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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012Β (2012), Article IDΒ 254791, 22 pages
Research Article

Subring Depth, Frobenius Extensions, and Towers

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

Received 24 March 2012; Accepted 23 April 2012

Academic Editor: TomaszΒ Brzezinski

Copyright Β© 2012 Lars Kadison. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The minimum depth 𝑑(𝐡,𝐴) of a subring π΅βŠ†π΄ introduced in the work of Boltje, Danz and KΓΌlshammer (2011) is studied and compared with the tower depth of a Frobenius extension. We show that 𝑑(𝐡,𝐴) < ∞ if 𝐴 is a finite-dimensional algebra and 𝐡𝑒 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 π΄βŠ‡π΅ is a QF extension, minimum left and right even subring depths are shown to coincide. If π΄βŠ‡π΅ 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.

1. Introduction and Preliminaries

A basic lemma in representation theory states that if a subalgebra 𝐡 of a finite-dimensional algebra 𝐴 has πœ‡βˆΆπ΄βŠ—π΅π΄β†’π΄,π‘ŽβŠ—π‘Žβ€²β†¦π‘Žπ‘Žβ€² a split epimorphism of 𝐴-𝐴-bimodules, then 𝐴 has finite representation type if 𝐡 has. Weakening the condition on πœ‡ to a split epimorphism of 𝐴-𝐡-bimodules does not place any restriction on π΅βŠ†π΄, but the opposite hypothesis that a split monomorphism exists from π΄βŠ—π΅π΄ into a multiple 𝑛𝐴=π΄βŠ•β‹―βŠ•π΄ 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 𝐡, where 𝐴𝐡 is a progenerator module (but 𝐴 and 𝐡 may be infinite-dimensional algebras), the β€œdepth two” condition as the opposite hypothesis is known as, implies that 𝐴 is a Galois extension of 𝐡, 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 𝐡) [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 π΅βŠ†π΄ to a tower of three rings πΆβŠ†π΅βŠ†π΄ [11]. This was applied to the tower of iterated right endomorphism rings above a Frobenius extension π΅βŠ†π΄βŠ†π΄1β†ͺ𝐴2β†ͺβ‹―, so that π΅βŠ†π΄ has (tower) depth 𝑛 if 𝐡β†ͺπ΄π‘›βˆ’3β†ͺπ΄π‘›βˆ’2 has the generalized depth two property (called a depth 3 tower in [11]). This yields a compact matrix inequality condition 𝑀[𝑛+1]β‰€π‘žπ‘€[π‘›βˆ’1](1.1) (some π‘žβˆˆβ„•) 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 𝐡β†ͺ𝐴 [3, 18]. Since 𝑀[2]=𝑀𝑀𝑑, 𝑀[3]=𝑀𝑀𝑑𝑀,…, 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 π΅βŠ†π΄, which has (right) depth 2𝑛 if the relative Hochschild 𝑛+1 bar resolution group 𝐢𝑛+1(𝐴,𝐡) maps as a split monomorphism into a multiple of a smaller group, π‘žπΆπ‘š(𝐴,𝐡) as 𝐴-𝐡-bimodules, and depth 2𝑛+1 if this condition only holds as natural 𝐡-𝐡-bimodules. Since subring π΅βŠ†π΄ having depth π‘š implies that it has depth π‘š+1, the minimum depth 𝑑(𝐡,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 π‘†π‘›βŠ‚π‘†π‘›+1 is 2π‘›βˆ’1 over any ground ring π‘˜ 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 π‘˜[𝐺]βŠ‡π‘˜[𝐻] of finite group algebras over any ground ring π‘˜ has finite depth, in fact bounded by twice the index [πΊβˆΆπ‘πΊ(𝐻)] of the normalizer subgroup.

The notion of subring depth 𝑑(𝐡,𝐴) in [12] is defined in equivalent terms in (1.7). In case 𝐡 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 π΄βŠ‡π΅ 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 groups 𝐢𝑛(𝐴,𝐡) (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 𝐡 of a finite-dimensional algebra 𝐴 has finite depth if its enveloping algebra 𝐡𝑒 has finite representation type.

1.1. π»βˆ’Equivalent Modules

Let 𝐴 be a ring. Two left 𝐴-modules, 𝐴𝑁 and 𝐴𝑀, are said to be β„Ž-equivalent, denoted by π΄π‘€β„ŽβˆΌπ΄π‘, if two conditions are met. First, for some positive integer π‘Ÿ, 𝑁 is isomorphic to a direct summand in the direct sum of π‘Ÿ copies of 𝑀, denoted by π΄π‘βŠ•βˆ—β‰…π΄π‘€π‘Ÿβ‡”π‘βˆ£π‘Ÿπ‘€βŸΊβˆƒπ‘“π‘–ξ€·βˆˆHom𝐴𝑀,𝐴𝑁,π‘”π‘–ξ€·βˆˆHom𝐴𝑁,π΄π‘€ξ€ΈβˆΆπ‘Ÿξ“π‘–=1π‘“π‘–βˆ˜π‘”π‘–=id𝑁.(1.2) Second, symmetrically there is π‘ βˆˆβ„€+ such that π‘€βˆ£π‘ π‘. It is easy to extend this definition of β„Ž-equivalence (sometimes referred to as similarity) to β„Ž-equivalence of two objects in an abelian category and to show that it is an equivalence relation.

If two modules are β„Ž-equivalent, π΄π‘β„ŽβˆΌπ΄π‘€, then they have Morita equivalent endomorphism rings, β„°π‘βˆΆ=End𝐴𝑁 and β„°π‘€βˆΆ=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, π‘€βˆ£π‘ π‘, any of the four Hom-functors such as Hom(π΄βˆ’,𝐴𝑀) from the category of left 𝐴-modules into the category of left 𝐸𝑀-modules. Then, the explicit conditions on mappings for β„Ž-equivalence show that 𝐻(𝑀,𝑁)βŠ—β„°π‘€π»(𝑁,𝑀)→ℰ𝑁 and the reverse mapping given by composition are surjections.

The theory of β„Ž-equivalent modules applies to bimodules π‘‡π‘€π΅β„ŽβˆΌπ‘‡π‘π΅ by letting 𝐴=π‘‡βŠ—β„€π΅op, which sets up an equivalence of abelian categories between 𝑇-𝐡-bimodules and left 𝐴-modules. Two additive functors 𝐹,πΊβˆΆπ’žβ†ͺπ’Ÿ are β„Ž-equivalent if there are natural split epis 𝐹(𝑋)𝑛β†ͺ𝐺(𝑋) and 𝐺(𝑋)π‘šβ†ͺ𝐹(𝑋) for all 𝑋 in π’ž. We leave the proof of the lemma below as an elementary exercise.

Lemma 1.1. Suppose two 𝐴-modules are β„Ž-equivalent, π‘€β„ŽβˆΌπ‘, and two additive functors from 𝐴-modules to an abelian category are β„Ž-equivalent, πΉβ„ŽβˆΌπΊ. Then, 𝐹(𝑀)β„ŽβˆΌπΊ(𝑁).

For example, the following substitution in equations involving the β„ŽβˆΌ-equivalence relation follows from the lemma: π΄π‘ƒπ‘‡β„ŽβˆΌπ΄π‘„π‘‡,π‘‡π‘ˆπ΅β„ŽβˆΌπ‘‡π‘‰π΅βŸΉπ΄π‘ƒβŠ—π‘‡π‘ˆπ΅β„ŽβˆΌπ΄π‘„βŠ—π‘‡π‘‰π΅.(1.3)

Example 1.2. Suppose 𝐴 is a finite-dimensional algebra with indecomposable 𝐴-modules {π‘ƒπ›Όβˆ£π›ΌβˆˆπΌ} (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)={π‘ƒπ›Όβˆ£[𝑃𝛼,𝑀]β‰ 0}, where [𝑃𝛼,𝑀] is the number of factors in 𝑀 isomorphic to 𝑃𝛼. Note that π‘€βˆ£π‘žπ‘ for some positive π‘ž if and only if Indec(𝑀)βŠ†Indec(𝑁). It follows that π‘€β„ŽβˆΌπ‘ if and only if Indec(𝑀)=Indec(𝑁).
Suppose 𝐴𝐴=𝑛1𝑃1βŠ•β‹―βŠ•π‘›π‘Ÿπ‘ƒπ‘Ÿ is the decomposition of the regular module into its projective indecomposables. Let 𝑃𝐴=𝑃1βŠ•β‹―βŠ•π‘ƒπ‘Ÿ. Then, 𝑃𝐴 and 𝐴𝐴 are β„Ž-equivalent, so that 𝐴 and End𝑃𝐴 are Morita equivalent. The algebra End𝑃𝐴 is the basic algebra of 𝐴.

1.2. Depth Two

A subring pair π΅βŠ†π΄ is said to have left depth 2 (or be a left depth two extension [4]) if π΄βŠ—π΅π΄β„ŽβˆΌπ΄ as natural 𝐡-𝐴-bimodules. Right depth 2 is defined similarly in terms of β„Ž-equivalence of natural 𝐴-𝐡-bimodules. In [4] it was noted that the left condition implies the right and conversely if 𝐴 is a Frobenius extension of 𝐡. Also in [4] a Galois theory of Hopf algebroids was defined on the endomorphism ring 𝐻∢=End𝐡𝐴𝐡 as total ring and the centralizer π‘…βˆΆ=𝐴𝐡 as base ring. The antipode is the natural anti-isomorphism stemming from following the arrows: Endπ΄π΅β‰…βŸΆπ΄βŠ—π΅π΄β‰…βŸΆξ€·End𝐡𝐴o𝑝(1.4) restricted to the intersection End𝐡𝐴𝐡=End𝐴𝐡∩End𝐡𝐴.

The Galois extension properties of a depth two extension π΄βŠ‡π΅ are as follows. If 𝐴𝐡 is faithfully flat, balanced or 𝐡 equals its double centralizer in 𝐴, the natural action of 𝐻 on 𝐴 has invariant subalgebra 𝐴𝐻 satisfying the Galois property of 𝐴𝐻=𝐡. Also the well-known Galois property of the endomorphism ring as a cross-product holds: the right endomorphism ring End𝐴𝐡≅𝐴#𝐻, where the latter has smash product ring structure on π΄βŠ—π‘…π» [4]. There is also a duality structure by going a step further along in the tower above π΅βŠ†π΄β†ͺEnd𝐴𝐡β†ͺEndπ΄βŠ—π΅π΄π΄, where the Hopf algebroid π»β€²βˆΆ=(π΄βŠ—π΅π΄)𝐡 is the 𝑅-dual of 𝐻 and acts naturally on End𝐴𝐡 in such a way that End(π΄βŠ—π΅π΄)𝐴 has a smash product ring structure [4].

Conversely, Galois extensions have depth 2. For example, an 𝐻-comodule algebra 𝐴 with invariant subalgebra 𝐡 and finite-dimensional Hopf algebra 𝐻 over a base field π‘˜, which has a Galois isomorphism from π΄βŠ—π΅π΄β‰…β†’π΄βŠ—π‘˜π» given by π‘Žξ…žβŠ—π‘Žβ†¦π‘Žξ…žπ‘Ž(0)βŠ—π‘Ž(1), satisfies (strongly) the depth two condition π΄βŠ—π΅π΄β‰…π΄dim𝐻 as 𝐴-𝐡-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-KΓΌlshammer [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 πœ€βˆΆπ΄β†’π‘˜ be an algebra homomorphism into the ground field π‘˜. Let 𝐴+ denote kerπœ€, and, for a subalgebra π΅βŠ†π΄, let 𝐡+ denote kerπœ€βˆ©π΅.

Proposition 1.3. Suppose π΅βŠ†π΄ is a subalgebra of an augmented algebra. If π΅βŠ†π΄ has right depth 2, then 𝐴𝐡+βŠ†π΅+𝐴.

The proof of this proposition is an exercise in tensoring both sides of π΄βŠ—π΅π΄βŠ•βˆ—β‰…π‘žπ΄ by the unit 𝐴-module π‘˜, 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. Let 𝐴=𝑇𝑛(π‘˜) be the algebra of 𝑛 by 𝑛 upper triangular matrices where 𝑛>1, and 𝐡=𝐷𝑛(π‘˜) the subalgebra of diagonal matrices. Note that there are 𝑛 augmentations πœ€π‘–βˆΆπ΄β†’π‘˜ given by πœ€π‘–(𝑋)=𝑋𝑖𝑖, and each of the 𝐡+𝑖 satisfies the inclusions above if left or right depth two. This is a clear contradiction, thus 𝑑(𝐡,𝐴)>2. We will see below that 𝑑(𝐡,𝐴)=3.

Also subalgebra pairs of semisimple complex algebras have depth 2 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). Suppose π΅βŠ†π΄ is a subalgebra pair of semisimple complex algebras. Then, π΅βŠ†π΄ has depth 2 if and only if, for every maximal ideal 𝐼 in 𝐴, one has 𝐴(𝐼∩𝐡)=(𝐼∩𝐡)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 centralizer 𝑅 mentioned above is semisimple (see Kaplansky's Fields and Rings for a πΆβˆ—-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, let 𝐴 be a unital associative ring and π΅βŠ†π΄ a subring where 1𝐡=1𝐴. Note the natural bimodules 𝐡𝐴𝐡 obtained by restriction of the natural 𝐴-𝐴-bimodule (briefly 𝐴-bimodule) 𝐴, also to the natural bimodules 𝐡𝐴𝐴, 𝐴𝐴𝐡 or 𝐡𝐴𝐡, which are referred to with no further ado.

Let 𝐢0(𝐴,𝐡)=𝐡, and, for 𝑛β‰₯1, 𝐢𝑛(𝐴,𝐡)=π΄βŠ—π΅β‹―βŠ—π΅π΄(𝑛times𝐴),(1.5) For 𝑛β‰₯1, 𝐢𝑛(𝐴,𝐡) has a natural 𝐴-bimodule structure, which restricts to 𝐡-𝐴-, 𝐴-𝐡-, and 𝐡-bimodule structures occurring in the next definition.

Definition 1.6. The subring π΅βŠ†π΄ has depth 2𝑛+1β‰₯1 if as 𝐡-bimodules 𝐢𝑛(𝐴,𝐡)β„ŽβˆΌπΆπ‘›+1(𝐴,𝐡). The subring π΅βŠ†π΄ has left (resp., right) depth 2𝑛β‰₯2 if 𝐢𝑛(𝐴,𝐡)β„ŽβˆΌπΆπ‘›+1(𝐴,𝐡) as 𝐡-𝐴-bimodules (resp., 𝐴-𝐡-bimodules).

It is clear that if π΅βŠ†π΄ has either left or right depth 2𝑛, it has depth 2𝑛+1 by restricting the β„Ž-equivalence condition to 𝐡-bimodules. If it has depth 2𝑛+1, it has depth 2𝑛+2 by tensoring the β„Ž-equivalence by βˆ’βŠ—π΅π΄ or π΄βŠ—π΅βˆ’. The minimum depth is denoted by 𝑑(𝐡,𝐴); if π΅βŠ†π΄ has no finite depth, write 𝑑(𝐡,𝐴)=∞.

Note that the minimum left and right minimum even depths may differ by 2 (in which case 𝑑(𝐡,𝐴) is the lesser of the two). In the next section we provide a general condition, which includes a Hopf subalgebra pair π΅βŠ†π΄ of symmetric (Frobenius) algebras, where the left and right minimum even depths coincide.

Also note that a subalgebra pair of Artin algebras π΅βŠ†π΄ have depth 2𝑛+1 if and only if the indecomposable module constituents of 𝐢𝑛+π‘š(𝐴,𝐡) remain the same for all π‘šβ‰₯0 as those already found in 𝐢𝑛(𝐴,𝐡) (see Example 1.2). This corresponds well with the classical notion of finite depth in subfactor theory.

Example 1.7. Again let 𝐴=𝑇𝑛(π‘˜) and 𝐡=𝐷𝑛(π‘˜)β‰…π‘˜π‘›, where 𝑛>1. Let 𝑒𝑖𝑗 denote the matrix units, π‘˜π‘– the 𝑛 simple 𝐡-modules, and π‘˜π‘–π‘— for 1≀𝑖≀𝑗≀𝑛 the 𝑛(𝑛+1)/2 simple components of 𝐡𝐴𝐡. Note that π΄βŠ—π΅π΄ as a 𝐡-bimodule has components π‘˜π‘’π‘–π‘ βŠ—π΅π‘’π‘ π‘—β‰…π‘˜π‘–π‘— where 𝑖≀𝑠≀𝑗, so π΄βŠ—π΅π΄βˆ£π‘›π΄ as 𝐡-bimodules. Thus, 𝑑(𝐡,𝐴)≀3. But 𝑑(𝐡,𝐴)β‰ 2 by the remark following Proposition 1.3; then 𝑑(𝐡,𝐴)=3.

1.4. β„‹-Depth

A subring π΅βŠ†π΄ has β„‹-depth 2π‘›βˆ’1 if 𝐢𝑛+1(𝐴,𝐡)β„ŽβˆΌπΆπ‘›(𝐴,𝐡) as 𝐴-𝐴-bimodules (𝑛=1,2,3,…). Note that 𝐡 has β„‹-depth 2π‘›βˆ’1 in 𝐴 implies that it has β„‹-depth 2𝑛+1 (also that it has depth 2𝑛). Thus, define the minimum β„‹-depth 𝑑ℋ(𝐡,𝐴) if it exists. Note that the definition of β„‹-depth 2π‘›βˆ’1 is equivalent to the condition on a subring π΅βŠ†π΄ that 𝐢𝑛+1(𝐴,𝐡)βˆ£π‘žπΆπ‘›(𝐴,𝐡) for some π‘žβˆˆβ„•. This is clear for 𝑛≠2 since 𝐢𝑛(𝐴,𝐡)βˆ£πΆπ‘›+1(𝐴,𝐡). For 𝑛=1, the 𝐻-separability condition π΄π΄βŠ—π΅π΄π΄βŠ•βˆ—β‰…π΄π΄π΄π‘ž(1.6) implies the separability condition π΄π΄π΄βŠ•βˆ—β‰…π΄π΄βŠ—π΅π΄π΄ as argued in the paper [13] by Hirata. The notion of β„‹-depth is studied in [14] where it is noted that |𝑑ℋ(𝐡,𝐴)βˆ’π‘‘(𝐡,𝐴)|≀2 if one or the other minimum depth is finite. See Section 2 for which Hopf subalgebras satisfy the 𝑑ℋ(𝐡,𝐴)=1 condition in (1.6).

Remark 1.8. Suppose 𝐡 is a subring of 𝐴. The minimum depth of the subring π΅βŠ†π΄ as defined in Boltje-Danz-KΓΌlshammer [12] coincides with 𝑑(𝐡,𝐴). In fact, for 𝑛>0, the depth 2𝑛+1 condition in [12] is that for some π‘žβˆˆβ„€+𝐢𝑛+1(𝐴,𝐡)βˆ£π‘žπΆπ‘›(𝐴,𝐡)(1.7) as 𝐡-bimodules. The left depth 2𝑛 condition in [12] is (1.7) more strongly as natural 𝐡-𝐴-bimodules (and as 𝐴-𝐡-bimodules for the right depth 2𝑛 condition). But (using a pair of classical face and degeneracy maps of homological algebra) we always have 𝐢𝑛(𝐴,𝐡)βˆ£πΆπ‘›+1(𝐴,𝐡) as 𝐴-𝐡-, 𝐡-𝐴-, or 𝐡-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 depth 1 in this paper is equivalent to the subring depth 1 notion in, for example, [4, 12, 15] since 𝐴 is β„Ž-equivalent to 𝐡 as 𝐡-bimodules if and only if 𝐴 is centrally projective over 𝐡 (i.e., π΄βˆ£π‘žπ΅ as 𝐡-bimodules). This follows from the lemma below.

Lemma 1.9. Suppose 𝐡 is a subring of ring 𝐴 such that π΅π΄π΅βˆ£π‘šπ΅π΅π΅ for some integer π‘šβ‰₯1. Then, 𝐡𝐡𝐡∣𝐡𝐴𝐡.

Proof. From the central projectivity condition on 𝐴, one obtains π‘š maps β„Žπ‘–βˆˆHom(𝐡𝐴𝐡,𝐡𝐡𝐡) and π‘š maps π‘”π‘–βˆˆHom(𝐡𝐡𝐡,𝐡𝐴𝐡)β‰…β†¦π‘£π‘–βˆˆπ΄π΅ such that βˆ‘π‘šπ‘–=1π‘£π‘–β„Žπ‘–(π‘Ž)=π‘Ž for every π‘Žβˆˆπ΄. It follows that π΄β‰…π΅βŠ—π‘(𝐡)𝐴𝐡 since β„Žπ‘–(𝐴𝐡)βŠ†π‘(𝐡). Note that restricting the equation to the centralizer 𝐴𝐡 shows that 𝐴𝐡 is a finitely generated projective 𝑍(𝐡)-module. But 𝑍(𝐡)βŠ†π΄π΅ is a commutative subring, whence 𝐴𝐡 is a generator 𝑍(𝐡)-module. From 𝑍(𝐡)βŠ•βˆ—β‰…π‘›π΄π΅ for some positive integer 𝑛, it follows from the tensor algebra decomposition of 𝐴 that π΅π΅π΅βˆ£π‘›π΅π΄π΅. Whence there are 𝑛 maps π‘“π‘–βˆˆHom(𝐡𝐴𝐡,𝐡𝐡𝐡) and 𝑛 elements π‘Ÿπ‘–βˆˆπ΄π΅ such that βˆ‘π‘›π‘–=1𝑓𝑖(π‘Ÿπ‘–)=1𝐴. Define a (condition expectation or) bimodule projection βˆ‘πΈ(π‘Ž)∢=𝑛𝑖=1𝑓𝑖(π‘Žπ‘Ÿπ‘–) of 𝐴 onto 𝐡.

Example 1.10. The paper [12] asks in its introduction about the depth 𝑑(𝐡,𝐴) of invariant subrings in classical invariant theory, where 𝐾 is a field, 𝐴=𝐾[𝑋1,…,𝑋𝑛], 𝐡=π‘˜[𝑋1,…,𝑋𝑛]𝐺 and 𝐺 is a finite group in 𝐺𝐿𝑛(𝐾) acting by linear substitution of the variables. In any case 𝐴𝐡 is finitely generated and 𝐡 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|, 𝐡 is itself an 𝑛-variable polynomial algebra and 𝐴 is a free 𝐡-module; consequences of the Shephard-Todd Theorem [16, 17]. Since 𝐴 is a commutative algebra, it follows that 𝑑(𝐡,𝐴)=1.

Example 1.11. Let π΅βŠ†π΄ be a subring pair of semisimple complex algebras. Then, the minimum depth 𝑑(𝐡,𝐴) may be computed from the inclusion matrix 𝑀, alternatively an π‘Ÿ-by-𝑠 induction-restriction table of π‘Ÿπ΅-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 𝐡-simples. The procedure to obtain 𝑑(𝐡,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(𝐡,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(𝐡,𝐴)≀2π‘‘βˆ’1 where 𝑀𝑀𝑑 has degree 𝑑 minimal polynomial [3]. A GAP subprogram exists to compute 𝑑(𝐡,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(𝐡,𝐴) being the least point of no decrease.
In terms of the bipartite graph of the inclusion π΅βŠ†π΄, 𝑑(𝐡,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 𝐡-simples be represented by π‘Ÿ 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, let 𝐴=ℂ𝑆4, the complex group algebra of the permutation group on four letters, and 𝐡=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, whence 𝑑(𝐡,𝐴)=5.

Example 1.12. The induction-restriction table 𝑀 of the inclusion of permutation groups π‘†π‘›Γ—π‘†π‘š<𝑆𝑛+π‘š via βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ (𝜎,𝜏)↦1⋯𝑛𝑛+1⋯𝑛+π‘šπœŽ(1)β‹―πœŽ(𝑛)𝑛+𝜏(1)⋯𝑛+𝜏(π‘š)(1.8) may be computed combinatorially from the Littlewood-Richardson coefficients π‘π›Ύπœ‡πœˆβˆˆβ„•, where πœ‡ is partition of 𝑛, 𝜈=(𝜈1,…,πœˆπ‘š) a partition of π‘š, and πœ† a partition of 𝑛+π‘š. Briefly, the coefficient number π‘π›Ύπœ‡πœˆ is zero if 𝛾 does not contain πœ‡ or is the number of Littlewood-Richardson fillings with content 𝜈 of 𝛾 with πœ‡ removed. A Littlewood-Richardson filling of a skew Young tableau is with integers 𝑖=1,2,…,π‘š occuring πœˆπ‘– 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 matrix 𝑀 for the subgroup 𝑆2×𝑆3<𝑆5 with respect to the ordered bases of irreducible characters of the subgroup πœ†(12)Γ—πœ‡(13), πœ†(12)Γ—πœ‡(2,1), πœ†(12)Γ—πœ‡(3), πœ†(2)Γ—πœ‡(13), πœ†(2)Γ—πœ‡(2,1), πœ†(2)Γ—πœ‡(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 βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ π‘€=111000001111000000110010010000111100001011.(1.9) 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)=2π‘›βˆ’1 [3, 12]; however, a formula for 𝑑(π‘†π‘›Γ—π‘†π‘š,𝑆𝑛+π‘š) is not known.

1.5. Finite Depth and Finite Representation Type

For the next proposition we adopt the notation 𝐡𝑒 for the (enveloping) algebra π΅βŠ—π‘˜π΅o𝑝 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 characteristic 𝑝 has finite representation type if and only if its Sylow 𝑝-subgroup is cyclic. Thus, 𝐡 having finite representation type does not imply that 𝐡𝑒 has finite representation type.

Proposition 1.13. Suppose π΅βŠ†π΄ is a subalgebra pair of finite-dimensional algebras where 𝐡𝑒 has in all π‘Ÿ indecomposable 𝐡𝑒-module isomorphism classes. Then, 𝑑(𝐡,𝐴)≀2π‘Ÿ+1.

Proof. This follows from the observation in Example 1.2 that since 𝐢𝑛(𝐴,𝐡) is the image of 𝐢𝑛+1(𝐴,𝐡) under an obvious split epimorphism of 𝐡𝑒-modules (equivalently, 𝐡-bimodules), there is an increasing chain of subset inclusions ξ€·Indec(𝐴)βŠ†Indecπ΄βŠ—π΅π΄ξ€Έξ€·βŠ†Indecπ΄βŠ—π΅π΄βŠ—π΅π΄ξ€ΈβŠ†β‹―,(1.10) which stops strictly increasing in at most π‘Ÿ steps. When Indec(𝐢𝑛(𝐴,𝐡))=Indec(𝐢𝑛+1(𝐴,𝐡)), then 𝐢𝑛(𝐴,𝐡)β„ŽβˆΌπΆπ‘›+1(𝐴,𝐡) as 𝐡𝑒-modules, whence π΄βŠ‡π΅ has depth 2𝑛+1≀2π‘Ÿ+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 π΅βŠ†π΄, 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 𝐡𝑒 is a Hopf algebra and semisimple if 𝐡 is so.) For example, we have the following corollary.

Corollary 1.14. Suppose 𝐡 is a semisimple Hopf subalgebra in a finite dimensional Hopf algebra 𝐴. Suppose that 𝐡 has 𝑛 nonisomorphic simple modules. Then, 𝑑(𝐡,𝐴)≀2𝑛2+1.

2. When Frobenius Extensions of the Second Kind Are Ordinary

A (proper) ring extension π΄βŠ‡π΅ is a subring or more generally a monomorphism πœ„βˆΆπ΅β†ͺ𝐴, which is equivalent to a subring πœ„(𝐡)βŠ†π΄. Restricted modules such as π΄πœ„(𝐡) and pullback modules 𝐴𝐡 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 𝐡→𝐴 and its pullback modules such as 𝐴𝐡; however, certain conditions needed below such as 𝐴𝐡 is a generator imply that 𝐡→𝐴 is monic.)

A ring extension π΄βŠ‡π΅ is a left QF extension if the module 𝐡𝐴 is finitely generated projective and the natural 𝐴-𝐡-bimodules satisfy π΄βˆ£π‘žHom(𝐡𝐴,𝐡𝐡) for some positive integer π‘ž. A right QF extension is oppositely defined. A QF extension π΄βŠ‡π΅ is both a left and right QF extension and may be characterized by both 𝐴𝐡 and 𝐡𝐴 being finite projective, and two β„Ž-equivalences of bimodules given by π΄π΄π΅β„ŽβˆΌπ΄Hom(𝐡𝐴,𝐡𝐡)𝐡 and (π΅π΄π΄β„ŽβˆΌπ΅Hom(𝐴𝐡,𝐡𝐡)𝐴 [21]. For example, a Frobenius extension π΄βŠ‡π΅ is a QF extension since it is left and right finite projective and satisfies the stronger conditions that 𝐴 is isomorphic to its right 𝐡-dual π΄βˆ— and its left 𝐡-dual βˆ—π΄ as natural 𝐡-𝐴-bimodules, respectively 𝐴-𝐡-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. Let 𝛽 be an automorphism of the ring 𝐡 and π΅βŠ†π΄ a subring pair. Denote the pullback module of a module 𝐡𝑀 along π›½βˆΆπ΅β†’π΅ by 𝛽𝑀, the so-called twisted module. A ring extension π΄βŠ‡π΅ is a 𝛽-Frobenius extension if 𝐴𝐡 is finite projective and there is a bimodule isomorphism 𝐡𝐴𝐴≅𝛽Hom(𝐴𝐡,𝐡𝐡). One shows that π΄βŠ‡π΅ is a Frobenius extension if and only if 𝛽 is an inner automorphism. A subring pair of Frobenius algebras π΅βŠ†π΄ is 𝛽-Frobenius extension so long as 𝐴𝐡 is finite projective and the Nakayama automorphism πœ‚π΄ of 𝐴 stabilizes 𝐡, in which case 𝛽=πœ‚π΅βˆ˜πœ‚π΄βˆ’1 [22]. For instance a finite-dimensional Hopf algebra 𝐴=𝐻 and 𝐡=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 π‘₯∈𝐾, 𝛽(π‘₯)=(π‘₯)π‘šπ»ξ€·π‘₯(1)ξ€Έπ‘šπΎξ€·π‘†ξ€·π‘₯(2)π‘₯ξ€Έξ€Έ(3),(2.1)

Given the bimodule isomorphism above 𝐡𝐴𝐴≅→𝛽Hom(𝐴𝐡,𝐡𝐡), apply it to 1𝐴 and let its value be πΈβˆΆπ΄β†’π΅, which then is a cyclic generator of 𝛽Hom(𝐴𝐡,𝐡𝐡)𝐴 satisfying 𝐸(𝑏1π‘Žπ‘2)=𝛽(𝑏1)𝐸(π‘Ž)𝑏2 for all 𝑏1,𝑏2∈𝐡,π‘Žβˆˆπ΄. If π‘₯1,…,π‘₯π‘šβˆˆπ΄ and πœ™1,…,πœ™π‘šβˆˆHom(𝐴𝐡,𝐡𝐡) are projective bases of 𝐴𝐡, and πΈπ‘¦π‘—βˆΆ=𝐸(π‘¦π‘—βˆ’)=πœ™π‘— the equations π‘šξ“π‘—=1π‘₯π‘—πΈξ€·π‘¦π‘—π‘Žξ€Έ=π‘Ž,π‘šξ“π‘—=1π›½βˆ’1ξ€·πΈξ€·π‘Žπ‘₯𝑗𝑦𝑗=π‘Ž(2.2) hold for all π‘Žβˆˆπ΄. Call (𝐸,π‘₯𝑗,𝑦𝑗) a 𝛽-πΉπ‘Ÿπ‘œπ‘π‘’π‘›π‘–π‘’π‘ π‘π‘œπ‘œπ‘Ÿπ‘‘π‘–π‘›π‘Žπ‘‘π‘’π‘ π‘¦π‘ π‘‘π‘’π‘š of π΄βŠ‡π΅. Note that also 𝐡𝐴 is finite projective, that a 𝛽-Frobenius coordinate system is equivalent to the ring extension 𝐴∣𝐡 being 𝛽-Frobenius and that 𝛽=id𝐡 if 𝐡 is in the center of 𝐴. Additionally, one notes that there is an automorphism πœ‚ of the centralizer subring 𝐴𝐡 such that 𝐸(π‘Žπ‘)=𝐸(πœ‚(𝑐)π‘Ž) for all π‘Žβˆˆπ΄ and π‘βˆˆπ΄π΅. Also an isomorphism π΄π›½βŠ—π΅π΄β‰…End𝐴𝐡 is easily defined from the data and equations above, where βˆ‘π‘—π‘₯π‘—βŠ—π‘¦π‘—β†¦id𝐴, so that if (𝐸,𝑧𝑖,𝑀𝑖) is another 𝛽-Frobenius coordinate system (sharing the same πΈβˆΆπ΄β†’π΅), then βˆ‘π‘–π‘§π‘–βŠ—π΅π‘€π‘–=βˆ‘π‘—π‘₯π‘—βŠ—π΅π‘¦π‘— in (π΄π›½βŠ—π΅π΄)𝐴.

When a 𝛽-Frobenius extension is a QF extension is addressed in the next proposition.

Proposition 2.1. A 𝛽-Frobenius extension π΄βŠ‡π΅ is a left QF extension if and only if there are 𝑒𝑖,π‘£π‘–βˆˆπ΄(𝑖=1,…,𝑛) such that 𝑠𝑒𝑖=𝑒𝑖𝛽(𝑠) and 𝑣𝑖𝑠=𝛽(𝑠)𝑣𝑖 for all 𝑖 and π‘ βˆˆπ΅, and π›½βˆ’1(𝑠)=𝑛𝑖=1𝑒𝑖𝑠𝑣𝑖.(2.3)

Proof. Suppose π΄βŠ‡π΅ is 𝛽-Frobenius extension with 𝛽-Frobenius system satisfying the equations above. Given the elements 𝑒𝑖,π‘£π‘–βˆˆπ΄ satisfying the equations above, let 𝐸𝑖=𝐸(π‘’π‘–βˆ’), which defines 𝑛 mappings in (the untwisted) Hom(𝐡𝐴𝐡,𝐡𝐡𝐡). Also define 𝑛 mappings πœ“π‘–βˆˆHom(𝐴(βˆ—π΄)𝐡,𝐴𝐴𝐡) by πœ“π‘–βˆ‘(𝑔)=π‘šπ‘—=1π‘₯𝑗𝑔(𝑣𝑖𝑦𝑗) where it is not hard to show using the 𝛽-Frobenius coordinate equations that βˆ‘π‘—π‘₯π‘—βŠ—π΅π‘£π‘–π‘¦π‘—βˆˆ(π΄βŠ—π΅π΄)𝐴 for each 𝑖 (a Casimir element). It follows that βˆ‘π‘›π‘–=1πœ“π‘–(𝐸𝑖)=1𝐴 and that π΄βˆ£π‘›(βˆ—π΄) as natural 𝐴-𝐡-bimodules, whence 𝐴 is a left QF extension of 𝐡.
Conversely, assume the left QF condition π΅π΄βˆ—π΄βˆ£π΄π‘›, equivalent to π΄π΄π΅βˆ£π‘›(βˆ—π΄) by applying the right 𝐡-dual functor and noting (βˆ—π΄)βˆ—β‰…π΄ as well βˆ—(π΄βˆ—)≅𝐴. Also assume the slightly rewritten 𝛽-Frobenius condition π›½βˆ’1𝐴𝐴≅𝐡(π΄βˆ—)𝐴, which then implies π›½βˆ’1π΄π΄βˆ£π‘›π΄. So there are 𝑛 mappings π‘”π‘–βˆˆHom(π›½βˆ’1𝐴𝐴,𝐡𝐴𝐴) and 𝑛 mappings π‘“π‘–βˆˆHom(𝐡𝐴𝐴,π›½βˆ’1𝐴𝐴) such that βˆ‘π‘›π‘–=1π‘“π‘–βˆ˜π‘”π‘–=id𝐴. Equivalently, with π‘’π‘–βˆΆ=𝑓(1𝐴) and π‘£π‘–βˆΆ=𝑔(1𝐴), βˆ‘π‘›π‘–=1𝑒𝑖𝑣𝑖=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. Let 𝐻 be a finite dimensional Hopf algebra and 𝐾 a Hopf subalgebra. In the notation of (2.1) the following are equivalent. (1)The automorphism 𝛽=id𝐾 and π»βŠ‡πΎ is a Frobenius extension. (2)The algebra extension π»βŠ‡πΎ is a QF extension. (3)The modular functions π‘šπ»(π‘₯)=π‘šπΎ(π‘₯) for all π‘₯∈𝐾.

Proof. (1β‡’2) A Frobenius extension is a QF extension. (2β‡’3) Set 𝑠=1 in (2.3), and apply the counit πœ€ to see that βˆ‘πœ€(𝑖𝑒𝑖𝑣𝑖)=1. Reapply πœ€ to (2.3) to obtain πœ€βˆ˜π›½=πœ€. Apply πœ€ to (2.1), and use uniqueness of inverse in convolution algebra Hom(𝐾,π‘˜), where π‘šπΎβˆ˜π‘†=π‘šπΎβˆ’1 and 1=πœ€, to show that π‘šπ»=π‘šπΎ on 𝐾. (3β‡’1) This follows from (2.1).

The following observation for a normal Hopf subalgebra πΎβŠ†π» has not been explicitly noted before in the literature.

Corollary 2.3. The modular function of a finite dimensional Hopf algebra 𝐻 restricts to the modular function of a Hopf subalgebra πΎβŠ†π» if 𝐾 has depth 𝑑(𝐾,𝐻)≀2.

Proof. If the Hopf subalgebra 𝐾 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 πΎβŠ†π» is an 𝐻-Galois extension: here 𝐻∢=𝐻/𝐻𝐾+ denotes the quotient Hopf algebra, 𝐻→𝐻, β„Žβ†¦β„Ž denotes the quotient map, and the Galois isomorphism canβˆΆπ»βŠ—πΎπ»β†’π»βŠ—π» is given by can(β„ŽβŠ—β„Žβ€²)=β„Žβ„Žβ€²(1)βŠ—β„Žβ€²(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 π‘šπΎ=π‘šπ»βˆ£πΎ.

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 π΅βŠ†π΄.

Lemma 2.4. Let (𝐴,πœ€) be an augmented Frobenius algebra with Nakayama automorphism πœ‚π΄, 𝐡 a subalgebra and Frobenius algebra where πœ‚π΄(𝐡)=𝐡, and 𝐴𝐡 finitely generated projective. It follows that π΄βŠ‡π΅ is a 𝛽-Frobenius extension where 𝛽=πœ‚π΅βˆ˜πœ‚π΄βˆ’1, a relative Nakayama automorphism [22, Satz 7], [29, Paragraph 5.1]. Then the modular automorphisms of 𝐴 and 𝐡 satisfy π‘šπ΄βˆ£π΅=π‘šπ΅βˆ˜π›½.(2.4)

Proof. Let (πœ™,π‘₯𝑖,𝑦𝑖,πœ‚π΄) be a Frobenius coordinate system for 𝐴, π‘‘π΄βˆˆπ΄ a right norm satisfying πœ™π‘‘π΄=πœ€, then 𝑑𝐴 is a right integral, satisfying 𝑑𝐴π‘₯=π‘‘π΄πœ€(π‘₯) for all π‘₯∈𝐴, spanning the one-dimensional space of integrals in 𝐴. Let π‘šπ΄ be the augmentation on 𝐴 defined by π‘₯𝑑𝐴=π‘šπ΄(π‘₯)𝑑𝐴 for π‘₯∈𝐴. It follows that πœ€=π‘šπ΄βˆ˜πœ‚π΄ by expressing 𝑑𝐴 in terms of dual bases, πœ€ and π‘šπ΄=π‘‘π΄πœ™ (and note that (πœ™,𝑦𝑖,πœ‚π΄(π‘₯𝑖)) are also dual bases) [29, Paragraph 3.2]. Similarly let (πœ“,𝑒𝑗,𝑣𝑗,πœ‚π΅) be a Frobenius coordinate system for 𝐡 and 𝑑𝐡 a right norm satisfying πœ“π‘‘π΅=πœ€βˆ£π΅, then 𝑑𝐡 is a right integral in 𝐡 and π‘₯𝑑𝐡=π‘šπ΅(π‘₯)𝑑𝐡 defines the π‘˜-valued algebra homomorphism π‘šπ΅, which satisfies πœ€βˆ£π΅=π‘šπ΅βˆ˜πœ‚π΅. It follows that π‘šπ΅βˆ˜π›½=π‘šπ΅βˆ˜πœ‚π΅βˆ˜πœ‚π΄βˆ’1=πœ€βˆ˜πœ‚π΄βˆ’1∣𝐡=π‘šπ΄βˆ£π΅.

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 algebra 𝐻 over its cyclic group subalgebra 𝐾 is a nontrivial 𝛽-Frobenius extension [23]. The algebra 𝐻 is generated over β„‚ by a grouplike 𝑔 of order 𝑛β‰₯2, a nilpotent π‘₯ of index 𝑛, and (𝑔,1)-primitive element where π‘₯𝑔=πœ“π‘”π‘₯ for πœ“βˆˆβ„‚ a primitive 𝑛th root of unity. This is a Hopf algebra having right integral 𝑑𝐻=π‘₯π‘›βˆ’1βˆ‘π‘›βˆ’1𝑗=0𝑔𝑗 with modular function π‘šπ»(𝑔)=πœ“ [23]. The Hopf subalgebra 𝐾 is generated by 𝑔. Then the twist automorphism of 𝐾 is given by 𝛽(𝑔𝑗)=πœ“π‘—π‘”π‘—. Of course, π‘šπ» restricted to 𝐾 is not equal to π‘šπΎ=πœ€βˆ£πΎ. The depth 𝑑(𝐾,𝐻)=3 is computed in [30].

Finally we note that unimodular Hopf algebra extensions are trivial if the β„‹-depth condition 𝑑ℋ(𝐡,𝐴)=1 is imposed.

Proposition 2.6. Suppose 𝐻 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. Since 𝐻 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 π»βŠ—πΎπ»β‰…Hom𝑍(𝐻)(𝐢𝐻(𝐾),𝐻) via π‘₯βŠ—πΎπ‘¦β†¦πœ†π‘₯βˆ˜πœŒπ‘¦, left and right multiplication [23, 31], where 𝐢𝐻(𝐾) denotes the centralizer subalgebra of 𝐾 in 𝐻. If π‘‘βˆˆπΆπ»(𝐢𝐻(𝐾)), then it is obvious from this that π‘‘βŠ—πΎ1𝐻=1π»βŠ—πΎπ‘‘, so that π‘‘βˆˆπΎ: it follows that 𝐾=𝐢𝐻𝐢𝐻(𝐾).(2.5)
Since 𝐻 is unimodular, it has a two-sided nonzero integral 𝑑. Note that π‘‘βˆˆπ‘(𝐻)βŠ†πΆπ»(𝐢𝐻(𝐾)), whence π‘‘βˆˆπΎ. Let πœ†βˆΆπ»β†’π‘˜ (where π‘˜ is the arbitrary ground field) be the left integral in the dual Hopf algebra π»βˆ— such that πœ†β†Όπ‘‘=πœ€. The bijective antipode π‘†βˆΆπ»β†’π» satisfies βˆ‘π‘†(π‘Ž)=(𝑑)𝑑(1)πœ†(π‘Žπ‘‘(2)) since βˆ‘(π‘Ž)π‘Ž(1)𝑆(π‘Ž(2))=1π»πœ€(π‘Ž) and πœ†β‡€π‘₯=πœ†(π‘₯)1𝐻 for all π‘₯,π‘Žβˆˆπ». Since βˆ‘Ξ”(𝑑)=(𝑑)𝑑(1)βŠ—π‘‘(2)βˆˆπΎβŠ—πΎ, it follows that 𝑆(π‘Ž)∈𝐾 for all π‘Žβˆˆπ». Thus 𝐻=𝐾.

3. Even Depth of QF Extensions

It is well known that for a Frobenius extension π΄βŠ‡π΅, coinduction of a module, 𝑀𝐡↦Hom(𝐴𝐡,𝑀𝐡) is naturally isomorphic as functors to induction (π‘€π΅β†¦π‘€βŠ—π΅π΄) (from the category of 𝐡-modules into the category of 𝐴-modules). Similarly, a QF extension has β„Ž-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. Suppose 𝑇𝑀𝐡 is a bimodule and π΄βŠ‡π΅ is a QF extension. Then, there is an β„Ž-equivalence of bimodules, π‘‡π‘€βŠ—π΅π΄π΄β„ŽβˆΌπ‘‡ξ€·π΄Hom𝐡,𝑀𝐡𝐴.(3.1)

Proof. Since 𝐴𝐡 is f.g. projective, it follows that there is a 𝑇-𝐴-bimodule isomorphism π‘€βŠ—π΅ξ€·π΄Hom𝐡,𝐡𝐡𝐴≅Hom𝐡,𝑀𝐡,(3.2) given by π‘šβŠ—π΅πœ™β†¦π‘šπœ™(βˆ’) with inverse constructed from projective bases for 𝐴𝐡. But the right 𝐡-dual of 𝐴 is β„Ž-equivalent to 𝐡𝐴𝐴, 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 Definition 1.6 where as before 𝐢𝑛(𝐴,𝐡)=π΄βŠ—π΅β‹―βŠ—π΅π΄, 𝑛 times 𝐴).

Theorem 3.2. If π΄βŠ‡π΅ is QF extension, then π΄βŠ‡π΅ has left depth 2𝑛 if and only if π΄βŠ‡π΅ has right depth 2𝑛.

Proof. The left depth 2𝑛 condition on π΄βŠ‡π΅ recall is 𝐢𝑛+1(𝐴,𝐡)β„ŽβˆΌπΆπ‘›(𝐴,𝐡) as 𝐡-𝐴-bimodules. To this apply the additive functor Hom(βˆ’π΄,𝐴𝐴) (into the category of 𝐴-𝐡-bimodules), noting that Hom(𝐢𝑛(𝐴,𝐡)𝐴,𝐴𝐴)β‰…Hom(πΆπ‘›βˆ’1(𝐴,𝐡)𝐡,𝐴𝐡) via 𝑓↦𝑓(βˆ’βŠ—π΅β‹―βˆ’βŠ—π΅1𝐴) for each integer 𝑛β‰₯1. It follows (from Lemma 1.1) that there is an 𝐴-𝐡-bimodule β„Ž-equivalence, 𝐢Hom𝑛(𝐴,𝐡)𝐡,π΄π΅ξ€Έβ„Žξ€·πΆβˆΌHomπ‘›βˆ’1(𝐴,𝐡)𝐡,𝐴𝐡.(3.3) (Then in the depth two case, the left depth two condition is equivalent to End π΄π΅β„ŽβˆΌπ΄ as natural 𝐴-𝐡-bimodules.)
Given bimodule 𝐴𝑀𝐡, we have π΄π‘€βŠ—π΅π΄π΄β„ŽβˆΌπ΄Hom(𝐴𝐡,𝑀𝐡)𝐴 by the previous lemma: apply this to 𝐢𝑛+1(𝐴,𝐡)=𝐢𝑛(𝐴,𝐡)βŠ—π΅π΄ using the hom-tensor adjoint relation: there are β„Ž-equivalences and isomorphisms of 𝐴-bimodules, 𝐢𝑛+1(𝐴,𝐡)β„Žξ€·π΄βˆΌHom𝐡,𝐢𝑛(𝐴,𝐡)π΅ξ€Έβ„Žξ€·π΄βˆΌHom𝐡𝐴,Hom𝐡,πΆπ‘›βˆ’1(𝐴,𝐡)𝐡𝐡≅Homπ΄βŠ—π΅π΄π΅,πΆπ‘›βˆ’1(𝐴,𝐡)π΅ξ€Έβ‹―β„Žξ€·πΆβˆΌHom𝑝(𝐴,𝐡)𝐡,πΆπ‘›βˆ’π‘+1(𝐴,𝐡)𝐡,(3.4) for each 𝑝=1,2,…,𝑛 and 𝑛=1,2,…. Compare (3.3) and (3.4) with 𝑝=𝑛 to get 𝐴𝐢𝑛+1(𝐴,𝐡)π΅β„ŽβˆΌπ΄πΆπ‘›(𝐴,𝐡)𝐡, 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 to 𝛽-Frobenius or even twisted QF extensions presents technical problems and is unknown to the author.

4. Frobenius Extensions

As noted above a Frobenius extension π΄βŠ‡π΅ is characterized by any of the following four conditions [23]. First, 𝐴𝐡 is finite projective and 𝐡𝐴𝐴≅Hom(𝐴𝐡,𝐡𝐡). Second, 𝐡𝐴 is finite projective and 𝐴𝐴𝐡≅Hom(𝐡𝐴,𝐡𝐡). Third, coinduction and induction of right (or left) 𝐡-modules is naturally equivalent. Fourth, there is a Frobenius coordinate system (πΈβˆΆπ΄β†’π΅;π‘₯1,…,π‘₯π‘š,𝑦1,…,π‘¦π‘šβˆˆπ΄), which satisfies ξ€·πΈβˆˆHom𝐡𝐴𝐡,𝐡𝐡𝐡,π‘šξ“π‘–=1πΈξ€·π‘Žπ‘₯𝑖𝑦𝑖=π‘Ž=π‘šξ“π‘–=1π‘₯π‘–πΈξ€·π‘¦π‘–π‘Žξ€Έ(βˆ€π‘Žβˆˆπ΄).(4.1) These (dual bases) equations may be used to show the useful fact that βˆ‘π‘–π‘₯π‘–βŠ—π‘¦π‘–βˆˆ(π΄βŠ—π΅π΄)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 module 𝐴𝐡 is a generator ⇔𝐡𝐴 is a generator ⇔ there are elements {π‘Žπ‘—}𝑛𝑗=1 and {𝑐𝑗}𝑛𝑗=1 such that βˆ‘π‘›π‘—=1𝐸(π‘Žπ‘—π‘π‘—)=1𝐡⇔𝐸 is surjective.

Proof. The bimodule isomorphism 𝐡𝐴𝐴≅→𝐡Hom(𝐴𝐡,𝐡𝐡)𝐴 is realized by π‘Žβ†¦πΈ(π‘Žβˆ’) (with inverse βˆ‘πœ™β†¦π‘šπ‘–=1πœ™(π‘₯𝑖)𝑦𝑖). If 𝐴𝐡 is a generator, then there are elements {𝑐𝑗}𝑛𝑗=1 of 𝐴 and mappings {πœ™π‘—}𝑛𝑗=1 of π΄βˆ— such that βˆ‘π‘›π‘—=1πœ™π‘—(𝑐𝑗)=1𝐡. Let πΈπ‘Žπ‘—=πœ™π‘—. Then, βˆ‘π‘›π‘—=1𝐸(π‘Žπ‘—π‘π‘—)=1𝐡.
Another bimodule isomorphism 𝐴𝐴𝐡≅→𝐴Hom(𝐡𝐴,𝐡𝐡)𝐡 is realized by π‘Žβ†¦πΈ(βˆ’π‘Ž)∢=π‘ŽπΈ. Then writing the last equation as βˆ‘π‘—π‘π‘—πΈ(π‘Žπ‘—)=1𝐡 exhibits 𝐡𝐴 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 by 𝐸𝑑 for some invertible π‘‘βˆˆπ΄π΅.

A Frobenius (or QF) extension π΄βŠ‡π΅ enjoys an endomorphism ring theorem [21, 32], which shows that β„°βˆΆ=Endπ΄π΅βŠ‡π΄ is a Frobenius (resp., QF) extension, where the default ring homomorphism 𝐴→ℰ is understood to be the left multiplication mapping πœ†βˆΆπ‘Žβ†¦πœ†π‘Ž where πœ†π‘Ž(π‘₯)=π‘Žπ‘₯. It is worth noting that πœ† is a left split 𝐴-monomorphism (by evaluation at 1𝐴) so 𝐴ℰ is a generator.

The tower of a Frobenius (resp., QF) extension is obtained by iteration of the endomorphism ring and πœ†, obtaining a tower of Frobenius (resp. QF) extensions where occasionally we need the notation 𝐡∢=β„°βˆ’1,𝐴=β„°0 and β„°=β„°1𝐡⟢𝐴β†ͺβ„°1β†ͺβ„°2β†ͺβ‹―β†ͺℰ𝑛β†ͺβ‹―(4.2) so β„°2=Endℰ𝐴, and so forth. By transitivity of Frobenius extension or QF extension [21, 22], all subextensions β„°π‘šβ†ͺβ„°π‘š+𝑛 in the tower are also Frobenius (resp. QF) extensions.

The rings ℰ𝑛 are β„Ž-equivalent to 𝐢𝑛+1(𝐴,𝐡)=π΄βŠ—π΅β‹―βŠ—π΅π΄ as 𝐴-bimodules in case π΄βŠ‡π΅ is a QF extension. This follows from noting the Endπ΄π΅β‰…π΄βŠ—π΅ξ€·π΄Hom𝐡,π΅π΅ξ€Έβ„ŽβˆΌπ΄βŠ—π΅π΄(4.3) also holding as natural β„°-𝐴-bimodules, obtained by substitution of π΄βˆ—β„ŽβˆΌπ΄. This observation is then iterated followed by cancellations of the type π΄βŠ—π΄π‘€β‰…π‘€.

4.1. Tower above Frobenius Extension

Specialize now to π΄βŠ‡π΅ a Frobenius extension with Frobenius coordinate system 𝐸 and {π‘₯𝑖}π‘šπ‘–=1,{𝑦𝑖}π‘šπ‘–=1. Then the β„Ž-equivalences above are replaced by isomorphisms, and ℰ𝑛≅𝐢𝑛+1(𝐴,𝐡) for each 𝑛β‰₯βˆ’1 as ring isomorphisms with respect to a certain induced β€œπΈ-multiplication.” The 𝐸-multiplication on π΄βŠ—π΅π΄ is induced from the endomorphism ring Endπ΄π΅β‰…β†’π΄βŠ—π΅π΄ given by βˆ‘π‘“β†¦π‘–π‘“(π‘₯𝑖)βŠ—π΅π‘¦π‘– with inverse π‘ŽβŠ—π‘Žβ€²β†¦πœ†π‘Žβˆ˜πΈβˆ˜πœ†π‘Žβ€². The outcome of 𝐸-multiplication on 𝐢2(𝐴,𝐡) is given by ξ€·π‘Ž1βŠ—π΅π‘Ž2π‘Žξ€Έξ€·3βŠ—π΅π‘Ž4ξ€Έ=π‘Ž1πΈξ€·π‘Ž2π‘Ž3ξ€ΈβŠ—π΅π‘Ž4(4.4) with unity element 11=βˆ‘π‘šπ‘–=1π‘₯π‘–βŠ—π΅π‘¦π‘–. Note that the 𝐴-bimodule structure on β„°1 induced by πœ†βˆΆπ΄β†ͺβ„° corresponds to the natural 𝐴-bimodule π΄βŠ—π΅π΄.

The 𝐸-multiplication is defined inductively on β„°π‘›β‰…β„°π‘›βˆ’1βŠ—β„°π‘›βˆ’2β„°π‘›βˆ’1(4.5) using the Frobenius homomorphism πΈπ‘›βˆ’1βˆΆβ„°π‘›βˆ’1β†’β„°π‘›βˆ’2 obtained by iterating the following natural Frobenius coordinate system on β„°1β‰…π΄βŠ—π΅π΄, given by 𝐸1(π‘ŽβŠ—π΅π‘Žβ€²)=π‘Žπ‘Žβ€² and {π‘₯π‘–βŠ—π΅1𝐴}π‘šπ‘–=1, {1π΄βŠ—π΅π‘¦π‘–}π‘šπ‘–=1 [23] as one checks.

The iterative 𝐸-multiplication on 𝐢𝑛(𝐴,𝐡) clearly exists as an associative algebra, but it seems worthwhile (and not available in the literature) to compute it explicitly. The multiplication on 𝐢2𝑛(𝐴,𝐡) is given by (βŠ—=βŠ—π΅,𝑛β‰₯1) ξ€·π‘Ž1βŠ—β‹―βŠ—π‘Ž2𝑛𝑐1βŠ—β‹―βŠ—π‘2𝑛=π‘Ž1βŠ—β‹―βŠ—π‘Žπ‘›πΈξ€·π‘Žπ‘›+1πΈξ€·ξ€·π‘Žβ‹―πΈ2π‘›βˆ’1πΈξ€·π‘Ž2𝑛𝑐1𝑐2ξ€Έβ‹―ξ€Έπ‘π‘›βˆ’1ξ€Έπ‘π‘›ξ€ΈβŠ—π‘π‘›+1βŠ—β‹―βŠ—π‘2𝑛.(4.6) The identity on 𝐢2𝑛(𝐴,𝐡) is in terms of the dual bases, 12π‘›βˆ’1=π‘šξ“π‘–1,…,𝑖𝑛=1π‘₯𝑖1βŠ—β‹―βŠ—π‘₯π‘–π‘›βŠ—π‘¦π‘–π‘›βŠ—β‹―βŠ—π‘¦π‘–1.(4.7)

The multiplication on 𝐢2𝑛+1(𝐴,𝐡) is given by ξ€·π‘Ž1βŠ—β‹―βŠ—π‘Ž2𝑛+1𝑐1βŠ—β‹―βŠ—π‘2𝑛+1ξ€Έ=π‘Ž1βŠ—β‹―βŠ—π‘Žπ‘›+1πΈξ€·π‘Žπ‘›+2πΈξ€·ξ€·π‘Žβ‹―πΈ2π‘›πΈξ€·π‘Ž2𝑛+1𝑐1𝑐2⋯𝑐𝑛𝑐𝑛+1βŠ—β‹―βŠ—π‘2𝑛+1(4.8) with identity 12𝑛=π‘šξ“π‘–1,…,𝑖𝑛=1π‘₯𝑖1βŠ—β‹―βŠ—π‘₯π‘–π‘›βŠ—1π΄βŠ—π‘¦π‘–π‘›βŠ—β‹―βŠ—π‘¦π‘–1.(4.9) Denote in brief notation the rings 𝐢𝑛(𝐴,𝐡)∢=𝐴𝑛 and distinguish them from the isomorphic rings β„°π‘›βˆ’1 (𝑛=0,1,…).

The inclusions 𝐴𝑛β†ͺ𝐴𝑛+1 are given by π‘Ž[𝑛]β†¦π‘Ž[𝑛]1𝑛, which works out in the odd and even cases to 𝐴2π‘›βˆ’1β†ͺ𝐴2𝑛,π‘Ž1βŠ—β‹―βŠ—π‘Ž2π‘›βˆ’1βŸΌξ“π‘–π‘Ž1βŠ—β‹―βŠ—π‘Žπ‘›π‘₯π‘–βŠ—π‘¦π‘–βŠ—π‘Žπ‘›+1βŠ—β‹―βŠ—π‘Ž2π‘›βˆ’1,𝐴2𝑛β†ͺ𝐴2𝑛+1,π‘Ž1βŠ—β‹―βŠ—π‘Ž2π‘›βŸΌπ‘Ž1βŠ—β‹―βŠ—π‘Žπ‘›βŠ—1π΄βŠ—π‘Žπ‘›+1βŠ—β‹―βŠ—π‘Ž2𝑛.(4.10)

The bimodule structure on 𝐴𝑛 over a subalgebra π΄π‘š (with π‘š<𝑛 via composition of left multiplication mappings πœ†) is just given in terms of the multiplication in π΄π‘š as follows: ξ€·π‘Ÿ1βŠ—β‹―βŠ—π‘Ÿπ‘šπ‘Žξ€Έξ€·1βŠ—β‹―βŠ—π‘Žπ‘›ξ€Έ=π‘Ÿξ€Ίξ€·1βŠ—β‹―βŠ—π‘Ÿπ‘šπ‘Žξ€Έξ€·1βŠ—β‹―βŠ—π‘Žπ‘šξ€Έξ€»βŠ—π‘Žπ‘š+1βŠ—β‹―βŠ—π‘Žπ‘›(4.11) with a similar formula for the right module structure.

The formulas for the successive Frobenius homomorphisms πΈπ‘šβˆΆπ΄π‘š+1β†’π΄π‘š are given in even degrees by 𝐸2π‘›ξ€·π‘Ž1βŠ—β‹―βŠ—π‘Ž2𝑛+1ξ€Έ=π‘Ž1βŠ—β‹―βŠ—π‘Žπ‘›πΈξ€·π‘Žπ‘›+1ξ€ΈβŠ—π‘Žπ‘›+2βŠ—β‹―βŠ—π‘Ž2𝑛+1(4.12) for 𝑛β‰₯0. The formula in the odd case is 𝐸2𝑛+1ξ€·π‘Ž1βŠ—β‹―βŠ—π‘Ž2𝑛+2ξ€Έ=π‘Ž1βŠ—β‹―βŠ—π‘Žπ‘›βŠ—π‘Žπ‘›+1π‘Žπ‘›+2βŠ—π‘Žπ‘›+3βŠ—β‹―βŠ—π‘Ž2𝑛+2(4.13) for 𝑛β‰₯0.

The dual bases of 𝐸𝑛 denoted by π‘₯𝑛𝑖 and 𝑦𝑛𝑖 are given by all-in-one formulas π‘₯𝑛𝑖=π‘₯π‘–βŠ—1π‘›βˆ’1,𝑦𝑛𝑖=1π‘›βˆ’1βŠ—π‘¦π‘–(4.14) for 𝑛β‰₯0 (where 10=1𝐴). Note that βˆ‘π‘–π‘₯π‘›π‘–βŠ—π΄π‘›π‘¦π‘›π‘–=1𝑛+1.

With another choice of Frobenius coordinate system (𝐹,𝑧𝑗,𝑀𝑗) for π΄βŠ‡π΅, there is in fact an invertible element 𝑑 in the centralizer subring 𝐴𝐡 of 𝐴 such that 𝐹=𝐸(π‘‘βˆ’) and βˆ‘π‘–π‘₯π‘–βŠ—π΅π‘¦π‘–=βˆ‘π‘—π‘§π‘—βŠ—π΅π‘‘βˆ’1𝑀𝑗 [22, 23]; whence an isomorphism of the 𝐸-multiplication onto the 𝐹-multiplication, both on π΄βŠ—π΅π΄, is given by π‘Ÿ1βŠ—π‘Ÿ2β†¦π‘Ÿ1βŠ—π‘‘βˆ’1π‘Ÿ2. If the tower with 𝐸-multiplication is denoted by 𝐴𝐸𝑛 and the tower with 𝐹-multiplication is denoted by 𝐴𝐹𝑛, there is a sequence of ring isomorphisms 𝐴𝐸≅2π‘›βŸΆπ΄πΉ2𝑛,π‘Ž1βŠ—β‹―βŠ—π‘Ž2π‘›βŸΌπ‘Ž1βŠ—β‹―βŠ—π‘Žπ‘›βŠ—π‘‘βˆ’1π‘Žπ‘›+1βŠ—β‹―βŠ—π‘‘βˆ’1π‘Ž2𝑛,𝐴(4.15)𝐸≅2𝑛+1⟢𝐴𝐹2𝑛+1π‘Ž1βŠ—β‹―βŠ—π‘Ž2𝑛+1βŸΌπ‘Ž1βŠ—β‹―βŠ—π‘Žπ‘›+1βŠ—π‘‘βˆ’1π‘Žπ‘›+2βŠ—β‹―βŠ—π‘‘βˆ’1π‘Ž2𝑛+1(4.16) which commute with the inclusions 𝐴𝑛𝐸,𝐹β†ͺ𝐴𝐸,𝐹𝑛+1.

Theorem 4.2. The multiplication, module, and Frobenius structures for the tower 𝐴𝑛=π΄βŠ—π΅β‹―βŠ—π΅π΄ (𝑛 times 𝐴) above a Frobenius extension π΄βŠ‡π΅ are given by formulas (4.4) to (4.16).

Proof. First define Temperley-Lieb generators iteratively by 𝑒𝑛=1π‘›βˆ’1βŠ—π΄π‘›βˆ’21π‘›βˆ’1βˆˆπ΄π‘›+1 for 𝑛=1,2,…, which results in the explicit formulas 𝑒2𝑛=𝑖1,…,𝑖𝑛+1π‘₯𝑖1βŠ—β‹―βŠ—π‘₯π‘–π‘›βŠ—π‘¦π‘–π‘›π‘₯𝑖𝑛+1βŠ—π‘¦π‘–π‘›+1βŠ—π‘¦π‘–π‘›βˆ’1βŠ—β‹―βŠ—π‘¦π‘–1,𝑒2𝑛+1=𝑖1,…,𝑖𝑛π‘₯𝑖1βŠ—β‹―βŠ—π‘₯π‘–π‘›βŠ—1π΄βŠ—1π΄βŠ—π‘¦π‘–π‘›βŠ—β‹―βŠ—π‘¦π‘–1.(4.17) These satisfy braid-like relations [4, page 106], namely, 𝑒𝑖𝑒𝑗=𝑒𝑗𝑒𝑖,||||π‘–βˆ’π‘—β‰₯2,𝑒𝑖+1𝑒𝑖𝑒𝑖+1=𝑒𝑖+1,𝑒𝑖𝑒𝑖+1𝑒𝑖=𝑒𝑖1𝑖+1.(4.18) (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 𝑒𝑛π‘₯𝑒𝑛=π‘’π‘›πΈπ‘›βˆ’1(π‘₯),βˆ€π‘₯βˆˆπ΄π‘›,𝑦𝑒𝑛=𝐸𝑛𝑦𝑒𝑛𝑒𝑛,βˆ€π‘¦βˆˆπ΄π‘›+1,𝐸𝑛𝑒𝑛=1π‘›βˆ’1,π‘₯𝑒𝑛=𝑒𝑛π‘₯,βˆ€π‘₯βˆˆπ΄π‘›βˆ’1.(4.19) 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 apparent π‘₯𝑛𝑖=π‘₯𝑖𝑒1𝑒2⋯𝑒𝑛 and 𝑦𝑛𝑖=𝑒𝑛⋯𝑒2𝑒1𝑦𝑖 and uniqueness of bases with respect to the same Frobenius homomorphism. In fact 𝑒𝑛⋯𝑒2𝑒1π‘Ž=1π‘›βˆ’1βŠ—π‘Ž for any π‘Žβˆˆπ΄,𝑛=1,2,… (a symmetrical formula holds as well) and 1𝑛=βˆ‘π‘–π‘₯𝑖𝑒1β‹―π‘’π‘›βˆ’1π‘’π‘›π‘’π‘›βˆ’1⋯𝑒1𝑦𝑖.
Since the inductive definitions of the ring and module structures on the 𝐴𝑛'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 π‘Ž1βŠ—β‹―βŠ—π‘Žπ‘›+1=ξ€·π‘Ž1βŠ—β‹―βŠ—π‘Žπ‘›1ξ€Έξ€·π‘›βˆ’1βŠ—π‘Žπ‘›+1ξ€Έ=ξ€·π‘Ž1βŠ—β‹―βŠ—π‘Žπ‘›βˆ’11ξ€Έξ€·π‘›βˆ’2βŠ—π‘Žπ‘›π‘’ξ€Έξ€·π‘›β‹―π‘’1π‘Žπ‘›+1ξ€Έ=β‹―=π‘Ž1𝑒1π‘Ž2𝑒2𝑒1π‘Ž3ξ€Έβ‹―ξ€·π‘’π‘›βˆ’1⋯𝑒1π‘Žπ‘›π‘’ξ€Έξ€·π‘›β‹―π‘’1π‘Žπ‘›+1ξ€Έ.(4.20) 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 the 𝐢𝑛(𝐴,𝐡)'s satisfying the hypotheses below (for one compares with (4.11)). This is true as well if 𝐴 is a ring with 𝐡 in its center, since the ordinary tensor algebra on π΄βŠ—π΅π΄ may be extended to an 𝑛-fold tensor product algebra π΄βŠ—π΅β‹―βŠ—π΅π΄.

Proposition 4.3. Let π΄βŠ‡π΅ be a ring extension. Suppose that there is a ring structure on each π΄π‘›βˆΆ=𝐢𝑛(𝐴,𝐡) for each 𝑛β‰₯0, a ring homomorphism π΄π‘›βˆ’1→𝐴𝑛 for each 𝑛β‰₯1, and that the composite 𝐡→𝐴𝑛 induces the natural bimodule given by 𝑏⋅(π‘Ž1βŠ—β‹―βŠ—π‘Žπ‘›)⋅𝑏′=π‘π‘Ž1βŠ—π‘Ž2βŠ—β‹―βŠ—π‘Žπ‘›π‘β€². Then, π΄βŠ‡π΅ has depth 2𝑛+1 if and only if π΄π‘›βˆ£π΅ has depth 3.

Proof. If π΄βŠ‡π΅ has depth 2𝑛+1, then π΄π‘›β„ŽβˆΌπ΄π‘›+1 as 𝐡-bimodules. By tensoring repeatedly by π΅π΄βŠ—π΅βˆ’, also π΄π‘›β„ŽβˆΌπ΄2𝑛 as 𝐡-bimodules. But 𝐴2π‘›β‰…π΄π‘›βŠ—π΅π΄π‘›. Then, π΄π‘›βŠ‡π΅ has depth three.
Conversely, if π΄π‘›βˆ£π΅ has depth 3, then π΄β„Ž2π‘›βˆΌπ΄π‘› as 𝐡-bimodules. But 𝐴𝑛+1∣𝐴2𝑛 via the split 𝐡-bimodule epi π‘Ž1βŠ—β‹―βŠ—π‘Ž2π‘›β†¦π‘Ž1β‹―π‘Žπ‘›βŠ—π‘Žπ‘›+1βŠ—β‹―βŠ—π‘Ž2𝑛. Then, 𝐴𝑛+1βˆ£π‘žπ΄π‘› for some π‘žβˆˆβ„€+. It follows that π΄βŠ‡π΅ 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. Suppose π΄βŠ‡π΅ is a QF extension. If π΄βŠ‡π΅ has depth 3, then β„°βŠ‡π΅ has depth 2. Conversely, if β„°βŠ‡π΅ has depth 2 and 𝐴𝐡 is a generator, then π΄βŠ‡π΅ has depth 3.

Proof. Since 𝐴 is a QF extension of 𝐡, we have β„°β„ŽβˆΌπ΄βŠ—π΅π΄ as β„°-𝐴-bimodules. Then, β„°βŠ—π΅β„°β„ŽβˆΌπ΄βŠ—π΅π΄βŠ—π΅π΄βŠ—π΅π΄ as β„°-𝐡-bimodules. Given the depth 3 condition, π΄βŠ—π΅π΄β„ŽβˆΌπ΄ as 𝐡-bimodules, it follows by two substitutions that β„°βŠ—π΅β„°β„ŽβˆΌπ΄βŠ—π΅π΄ as β„°-𝐡-bimodules. Consequently, β„°βŠ—π΅β„°β„ŽβˆΌβ„° as β„°-𝐡-bimodules. Hence, β„°βŠ‡π΅ has right depth 2, and since it is a QF extension by the endomorphism ring theorem and transitivity, β„°βŠ‡π΅ also has left depth 2.
Conversely, we are given 𝐴𝐡 a progenerator, so that β„° and 𝐡 are Morita equivalent rings, where 𝐡Hom(𝐴𝐡,𝐡𝐡)β„° and ℰ𝐴𝐡 are the context bimodules. If β„°βŠ‡π΅ has depth two, then β„°βŠ—π΅β„°β„ŽβˆΌβ„° as β„°-𝐡-bimodules. Then π΄βŠ—π΅π΄βŠ—π΅π΄βŠ—π΅π΄β„ŽβˆΌπ΄βŠ—π΅π΄ as β„°-𝐡-bimodules. Since Hom(𝐴𝐡,𝐡𝐡)βŠ—β„°π΄β‰…π΅ as 𝐡-bimodules, a cancellation of the bimodules ℰ𝐴𝐡 follows, so π΄βŠ—π΅π΄βŠ—π΅π΄β„ŽβˆΌπ΄ as 𝐡-bimodules. Since π΄βŠ—π΅π΄βˆ£π΄βŠ—π΅π΄βŠ—π΅π΄, it follows that π΄βŠ—π΅π΄βˆ£π‘žπ΄ for some π‘žβˆˆβ„€+. Then π΄βŠ‡π΅ has depth 3.

Example 4.5. To illustrate that the theorem does not extend to when π΄βŠ‡π΅ is not a QF extension, consider 𝐴=𝑇𝑛(π‘˜), 𝑛β‰₯2 (a hereditary algebra) and 𝐡=𝐷𝑛(π‘˜) (a semisimple algebra), and left π‘˜ be an algebraically closed field of characteristic zero. (Since 𝐡,𝐴 is, is not a QF-algebra it follows by transitivity that π΄βŠ‡π΅ is not a QF extension.) It was computed that 𝑑(𝐡,𝐴)=3 in Example 1.7. Thinking of the columns of 𝐴 as 𝐴𝑒𝑖𝑖, it is quite easy to see that End𝐴𝐡≅𝑀1(π‘˜)×𝑀2(π‘˜)×⋯×𝑀𝑛(π‘˜) and that the inclusion of 𝐴β†ͺEnd𝐴𝐡 is given by βŽ›βŽœβŽœβŽπ‘‹π‘‹βŸΌ11,βŽ›βŽœβŽœβŽπ‘‹11𝑋12𝑋12𝑋22⎞⎟⎟⎠⎞⎟⎟⎠,…,𝑋.(4.21) Its restriction to 𝐡 is given by ξ€·πœ‡Diag1,…,πœ‡π‘›ξ€ΈβŸΌξ€·πœ‡1ξ€·πœ‡,Diag1,πœ‡2ξ€Έξ€·πœ‡,…,Diag1,…,πœ‡π‘›ξ€Έξ€Έ(4.22) with inclusion matrix βˆ‘π‘€=𝑖≀𝑗𝑒𝑖𝑗. Then, 𝑀𝑀𝑑>0, and from (1.1) we see that 𝑑(𝐡,β„°)=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 π΄βŠ‡π΅βŠ‡π‘‡, where π΄βŠ‡π΅ and π΅βŠ‡π‘‡ are ring extensions, has generalized right depth 2 if π΄βŠ—π΅π΄β„Žξ‚π΄ as natural 𝐴-𝑇-bimodules. (Note that if 𝑇=𝐡, this is the definition of the ring extension π΄βŠ‡π΅ having right depth 2.)

Throughout the section below we suppose π΄βŠ‡π΅ is a Frobenius extension and ℰ𝑖β†ͺℰ𝑖+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 π΄βŠ‡π΅ has right tower depth 𝑛β‰₯2 if the subtower of composite ring extensions π΅β†’β„°π‘›βˆ’3β†ͺβ„°π‘›βˆ’2 has generalized right depth 2; equivalently, as natural β„°π‘›βˆ’2-𝐡-bimodules, β„°π‘›βˆ’2βŠ—β„°π‘›βˆ’3β„°π‘›βˆ’2βŠ•βˆ—β‰…π‘žβ„°π‘›βˆ’2(5.1) for some positive integer π‘ž, since the reverse condition is always satisfied. Since β„°βˆ’1=𝐡 and β„°0=𝐴, this recovers the right depth two condition on a subring 𝐡 of 𝐴. To this definition we add that a Frobenius extension π΄βŠ‡π΅ has tower depth 1 if it is a centrally projective ring extension; that is, π΅π΄π΅βˆ£π‘žπ΅ for some