitive nth root of unity. This is a Hopf algebra having right integral tH=xn-1∑j=0n-1‍gj with modular function mH(g)=ψ [23]. The Hopf subalgebra K is generated by g. Then the twist automorphism of K is given by β(gj)=ψjgj. Of course, mH restricted to K is not equal to mK=ε∣K. The depth d(K,H)=3 is computed in [30].Finally we note that unimodular Hopf algebra extensions are trivial if theℋ-depth condition dℋ(B,A)=1 is imposed.Proposition 2.6.
SupposeH is a finite-dimensional Hopf algebra and K is a Hopf subalgebra of H. If dℋ(K,H)=1, then K satisfies a double centralizer result; in particular, if H is unimodular, then K=H.Proof.
SinceH is a finite-dimensional Hopf algebra, it is a free extension of the Hopf subalgebra K, therefore faithfully flat. If dℋ(K,H)=1, then the ring extension satisfies the generalized Azumaya condition H⊗KH≅HomZ(H)(CH(K),H) via x⊗Ky↦λx∘ρy, left and right multiplication [23, 31], where CH(K) denotes the centralizer subalgebra of K in H. If d∈CH(CH(K)), then it is obvious from this that d⊗K1H=1H⊗Kd, so that d∈K: it follows that
(2.5)K=CH(CH(K)).
SinceH is unimodular, it has a two-sided nonzero integral t. Note that t∈Z(H)⊆CH(CH(K)), whence t∈K. Let λ:H→k (where k is the arbitrary ground field) be the left integral in the dual Hopf algebra H* such that λ↼t=ε. The bijective antipode S:H→H satisfies S(a)=∑(t)‍t(1)λ(at(2)) since ∑(a)‍a(1)S(a(2))=1Hε(a) and λ⇀x=λ(x)1H for all x,a∈H. Since Δ(t)=∑(t)‍t(1)⊗t(2)∈K⊗K, it follows that S(a)∈K for all a∈H. Thus H=K.


## 3.   Even Depth of QF Extensions

It is well known that for a Frobenius extensionA⊇B, coinduction of a module, MB↦Hom(AB,MB) is naturally isomorphic as functors to induction (MB↦M⊗BA) (from the category of B-modules into the category of A-modules). Similarly, a QF extension has h-equivalent coinduction and induction functors, which is seen from the naturality of the mappings in the next proof. Let T be an arbitrary third ring.Proposition 3.1.
SupposeTMB is a bimodule and A⊇B is a QF extension. Then, there is an h-equivalence of bimodules,
(3.1)TM⊗BAA~hTHom(AB,MB)A.Proof.
SinceAB is f.g. projective, it follows that there is a T-A-bimodule isomorphism
(3.2)M⊗BHom(AB,BB)≅Hom(AB,MB),
given by m⊗Bϕ↦mϕ(-) with inverse constructed from projective bases for AB. But the right B-dual of A is h-equivalent to BAA, so  (3.1) holds by Lemma 1.1.The next theorem shows that minimum right and left even depth of a QF extension are equal (see Definition1.6 where as before Cn(A,B)=A⊗B⋯⊗BA, n times A).Theorem 3.2.
IfA⊇B is QF extension, then A⊇B has left depth 2n if and only if A⊇B has right depth 2n.Proof.
The left depth2n condition on A⊇B recall is Cn+1(A,B)~hCn(A,B) as B-A-bimodules. To this apply the additive functor Hom(-A,AA) (into the category of A-B-bimodules), noting that Hom(Cn(A,B)A,AA)≅Hom(Cn-1(A,B)B,AB) via f↦f(-⊗B⋯-⊗B1A) for each integer n≥1. It follows (from Lemma 1.1) that there is an A-B-bimodule h-equivalence,
(3.3)Hom(Cn(A,B)B,AB)~hHom(Cn-1(A,B)B,AB).
(Then in the depth two case, the left depth two condition is equivalent to End AB~hA as natural A-B-bimodules.)
Given bimoduleAMB, we have AM⊗BAA~hAHom(AB,MB)A by the previous lemma: apply this to Cn+1(A,B)=Cn(A,B)⊗BA using the hom-tensor adjoint relation: there are h-equivalences and isomorphisms of A-bimodules,
(3.4)Cn+1(A,B)~hHom(AB,Cn(A,B)B)~hHom(AB,Hom(AB,Cn-1(A,B)B)B)≅Hom(A⊗BAB,Cn-1(A,B)B)⋯~hHom(Cp(A,B)B,Cn-p+1(A,B)B),
for each p=1,2,…,n and n=1,2,…. Compare  (3.3) and (3.4) with p=n to get ACn+1(A,B)B~hACn(A,B)B, which is the right depth 2n condition.
The converse is proven similarly from the symmetric conditions of the QF hypothesis.The extent to which the theorem (and most of the results in the next section) extends toβ-Frobenius or even twisted QF extensions presents technical problems and is unknown to the author.


## 4. Frobenius Extensions

As noted above a Frobenius extensionA⊇B is characterized by any of the following four conditions [23]. First,  AB is finite projective and BAA≅Hom(AB,BB). Second, BA is finite projective and AAB≅Hom(BA,BB). Third, coinduction and induction of right (or left) B-modules is naturally equivalent. Fourth, there is a Frobenius coordinate system (E:A→B;x1,…,xm,y1,…,ym∈A), which satisfies
(4.1)E∈Hom(BAB,BBB),∑i=1mE(axi)yi=a=∑i=1mxiE(yia)(∀a∈A).
These (dual bases) equations may be used to show the useful fact that ∑i‍xi⊗yi∈(A⊗BA)A.We continue this notation in the next lemma. Although most Frobenius extensions in the literature are generator extensions, by the lemma equivalent to having a surjective Frobenius homomorphism, Example 2.7 in [23] provides a somewhat pathological example of a matrix algebra Frobenius extension with a nonsurjective Frobenius homomorphism.Lemma 4.1.
The natural moduleAB is a generator ⇔BA is a generator ⇔ there are elements {aj}j=1n and {cj}j=1n such that ∑j=1n‍E(ajcj)=1B⇔E is surjective.Proof.
The bimodule isomorphismBAA→≅BHom(AB,BB)A is realized by a↦E(a-) (with inverse ϕ↦∑i=1m‍ϕ(xi)yi). If AB is a generator, then there are elements {cj}j=1n of A and mappings {ϕj}j=1n of A* such that ∑j=1n‍ϕj(cj)=1B. Let Eaj=ϕj. Then, ∑j=1n‍E(ajcj)=1B.
Another bimodule isomorphismAAB→≅AHom(BA,BB)B is realized by a↦E(-a):=aE. Then writing the last equation as ∑j‍cjE(aj)=1B exhibits BA as a generator.
The last of the equivalent conditions is implied by the previous condition and implies the first condition. Also note that any other Frobenius homomorphism is given byEd for some invertible d∈AB.A Frobenius (or QF) extensionA⊇B enjoys an endomorphism ring theorem [21, 32], which shows that ℰ:=EndAB⊇A is a Frobenius (resp., QF) extension, where the default ring homomorphism A→ℰ is understood to be the left multiplication mapping λ:a↦λa where λa(x)=ax. It is worth noting that λ is a left split A-monomorphism (by evaluation at 1A) so Aℰ is a generator.Thetower of a Frobenius (resp., QF) extension is obtained by iteration of the endomorphism ring and λ, obtaining a tower of Frobenius (resp. QF) extensions where occasionally we need the notation B:=ℰ-1,A=ℰ0 and ℰ=ℰ1(4.2)B⟶A↪E1↪E2↪⋯↪En↪⋯
so ℰ2=EndℰA, and so forth. By transitivity of Frobenius extension or QF extension [21, 22], all subextensions ℰm↪ℰm+n in the tower are also Frobenius (resp. QF) extensions.The ringsℰn are h-equivalent to Cn+1(A,B)=A⊗B⋯⊗BA as A-bimodules in case A⊇B is a QF extension. This follows from noting the
(4.3)EndAB≅A⊗BHom(AB,BB)~hA⊗BA
also holding as natural ℰ-A-bimodules, obtained by substitution of A*~hA. This observation is then iterated followed by cancellations of the type A⊗AM≅M.


### 4.1. Tower above Frobenius Extension

Specialize now toA⊇B a Frobenius extension with Frobenius coordinate system E and {xi}i=1m,{yi}i=1m. Then the h-equivalences above are replaced by isomorphisms, and ℰn≅Cn+1(A,B) for each n≥-1 as ring isomorphisms with respect to a certain induced “E-multiplication.” The E-multiplication on A⊗BA is induced from the endomorphism ring EndAB→≅A⊗BA given by f↦∑i‍f(xi)⊗Byi with inverse a⊗a′↦λa∘E∘λa′. The outcome of E-multiplication on C2(A,B) is given by
(4.4)(a1⊗Ba2)(a3⊗Ba4)=a1E(a2a3)⊗Ba4
with unity element 11=∑i=1m‍xi⊗Byi. Note that the A-bimodule structure on ℰ1 induced by λ:A↪ℰ corresponds to the natural A-bimodule A⊗BA.TheE-multiplication is defined inductively on
(4.5)En≅En-1⊗En-2En-1
using the Frobenius homomorphism En-1:ℰn-1→ℰn-2 obtained by iterating the following natural Frobenius coordinate system on ℰ1≅A⊗BA, given by E1(a⊗Ba′)=aa′ and {xi⊗B1A}i=1m, {1A⊗Byi}i=1m [23] as one checks.The iterativeE-multiplication on Cn(A,B) clearly exists as an associative algebra, but it seems worthwhile (and not available in the literature) to compute it explicitly. The multiplication on C2n(A,B) is given by (⊗=⊗B,n≥1)
(4.6)(a1⊗⋯⊗a2n)(c1⊗⋯⊗c2n)=a1⊗⋯⊗anE(an+1E(⋯E(a2n-1E(a2nc1)c2)⋯)cn-1)cn)⊗cn+1⊗⋯⊗c2n.
The identity on C2n(A,B) is in terms of the dual bases,
(4.7)12n-1=∑i1,…,in=1mxi1⊗⋯⊗xin⊗yin⊗⋯⊗yi1.The multiplication onC2n+1(A,B) is given by
(4.8)(a1⊗⋯⊗a2n+1)(c1⊗⋯⊗c2n+1)=a1⊗⋯⊗an+1E(an+2E(⋯E(a2nE(a2n+1c1)c2)⋯)cn)cn+1⊗⋯⊗c2n+1
with identity
(4.9)12n=∑i1,…,in=1mxi1⊗⋯⊗xin⊗1A⊗yin⊗⋯⊗yi1.
Denote in brief notation the rings Cn(A,B):=An and distinguish them from the isomorphic rings ℰn-1 (n=0,1,…).The inclusionsAn↪An+1 are given by a[n]↦a[n]1n, which works out in the odd and even cases to
(4.10)A2n-1↪A2n,a1⊗⋯⊗a2n-1⟼∑ia1⊗⋯⊗anxi⊗yi⊗an+1⊗⋯⊗a2n-1,A2n↪A2n+1,a1⊗⋯⊗a2n⟼a1⊗⋯⊗an⊗1A⊗an+1⊗⋯⊗a2n.The bimodule structure onAn over a subalgebra Am (with m<n via composition of left multiplication mappings λ) is just given in terms of the multiplication in Am as follows:
(4.11)(r1⊗⋯⊗rm)(a1⊗⋯⊗an)=[(r1⊗⋯⊗rm)(a1⊗⋯⊗am)]⊗am+1⊗⋯⊗an
with a similar formula for the right module structure.The formulas for the successive Frobenius homomorphismsEm:Am+1→Am are given in even degrees by
(4.12)E2n(a1⊗⋯⊗a2n+1)=a1⊗⋯⊗anE(an+1)⊗an+2⊗⋯⊗a2n+1
for n≥0. The formula in the odd case is
(4.13)E2n+1(a1⊗⋯⊗a2n+2)=a1⊗⋯⊗an⊗an+1an+2⊗an+3⊗⋯⊗a2n+2
for n≥0.The dual bases ofEn denoted by xin and yin are given by all-in-one formulas
(4.14)xin=xi⊗1n-1,yin=1n-1⊗yi
for n≥0 (where 10=1A). Note that ∑i‍xin⊗Anyin=1n+1.With another choice of Frobenius coordinate system(F,zj,wj) for A⊇B, there is in fact an invertible element d in the centralizer subring AB of A such that F=E(d-) and ∑i‍xi⊗Byi=∑j‍zj⊗Bd-1wj [22, 23]; whence an isomorphism of the E-multiplication onto the F-multiplication, both on A⊗BA, is given by r1⊗r2↦r1⊗d-1r2. If the tower with E-multiplication is denoted by AnE and the tower with F-multiplication is denoted by AnF, there is a sequence of ring isomorphisms
(4.15)A2nE⟶≅A2nF,a1⊗⋯⊗a2n⟼a1⊗⋯⊗an⊗d-1an+1⊗⋯⊗d-1a2n,(4.16)A2n+1E⟶≅A2n+1Fa1⊗⋯⊗a2n+1⟼a1⊗⋯⊗an+1⊗d-1an+2⊗⋯⊗d-1a2n+1
which commute with the inclusions AnE,F↪An+1E,F.Theorem 4.2.
The multiplication, module, and Frobenius structures for the towerAn=A⊗B⋯⊗BA (n times A) above a Frobenius extension A⊇B are given by  formulas (4.4) to (4.16).Proof.
First define Temperley-Lieb generators iteratively byen=1n-1⊗An-21n-1∈An+1 for n=1,2,…, which results in the explicit formulas
(4.17)e2n=∑i1,…,in+1xi1⊗⋯⊗xin⊗yinxin+1⊗yin+1⊗yin-1⊗⋯⊗yi1,e2n+1=∑i1,…,inxi1⊗⋯⊗xin⊗1A⊗1A⊗yin⊗⋯⊗yi1.
These satisfy braid-like relations [4, page 106], namely,
(4.18)eiej=ejei,|i-j|≥2,ei+1eiei+1=ei+1,eiei+1ei=ei1i+1.
(The generators above fail to be idempotents to the extent that E(1) differs from 1.) The proof that the formulas above are the correct outcomes of the inductive definitions may be given in terms of Temperley-Lieb generators, braid-like relations and important relations
(4.19)enxen=enEn-1(x),∀x∈An,yen=En(yen)en,∀y∈An+1,En(en)=1n-1,xen=enx,∀x∈An-1.
Reference [4, page 106] (for background see [33]) as well as the symmetric left-right relations. These relations and the Frobenius equations (4.1) may be checked to hold in terms of the equations above in a series of exercises left to the reader.
The formulas for the Frobenius bases follow from the iteratively apparentxin=xie1e2⋯en and yin=en⋯e2e1yi and uniqueness of bases with respect to the same Frobenius homomorphism. In fact en⋯e2e1a=1n-1⊗a for any a∈A,n=1,2,… (a symmetrical formula holds as well) and 1n=∑i‍xie1⋯en-1enen-1⋯e1yi.
Since the inductive definitions of the ring and module structures on theAn's also satisfy the relations listed above and agree on and below A2, the proof is finished with an induction argument based on expressing tensors as words in Temperley-Lieb generators and elements of A.
We note that(4.20)a1⊗⋯⊗an+1=(a1⊗⋯⊗an)(1n-1⊗an+1)=(a1⊗⋯⊗an-1)(1n-2⊗an)(en⋯e1an+1)=⋯=a1(e1a2)(e2e1a3)⋯(en-1⋯e1an)(en⋯e1an+1).
The formulas for multiplication (4.8), (4.6), and (4.11) follow from induction and applying the relations (4.18) through (4.20).For the next proposition the main point is that given a Frobenius extension there is a ring structure on theCn(A,B)'s satisfying the hypotheses below (for one compares with (4.11)). This is true as well if A is a ring with B in its center, since the ordinary tensor algebra on A⊗BA may be extended to an n-fold tensor product algebra A⊗B⋯⊗BA.Proposition 4.3.
LetA⊇B be a ring extension. Suppose that there is a ring structure on each An:=Cn(A,B) for each n≥0, a ring homomorphism An-1→An for each n≥1, and that the composite B→An induces the natural bimodule given by b·(a1⊗⋯⊗an)·b′=ba1⊗a2⊗⋯⊗anb′. Then, A⊇B has depth 2n+1 if and only if An∣B has depth 3.Proof.
IfA⊇B has depth 2n+1, then An~hAn+1 as B-bimodules. By tensoring repeatedly by BA⊗B-, also An~hA2n as B-bimodules. But A2n≅An⊗BAn. Then, An⊇B has depth three.
Conversely, ifAn∣B has depth 3, then A2n~hAn as B-bimodules. But An+1∣A2n via the split B-bimodule epi a1⊗⋯⊗a2n↦a1⋯an⊗an+1⊗⋯⊗a2n. Then, An+1∣qAn for some q∈ℤ+. It follows that A⊇B has depth 2n+1.One may in turn embed a depth three extension into a ring extension having depth two. The proof requires the QF condition. Retain the notation for the endomorphism ring introduced earlier in this section.Theorem 4.4.
SupposeA⊇B is a QF extension. If A⊇B has depth 3, then ℰ⊇B has depth 2. Conversely, if ℰ⊇B has depth 2 and AB is a generator, then A⊇B has depth 3.Proof.
SinceA is a QF extension of B, we have ℰ~hA⊗BA as ℰ-A-bimodules. Then, ℰ⊗Bℰ~hA⊗BA⊗BA⊗BA as ℰ-B-bimodules. Given the depth 3 condition, A⊗BA~hA as B-bimodules, it follows by two substitutions that ℰ⊗Bℰ~hA⊗BA as ℰ-B-bimodules. Consequently, ℰ⊗Bℰ~hℰ as ℰ-B-bimodules. Hence, ℰ⊇B has right depth 2, and since it is a QF extension by the endomorphism ring theorem and transitivity, ℰ⊇B also has left depth 2.
Conversely, we are givenAB a progenerator, so that ℰ and B are Morita equivalent rings, where BHom(AB,BB)ℰ and ℰAB are the context bimodules. If ℰ⊇B has depth two, then ℰ⊗Bℰ~hℰ as ℰ-B-bimodules. Then A⊗BA⊗BA⊗BA~hA⊗BA as ℰ-B-bimodules. Since Hom(AB,BB)⊗ℰA≅B as B-bimodules, a cancellation of the bimodules ℰAB follows, so A⊗BA⊗BA~hA as B-bimodules. Since A⊗BA∣A⊗BA⊗BA, it follows that A⊗BA∣qA for some q∈ℤ+. Then A⊇B has depth 3.Example 4.5.
To illustrate that the theorem does not extend to whenA⊇B is not a QF extension, consider A=Tn(k), n≥2 (a hereditary algebra) and B=Dn(k) (a semisimple algebra), and left k be an algebraically closed field of characteristic zero. (Since B,A is, is not a QF-algebra it follows by transitivity that A⊇B is not a QF extension.) It was computed that d(B,A)=3 in Example 1.7. Thinking of the columns of A as Aeii, it is quite easy to see that EndAB≅M1(k)×M2(k)×⋯×Mn(k) and that the inclusion of A↪EndAB is given by
(4.21)X⟼(X11,(X11X12X12X22),…,X).
Its restriction to B is given by
(4.22)Diag(μ1,…,μn)⟼(μ1,Diag(μ1,μ2),…,Diag(μ1,…,μn))
with inclusion matrix M=∑i≤j‍eij. Then, MMt>0, and from (1.1) we see that d(B,ℰ)=3.


## 4.1. Tower above Frobenius Extension

Specialize now toA⊇B a Frobenius extension with Frobenius coordinate system E and {xi}i=1m,{yi}i=1m. Then the h-equivalences above are replaced by isomorphisms, and ℰn≅Cn+1(A,B) for each n≥-1 as ring isomorphisms with respect to a certain induced “E-multiplication.” The E-multiplication on A⊗BA is induced from the endomorphism ring EndAB→≅A⊗BA given by f↦∑i‍f(xi)⊗Byi with inverse a⊗a′↦λa∘E∘λa′. The outcome of E-multiplication on C2(A,B) is given by
(4.4)(a1⊗Ba2)(a3⊗Ba4)=a1E(a2a3)⊗Ba4
with unity element 11=∑i=1m‍xi⊗Byi. Note that the A-bimodule structure on ℰ1 induced by λ:A↪ℰ corresponds to the natural A-bimodule A⊗BA.TheE-multiplication is defined inductively on
(4.5)En≅En-1⊗En-2En-1
using the Frobenius homomorphism En-1:ℰn-1→ℰn-2 obtained by iterating the following natural Frobenius coordinate system on ℰ1≅A⊗BA, given by E1(a⊗Ba′)=aa′ and {xi⊗B1A}i=1m, {1A⊗Byi}i=1m [23] as one checks.The iterativeE-multiplication on Cn(A,B) clearly exists as an associative algebra, but it seems worthwhile (and not available in the literature) to compute it explicitly. The multiplication on C2n(A,B) is given by (⊗=⊗B,n≥1)
(4.6)(a1⊗⋯⊗a2n)(c1⊗⋯⊗c2n)=a1⊗⋯⊗anE(an+1E(⋯E(a2n-1E(a2nc1)c2)⋯)cn-1)cn)⊗cn+1⊗⋯⊗c2n.
The identity on C2n(A,B) is in terms of the dual bases,
(4.7)12n-1=∑i1,…,in=1mxi1⊗⋯⊗xin⊗yin⊗⋯⊗yi1.The multiplication onC2n+1(A,B) is given by
(4.8)(a1⊗⋯⊗a2n+1)(c1⊗⋯⊗c2n+1)=a1⊗⋯⊗an+1E(an+2E(⋯E(a2nE(a2n+1c1)c2)⋯)cn)cn+1⊗⋯⊗c2n+1
with identity
(4.9)12n=∑i1,…,in=1mxi1⊗⋯⊗xin⊗1A⊗yin⊗⋯⊗yi1.
Denote in brief notation the rings Cn(A,B):=An and distinguish them from the isomorphic rings ℰn-1 (n=0,1,…).The inclusionsAn↪An+1 are given by a[n]↦a[n]1n, which works out in the odd and even cases to
(4.10)A2n-1↪A2n,a1⊗⋯⊗a2n-1⟼∑ia1⊗⋯⊗anxi⊗yi⊗an+1⊗⋯⊗a2n-1,A2n↪A2n+1,a1⊗⋯⊗a2n⟼a1⊗⋯⊗an⊗1A⊗an+1⊗⋯⊗a2n.The bimodule structure onAn over a subalgebra Am (with m<n via composition of left multiplication mappings λ) is just given in terms of the multiplication in Am as follows:
(4.11)(r1⊗⋯⊗rm)(a1⊗⋯⊗an)=[(r1⊗⋯⊗rm)(a1⊗⋯⊗am)]⊗am+1⊗⋯⊗an
with a similar formula for the right module structure.The formulas for the successive Frobenius homomorphismsEm:Am+1→Am are given in even degrees by
(4.12)E2n(a1⊗⋯⊗a2n+1)=a1⊗⋯⊗anE(an+1)⊗an+2⊗⋯⊗a2n+1
for n≥0. The formula in the odd case is
(4.13)E2n+1(a1⊗⋯⊗a2n+2)=a1⊗⋯⊗an⊗an+1an+2⊗an+3⊗⋯⊗a2n+2
for n≥0.The dual bases ofEn denoted by xin and yin are given by all-in-one formulas
(4.14)xin=xi⊗1n-1,yin=1n-1⊗yi
for n≥0 (where 10=1A). Note that ∑i‍xin⊗Anyin=1n+1.With another choice of Frobenius coordinate system(F,zj,wj) for A⊇B, there is in fact an invertible element d in the centralizer subring AB of A such that F=E(d-) and ∑i‍xi⊗Byi=∑j‍zj⊗Bd-1wj [22, 23]; whence an isomorphism of the E-multiplication onto the F-multiplication, both on A⊗BA, is given by r1⊗r2↦r1⊗d-1r2. If the tower with E-multiplication is denoted by AnE and the tower with F-multiplication is denoted by AnF, there is a sequence of ring isomorphisms
(4.15)A2nE⟶≅A2nF,a1⊗⋯⊗a2n⟼a1⊗⋯⊗an⊗d-1an+1⊗⋯⊗d-1a2n,(4.16)A2n+1E⟶≅A2n+1Fa1⊗⋯⊗a2n+1⟼a1⊗⋯⊗an+1⊗d-1an+2⊗⋯⊗d-1a2n+1
which commute with the inclusions AnE,F↪An+1E,F.Theorem 4.2.
The multiplication, module, and Frobenius structures for the towerAn=A⊗B⋯⊗BA (n times A) above a Frobenius extension A⊇B are given by  formulas (4.4) to (4.16).Proof.
First define Temperley-Lieb generators iteratively byen=1n-1⊗An-21n-1∈An+1 for n=1,2,…, which results in the explicit formulas
(4.17)e2n=∑i1,…,in+1xi1⊗⋯⊗xin⊗yinxin+1⊗yin+1⊗yin-1⊗⋯⊗yi1,e2n+1=∑i1,…,inxi1⊗⋯⊗xin⊗1A⊗1A⊗yin⊗⋯⊗yi1.
These satisfy braid-like relations [4, page 106], namely,
(4.18)eiej=ejei,|i-j|≥2,ei+1eiei+1=ei+1,eiei+1ei=ei1i+1.
(The generators above fail to be idempotents to the extent that E(1) differs from 1.) The proof that the formulas above are the correct outcomes of the inductive definitions may be given in terms of Temperley-Lieb generators, braid-like relations and important relations
(4.19)enxen=enEn-1(x),∀x∈An,yen=En(yen)en,∀y∈An+1,En(en)=1n-1,xen=enx,∀x∈An-1.
Reference [4, page 106] (for background see [33]) as well as the symmetric left-right relations. These relations and the Frobenius equations (4.1) may be checked to hold in terms of the equations above in a series of exercises left to the reader.
The formulas for the Frobenius bases follow from the iteratively apparentxin=xie1e2⋯en and yin=en⋯e2e1yi and uniqueness of bases with respect to the same Frobenius homomorphism. In fact en⋯e2e1a=1n-1⊗a for any a∈A,n=1,2,… (a symmetrical formula holds as well) and 1n=∑i‍xie1⋯en-1enen-1⋯e1yi.
Since the inductive definitions of the ring and module structures on theAn's also satisfy the relations listed above and agree on and below A2, the proof is finished with an induction argument based on expressing tensors as words in Temperley-Lieb generators and elements of A.
We note that(4.20)a1⊗⋯⊗an+1=(a1⊗⋯⊗an)(1n-1⊗an+1)=(a1⊗⋯⊗an-1)(1n-2⊗an)(en⋯e1an+1)=⋯=a1(e1a2)(e2e1a3)⋯(en-1⋯e1an)(en⋯e1an+1).
The formulas for multiplication (4.8), (4.6), and (4.11) follow from induction and applying the relations (4.18) through (4.20).For the next proposition the main point is that given a Frobenius extension there is a ring structure on theCn(A,B)'s satisfying the hypotheses below (for one compares with (4.11)). This is true as well if A is a ring with B in its center, since the ordinary tensor algebra on A⊗BA may be extended to an n-fold tensor product algebra A⊗B⋯⊗BA.Proposition 4.3.
LetA⊇B be a ring extension. Suppose that there is a ring structure on each An:=Cn(A,B) for each n≥0, a ring homomorphism An-1→An for each n≥1, and that the composite B→An induces the natural bimodule given by b·(a1⊗⋯⊗an)·b′=ba1⊗a2⊗⋯⊗anb′. Then, A⊇B has depth 2n+1 if and only if An∣B has depth 3.Proof.
IfA⊇B has depth 2n+1, then An~hAn+1 as B-bimodules. By tensoring repeatedly by BA⊗B-, also An~hA2n as B-bimodules. But A2n≅An⊗BAn. Then, An⊇B has depth three.
Conversely, ifAn∣B has depth 3, then A2n~hAn as B-bimodules. But An+1∣A2n via the split B-bimodule epi a1⊗⋯⊗a2n↦a1⋯an⊗an+1⊗⋯⊗a2n. Then, An+1∣qAn for some q∈ℤ+. It follows that A⊇B has depth 2n+1.One may in turn embed a depth three extension into a ring extension having depth two. The proof requires the QF condition. Retain the notation for the endomorphism ring introduced earlier in this section.Theorem 4.4.
SupposeA⊇B is a QF extension. If A⊇B has depth 3, then ℰ⊇B has depth 2. Conversely, if ℰ⊇B has depth 2 and AB is a generator, then A⊇B has depth 3.Proof.
SinceA is a QF extension of B, we have ℰ~hA⊗BA as ℰ-A-bimodules. Then, ℰ⊗Bℰ~hA⊗BA⊗BA⊗BA as ℰ-B-bimodules. Given the depth 3 condition, A⊗BA~hA as B-bimodules, it follows by two substitutions that ℰ⊗Bℰ~hA⊗BA as ℰ-B-bimodules. Consequently, ℰ⊗Bℰ~hℰ as ℰ-B-bimodules. Hence, ℰ⊇B has right depth 2, and since it is a QF extension by the endomorphism ring theorem and transitivity, ℰ⊇B also has left depth 2.
Conversely, we are givenAB a progenerator, so that ℰ and B are Morita equivalent rings, where BHom(AB,BB)ℰ and ℰAB are the context bimodules. If ℰ⊇B has depth two, then ℰ⊗Bℰ~hℰ as ℰ-B-bimodules. Then A⊗BA⊗BA⊗BA~hA⊗BA as ℰ-B-bimodules. Since Hom(AB,BB)⊗ℰA≅B as B-bimodules, a cancellation of the bimodules ℰAB follows, so A⊗BA⊗BA~hA as B-bimodules. Since A⊗BA∣A⊗BA⊗BA, it follows that A⊗BA∣qA for some q∈ℤ+. Then A⊇B has depth 3.Example 4.5.
To illustrate that the theorem does not extend to whenA⊇B is not a QF extension, consider A=Tn(k), n≥2 (a hereditary algebra) and B=Dn(k) (a semisimple algebra), and left k be an algebraically closed field of characteristic zero. (Since B,A is, is not a QF-algebra it follows by transitivity that A⊇B is not a QF extension.) It was computed that d(B,A)=3 in Example 1.7. Thinking of the columns of A as Aeii, it is quite easy to see that EndAB≅M1(k)×M2(k)×⋯×Mn(k) and that the inclusion of A↪EndAB is given by
(4.21)X⟼(X11,(X11X12X12X22),…,X).
Its restriction to B is given by
(4.22)Diag(μ1,…,μn)⟼(μ1,Diag(μ1,μ2),…,Diag(μ1,…,μn))
with inclusion matrix M=∑i≤j‍eij. Then, MMt>0, and from (1.1) we see that d(B,ℰ)=3.


## 5. When Tower Depth Equals Subring Depth

In this section we review tower depth from [11] and find a general case when it is the same as subring depth defined in (1.7) and in [12]. We first require a generalization of left and right depth 2 to a tower of three rings. We say that a tower A⊇B⊇T, where A⊇B and B⊇T are ring extensions, has generalized right depth2 if A⊗BAh~A as natural A-T-bimodules. (Note that if T=B, this is the definition of the ring extension A⊇B having right depth 2.)Throughout the section below we supposeA⊇B is a Frobenius extension and ℰi↪ℰi+1 is its tower above it, as defined in (4.2) and the ensuing discussion in Section 4. Following [11] (with a small change in vocabulary), we say that A⊇B has right tower depthn≥2 if the subtower of composite ring extensions B→ℰn-3↪ℰn-2 has generalized right depth 2; equivalently, as natural ℰn-2-B-bimodules,
(5.1)En-2⊗En-3En-2⊕*≅qEn-2
for some positive integer q, since the reverse condition is always satisfied. Since ℰ-1=B and ℰ0=A, this recovers the right depth two condition on a subring B of A. To this definition we add that a Frobenius extension A⊇B has tower depth 1 if it is a centrally projective ring extension; that is, BAB∣qB for some q∈ℤ+. Left tower depth n is just defined using (5.1) but as natural B-ℰn-2-bimodules. By [11, Theorem 2.7] the left and right tower depth n conditions are equivalent on Frobenius extensions.From the definition of tower depth and a comparison of (4.5) and Definition 1.6 we note that if A is a Frobenius extension of B of tower depth n>1, then B⊆A has subring depth 2n-2; from (5.1) we obtain An∣qAn-1 as A-B-bimodules, since An≅ℰn-1≅ℰn-2⊗ℰn-3ℰn-2.From [11, Lemma 8.3], it follows that if A⊇B has tower depth n, it has tower depth n+1. Define dF(A,B) to be the minimum tower depth if A⊇B has tower depth n for some integer n, dF(A,B)=∞ if the condition (5.1) is not satisfied for any n≥2 nor is it depth 1. Notice that dF(A,B)=d(B,A) if  d(B,A)≤2 or dF(A,B)≤2. This is extended to dF(A,B)=d(B,A) if  d(B,A)ordF(A,B)≤3 in the next lemma.Notice that tower depthn makes sense for a QF extension A⊇B: by elementary considerations, it has right tower depth 3 if B→A↪ℰ satisfies ℰ⊗Aℰ~hℰ as ℰ-B-bimodules. It has been noted elsewhere that a QF extension has right tower depth 3 if and only if it has left tower depth 3 by an argument essentially identical to that in [11, Theorem 2.8] but replacing Frobenius isomorphisms with quasi-Frobenius h-equivalences.Lemma 5.1.
A QF extensionA⊇B such that AB is a generator has tower depth 3 if and only if B has depth 3 as a subring in A.Proof.
(
⇒
) By the QF property, ℰ~hA⊗BA as ℰ-B-bimodules. By the tower depth 3 condition, ℰ⊗Aℰ~hℰ as ℰ-B-bimodules. Then, A⊗BA⊗BA~hA⊗BA as ℰ-B-bimodules. Since AB is a progenerator, we cancel bimodules ℰAB as in the proof of Theorem 4.4 to obtain A⊗BA~hA as B-bimodules. Hence, B⊆A has depth 3.
(
⇐
) Given BAB~hBA⊗BAB, by tensoring with ℰA⊗B- we get A⊗BA~hA⊗BA⊗BA as ℰ-B-bimodules. By the QF property, ℰ⊗Aℰ~hℰ as ℰ-B-bimodules follows, whence A⊇B has tower depth 3.The theorem below proves that subring depth and tower depth coincide on Frobenius generator extensions, which are the most common Frobenius extensions, for example, including all group algebra extensions: the endomorphism ring extension of any Frobenius extension is a Frobenius generator extension. At a certain point in the proof, we use the following fundamental fact about the towerAn above a Frobenius extension A⊇B: since the compositions of the Frobenius extensions remain Frobenius, the iterative construction of E-multiplication on tensor-squares isomorphic to endomorphism rings applies but gives isomorphic ring structures to those on the An. For example, the composite extension B→An is Frobenius with End(An)B≅An⊗BAn≅A2n, isomorphic in its E∘E1∘⋯∘En-1-multiplication or its E-multiplication given in (4.6) [10].Theorem 5.2.
SupposeA is a Frobenius extension of B and AB is a generator. Then, A⊇B has tower depth m for m=1,2,… if and only if the subring B⊆A has depth m. Consequently, dF(A,B)=d(B,A).Proof.
The casesm=1,2,3 have been dealt with above. We divide the rest of the proof into odd m and even m. The proof for odd m=2n+1: (⇒) if A⊇B has tower depth 2n+1, then A2n⊗A2n-1A2n∣qA2n as A2n-B-bimodules. Continuing with A2n≅A2n-1⊗A2n-2A2n-1, iterating and performing standard cancellations, we obtain
(5.2)A2n+1∣qA2n
as End(An)B-B-bimodules. But the module (An)B is a generator for all n by Lemma 4.1, the endomorphism ring theorem for Frobenius generator extensions and transitivity of generator property for modules (if MA and AB are generators, then restricted module MB is clearly a generator). It follows that (An)B is a progenerator and cancellable as an End(An)B-B-bimodule (applying the Morita theorem as in the proof of Theorem 4.4). Then, B(An+1)B∣B(An)B after cancellation of An from (5.2), which is the depth 2n+1 condition in (1.7).
(
⇐
) Suppose An+1⊕*≅An as B-bimodules. Apply to this the additive functor An⊗B- from category of B-bimodules into the category of End(An)B-B-bimodules. We obtain (5.2), which is equivalent to the tower depth 2n+1 condition of A⊇B.
The proof in the even case,m=2n, does not need the generator condition (since even nongenerator Frobenius extensions have endomorphism ring extensions that are generators).
(
⇒
) Given the tower depth 2n condition A2n-1⊗A2n-2A2n-1≅A2n is isomorphic as A2n-1-B-bimodules to a direct summand in qA2n-1 for some positive integer q, introduce a cancellable extra term in A2n≅An⊗AAn+1 and in A2n-1≅An⊗AAn. Now note that A2n-1≅End(An)A, which is Morita equivalent to A. After cancellation of the End(An)A-A-bimodule An, we obtain An+1∣An as A-B-bimodules as required by (1.7).
(
⇐
) Given A(An+1)B∣A(An)B, we apply End(An)AAn⊗A- obtaining A2n∣A2n-1 as A2n-1-B-bimodules, which is equivalent to the tower depth 2n condition.A depth2 extension A⊇B may have easier equivalent conditions, for example, a normality condition, to fulfill than the B-A-bimodule condition A⊗BA∣qA [2]. Thus, the next corollary (or one like it stated more generally for Frobenius extensions) presents a simplification in determining whether a special type of ring extension has finite depth. The corollary follows from the theorem above as well as [11, 8.6], Corollary 2.2, Proposition 4.3 and Theorem 4.4.Corollary 5.3.
LetK⊆H be a Hopf subalgebra pair of finite-dimensional unimodular Hopf algebras. Then, K has finite depth in H if and only if there is a tower algebra Hm such that K⊆Hm has depth 2.



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