International Journal of Mathematics and Mathematical Sciences

VolumeΒ 2012, Article IDΒ 254791, 22 pages

http://dx.doi.org/10.1155/2012/254791

## 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.

#### Abstract

The minimum depth of a subring 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 < β if is a finite-dimensional algebra and 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 is a QF extension, minimum left and right even subring depths are shown to coincide. If 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 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 of a finite-dimensional algebra has a split epimorphism of --bimodules, then has finite representation type if has. Weakening the condition on to a split epimorphism of --bimodules does not place any restriction on , but the opposite hypothesis that a split monomorphism exists from into a multiple captures the notion of normality of a subalgebra in the context of group algebras [1], Hopf algebras [2], and semisimple algebras [3]. If is a Frobenius extension of , where is a progenerator module (but and may be infinite-dimensional algebras), the βdepth twoβ condition as the opposite hypothesis is known as, implies that is a Galois extension of , where the bimodule endomorphism ring of the extension may be given the structure of a Hopf algebroid (which acts naturally on with invariant subalgebra ) [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 to a tower of three rings [11]. This was applied to the tower of iterated right endomorphism rings above a Frobenius extension , so that has (tower) depth if has the generalized depth two property (called a depth 3 tower in [11]). This yields a compact matrix inequality condition (some ) for when a subalgebra pair of semisimple complex algebras has depth in terms of the inclusion matrix , equivalently the incidence matrix of the Bratteli diagram of the inclusion [3, 18]. Since , , 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 , which has (right) depth if the relative Hochschild bar resolution group maps as a split monomorphism into a multiple of a smaller group, as --bimodules, and depth if this condition only holds as natural --bimodules. Since subring having depth implies that it has depth , the minimum depth 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 is over any ground ring and depends only on a combinatorial depth of a subgroup defined in terms of -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 of finite group algebras over any ground ring has finite depth, in fact bounded by twice the index of the normalizer subgroup.

The notion of subring depth in [12] is defined in equivalent terms in (1.7). In case and 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 in [3] and given in (1.1). Such a pair 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 () 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 are known to have depth 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 of a finite-dimensional algebra has finite depth if its enveloping algebra has finite representation type.

##### 1.1. Equivalent Modules

Let be a ring. Two left -modules, and , are said to be *-equivalent*, denoted by , if two conditions are met. First, for some positive integer , is isomorphic to a direct summand in the direct sum of copies of , denoted by
Second, symmetrically there is such that . It is easy to extend this definition of -equivalence (sometimes referred to as similarity) to -equivalence of two objects in an abelian category and to show that it is an equivalence relation.

If two modules are -equivalent, , then they have Morita equivalent endomorphism rings, and , since a Morita context of bimodules is given by , which is an --bimodule via composition, and the bimodule ; these are progenerator modules, by applying to (1.2) or its reverse, , any of the four Hom-functors such as from the category of left -modules into the category of left -modules. Then, the explicit conditions on mappings for -equivalence show that and the reverse mapping given by composition are surjections.

The theory of -equivalent modules applies to bimodules by letting , which sets up an equivalence of abelian categories between --bimodules and left -modules. Two additive functors are -equivalent if there are natural split epis and for all in . We leave the proof of the lemma below as an elementary exercise.

Lemma 1.1. * Suppose two -modules are -equivalent, , and two additive functors from -modules to an abelian category are -equivalent, . Then, . *

For example, the following substitution in equations involving the -equivalence relation follows from the lemma:

*Example 1.2. * Suppose is a finite-dimensional algebra with indecomposable -modules (representatives from each isomorphism class for some index set ). By Krull-Schmidt finitely generated modules and have a unique factorization into a direct sum of multiples of finitely many indecomposable module components. Denote the indecomposable constituents of by , where is the number of factors in isomorphic to . Note that for some positive if and only if . It follows that if and only if .

Suppose is the decomposition of the regular module into its projective indecomposables. Let . Then, and are -equivalent, so that and are Morita equivalent. The algebra is the basic algebra of .

##### 1.2. Depth Two

A subring pair is said to have left depth (or be a left depth two extension [4]) if as natural --bimodules. Right depth is defined similarly in terms of -equivalence of natural --bimodules. In [4] it was noted that the left condition implies the right and conversely if is a Frobenius extension of . Also in [4] a Galois theory of Hopf algebroids was defined on the endomorphism ring as total ring and the centralizer as base ring. The antipode is the natural anti-isomorphism stemming from following the arrows: restricted to the intersection .

The Galois extension properties of a depth two extension are as follows. If is faithfully flat, balanced or equals its double centralizer in , the natural action of on has invariant subalgebra satisfying the Galois property of . Also the well-known Galois property of the endomorphism ring as a cross-product holds: the right endomorphism ring , where the latter has smash product ring structure on [4]. There is also a duality structure by going a step further along in the tower above , where the Hopf algebroid is the -dual of and acts naturally on in such a way that has a smash product ring structure [4].

Conversely, Galois extensions have depth . For example, an -comodule algebra with invariant subalgebra and finite-dimensional Hopf algebra over a base field , which has a Galois isomorphism from given by , satisfies (strongly) the depth two condition as --bimodules. The Hopf subalgebras within a finite-dimensional Hopf algebra, which have depth , are precisely the normal Hopf subalgebras; if normal, it has depth 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 ), where left normal is invariance under the left adjoint action. In the context of an augmented algebra their results extend to the following proposition. Let be an algebra homomorphism into the ground field . Let denote , and, for a subalgebra , let denote .

Proposition 1.3. * Suppose is a subalgebra of an augmented algebra. If has right depth 2, then . *

The proof of this proposition is an exercise in tensoring both sides of by the unit -module , 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 be the algebra of by upper triangular matrices where , and the subalgebra of diagonal matrices. Note that there are augmentations given by , and each of the satisfies the inclusions above if left or right depth two. This is a clear contradiction, thus . We will see below that .

Also subalgebra pairs of semisimple complex algebras have depth 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 is a subalgebra pair of semisimple complex algebras. Then, has depth if and only if, for every maximal ideal in , one has . *

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 mentioned above is semisimple (see Kaplansky's Fields and Rings for a -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 be a unital associative ring and a subring where . Note the natural bimodules obtained by restriction of the natural --bimodule (briefly -bimodule) , also to the natural bimodules , or , which are referred to with no further ado.

Let , and, for , For , has a natural -bimodule structure, which restricts to --, --, and -bimodule structures occurring in the next definition.

*Definition 1.6. * The subring has depth if as -bimodules . The subring has left (resp., right) depth if as --bimodules (resp., --bimodules).

It is clear that if has either left or right depth , it has depth by restricting the -equivalence condition to -bimodules. If it has depth , it has depth by tensoring the -equivalence by or . The *minimum depth* is denoted by ; if has no finite depth, write .

Note that the minimum left and right minimum *even* depths may differ by (in which case is the lesser of the two). In the next section we provide a general condition, which includes a Hopf subalgebra pair of symmetric (Frobenius) algebras, where the left and right minimum even depths coincide.

Also note that a subalgebra pair of Artin algebras have depth if and only if the indecomposable module constituents of remain the same for all as those already found in (see Example 1.2). This corresponds well with the classical notion of finite depth in subfactor theory.

*Example 1.7. * Again let and , where . Let denote the matrix units, the simple -modules, and for the simple components of . Note that as a -bimodule has components where , so as -bimodules. Thus, . But by the remark following Proposition 1.3; then .

##### 1.4. -Depth

A subring has -depth if as --bimodules (). Note that has -depth in implies that it has -depth (also that it has depth ). Thus, define the minimum -depth if it exists. Note that the definition of -depth is equivalent to the condition on a subring that for some . This is clear for since . For , the -separability condition implies the separability condition as argued in the paper [13] by Hirata. The notion of -depth is studied in [14] where it is noted that if one or the other minimum depth is finite. See Section 2 for which Hopf subalgebras satisfy the condition in (1.6).

*Remark 1.8. * Suppose is a subring of . The minimum depth of the subring as defined in Boltje-Danz-KΓΌlshammer [12] coincides with . In fact, for , the depth condition in [12] is that for some
as -bimodules. The left depth condition in [12] is (1.7) more strongly as natural --bimodules (and as --bimodules for the right depth condition). But (using a pair of classical face and degeneracy maps of homological algebra) we always have as --, --, or -bimodules, so that the depth as well as conditions coincide in the case of subring having depth and conditions above.

Note that depth in this paper is equivalent to the subring depth notion in, for example, [4, 12, 15] since is -equivalent to as -bimodules if and only if is centrally projective over (i.e., as -bimodules). This follows from the lemma below.

Lemma 1.9. * Suppose is a subring of ring such that for some integer . Then, . *

*Proof. * From the central projectivity condition on , one obtains maps and maps such that for every . It follows that since . Note that restricting the equation to the centralizer shows that is a finitely generated projective -module. But is a commutative subring, whence is a generator -module. From for some positive integer , it follows from the tensor algebra decomposition of that . Whence there are maps and elements such that . Define a (condition expectation or) bimodule projection of onto .

*Example 1.10. * The paper [12] asks in its introduction about the depth of invariant subrings in classical invariant theory, where is a field, , and is a finite group in acting by linear substitution of the variables. In any case is finitely generated and is a finitely generated affine -algebra. We note here that if is generated by pseudoreflections (such as , the symmetric group) and the characteristic of is coprime to , is itself an -variable polynomial algebra and is a free -module; consequences of the Shephard-Todd Theorem [16, 17]. Since is a commutative algebra, it follows that .

*Example 1.11. * Let be a subring pair of semisimple complex algebras. Then, the minimum depth may be computed from the inclusion matrix , alternatively an -by- induction-restriction table of -simples induced to nonnegative integer linear combination of -simples along rows, and by Frobenius reciprocity, columns show restriction of -simples in terms of -simples. The procedure to obtain given in the paper [3] is to compute the bracketed powers of given in Section 1, and check for which th power of satisfies the matrix inequality in (1.1): is the least such by results in [12, appendix] (or Theorem 5.2 below combined with [3, 18]). One may note that where has degree minimal polynomial [3]. A GAP subprogram exists to compute for a complex group algebra extension by converting character tables to an induction-restriction table , then counting the number of zero entries in the bracketed powers of , which decreases nonstrictly with increasing even and odd powers of , being the least point of no decrease.

In terms of the bipartite graph of the inclusion , is the lesser of the minimum odd depth and the minimum even depth [3]. The matrix is an incidence matrix of this bipartite graph if all entries greater than are changed to , while zero entries are retained as : let the -simples be represented by black dots in a bottom row of the graph and -simples by white dots in a top row, connected by edges joining black and white dots (or not) according to the --matrix entries obtained from . The minimum odd depth of the bipartite graph is plus the diameter in edges of the row of black dots (indeed an odd number), while the minimum even depth is 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 , the complex group algebra of the permutation group on four letters, and . The inclusion diagram pictured in Figure with the degrees of the irreducible representations is determined from the character tables of and or the branching rule (for the Young diagrams labelled by the partitions of and representing the irreducibles of ).

This graph has minimum odd depth 5 and minimum even depth 6, whence .

*Example 1.12. * The induction-restriction table of the inclusion of permutation groups via
may be computed combinatorially from the Littlewood-Richardson coefficients , where is partition of , a partition of , and a partition of . Briefly, the coefficient number is zero if does not contain or is the number of Littlewood-Richardson fillings with content of with removed. A Littlewood-Richardson filling of a skew Young tableau is with integers occuring 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 for the subgroup with respect to the ordered bases of irreducible characters of the subgroup , , , , , and of the group ,, , , , , yields
The bracketed powers of satisfy a minimum depth inequality (1.1) so that . We mentioned before that [3, 12]; however, a formula for is not known.

##### 1.5. Finite Depth and Finite Representation Type

For the next proposition we adopt the notation for the (enveloping) algebra 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 has finite representation type if and only if its Sylow -subgroup is cyclic. Thus, having finite representation type does not imply that has finite representation type.

Proposition 1.13. *Suppose is a subalgebra pair of finite-dimensional algebras where has in all indecomposable -module isomorphism classes. Then, . *

*Proof. *This follows from the observation in Example 1.2 that since is the image of under an obvious split epimorphism of -modules (equivalently, -bimodules), there is an increasing chain of subset inclusions
which stops strictly increasing in at most steps. When , then as -modules, whence has depth .

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 , 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 is a Hopf algebra and semisimple if is so.) For example, we have the following corollary.

Corollary 1.14. *Suppose is a semisimple Hopf subalgebra in a finite dimensional Hopf algebra . Suppose that has nonisomorphic simple modules. Then, . *

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

A (proper) ring extension is a subring or more generally a monomorphism , which is equivalent to a subring . Restricted modules such as and pullback modules 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 and its pullback modules such as ; however, certain conditions needed below such as is a generator imply that is monic.)

A ring extension is a *left QF extension* if the module is finitely generated projective and the natural --bimodules satisfy for some positive integer . A right QF extension is oppositely defined. A *QF extension * is both a left and right QF extension and may be characterized by both and being finite projective, and two -equivalences of bimodules given by and [21]. For example, a Frobenius extension is a QF extension since it is left and right finite projective and satisfies the stronger conditions that is *isomorphic* to its right -dual and its left -dual as natural --bimodules, respectively --bimodules; the more precise definition is given in the next section.

##### 2.1. -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 be an automorphism of the ring and a subring pair. Denote the pullback module of a module along by , the so-called twisted module. A ring extension is a -*Frobenius extension* if is finite projective and there is a bimodule isomorphism . One shows that is a Frobenius extension if and only if is an inner automorphism. A subring pair of Frobenius algebras is -Frobenius extension so long as is finite projective and the Nakayama automorphism of stabilizes , in which case [22]. For instance a finite-dimensional Hopf algebra and a Hopf subalgebra of are a pair of Frobenius algebras satisfying the conditions just given: the formula for reduces to the following given in terms of the modular functions of and and the antipode [23, 7.8]: for ,

Given the bimodule isomorphism above , apply it to and let its value be , which then is a cyclic generator of satisfying for all . If and are projective bases of , and the equations hold for all . Call a - of . Note that also is finite projective, that a -Frobenius coordinate system is equivalent to the ring extension being -Frobenius and that if is in the center of . Additionally, one notes that there is an automorphism of the centralizer subring such that for all and . Also an isomorphism is easily defined from the data and equations above, where , so that if is another -Frobenius coordinate system (sharing the same ), then in .

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

Proposition 2.1. *A -Frobenius extension is a left QF extension if and only if there are such that and for all and , and
*

*Proof. *Suppose is -Frobenius extension with -Frobenius system satisfying the equations above. Given the elements satisfying the equations above, let , which defines mappings in (the untwisted) . Also define mappings by where it is not hard to show using the -Frobenius coordinate equations that for each (a Casimir element). It follows that and that as natural --bimodules, whence is a left QF extension of .

Conversely, assume the left QF condition , equivalent to by applying the right -dual functor and noting as well . Also assume the slightly rewritten -Frobenius condition , which then implies . So there are mappings and mappings such that . Equivalently, with and , , 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 be a finite dimensional Hopf algebra and a Hopf subalgebra. In the notation of (2.1) the following are equivalent. *(1)*The automorphism and is a Frobenius extension. *(2)*The algebra extension is a QF extension. *(3)*The modular functions for all . *

*Proof. *() A Frobenius extension is a QF extension. () Set in (2.3), and apply the counit to see that . Reapply to (2.3) to obtain . Apply to (2.1), and use uniqueness of inverse in convolution algebra , where and , to show that on . () This follows from (2.1).

The following observation for a normal Hopf subalgebra has not been explicitly noted before in the literature.

Corollary 2.3. * The modular function of a finite dimensional Hopf algebra restricts to the modular function of a Hopf subalgebra if has depth . *

*Proof. * If the Hopf subalgebra has depth in , it has depth . If it has depth , it is equivalently a normal Hopf subalgebra by the result of [2]. But a normal Hopf subalgebra is an -Galois extension: here denotes the quotient Hopf algebra, , denotes the quotient map, and the Galois isomorphism is given by [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, in the corollary above, so .

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 .

Lemma 2.4. *Let be an augmented Frobenius algebra with Nakayama automorphism , a subalgebra and Frobenius algebra where , and finitely generated projective. It follows that is a -Frobenius extension where , a relative Nakayama automorphism [22, Satz 7], [29, Paragraph 5.1]. Then the modular automorphisms of and satisfy
*

*Proof. *Let be a Frobenius coordinate system for , a right norm satisfying , then is a right integral, satisfying for all , spanning the one-dimensional space of integrals in . Let be the augmentation on defined by for . It follows that by expressing in terms of dual bases, and (and note that are also dual bases) [29, Paragraph 3.2]. Similarly let be a Frobenius coordinate system for and a right norm satisfying , then is a right integral in and defines the -valued algebra homomorphism , which satisfies . It follows that .

Note that (2.4) for Hopf subalgebras also follows from (2.1). Corollary 2.3 does not extend to depth Hopf subalgebras by the next example.

*Example 2.5. * The Taft-Hopf algebra over its cyclic group subalgebra is a nontrivial -Frobenius extension [23]. The algebra is generated over by a grouplike of order , a nilpotent of index , and -primitive element where for a primitive th root of unity. This is a Hopf algebra having right integral with modular function [23]. The Hopf subalgebra is generated by . Then the twist automorphism of is given by . Of course, restricted to is not equal to . The depth is computed in [30].

Finally we note that unimodular Hopf algebra extensions are trivial if the -depth condition is imposed.

Proposition 2.6. *Suppose is a finite-dimensional Hopf algebra and is a Hopf subalgebra of . If , then satisfies a double centralizer result; in particular, if is unimodular, then . *

*Proof. *Since is a finite-dimensional Hopf algebra, it is a free extension of the Hopf subalgebra , therefore faithfully flat. If , then the ring extension satisfies the generalized Azumaya condition via , left and right multiplication [23, 31], where denotes the centralizer subalgebra of in . If , then it is obvious from this that , so that : it follows that

Since is unimodular, it has a two-sided nonzero integral . Note that , whence . Let (where is the arbitrary ground field) be the left integral in the dual Hopf algebra such that . The bijective antipode satisfies since and for all . Since , it follows that for all . Thus .

#### 3. Even Depth of QF Extensions

It is well known that for a Frobenius extension , coinduction of a module, is naturally isomorphic as functors to induction (from the category of -modules into the category of -modules). Similarly, a QF extension has -equivalent coinduction and induction functors, which is seen from the naturality of the mappings in the next proof. Let be an arbitrary third ring.

Proposition 3.1. * Suppose is a bimodule and is a QF extension. Then, there is an -equivalence of bimodules,
*

*Proof. *Since is f.g. projective, it follows that there is a --bimodule isomorphism
given by with inverse constructed from projective bases for . But the right -dual of is -equivalent to , 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 , times ).

Theorem 3.2. * If is QF extension, then has left depth if and only if has right depth . *

*Proof. * The left depth condition on recall is as --bimodules. To this apply the additive functor (into the category of --bimodules), noting that via for each integer . It follows (from Lemma 1.1) that there is an --bimodule -equivalence,
(Then in the depth two case, the left depth two condition is equivalent to End as natural --bimodules.)

Given bimodule , we have by the previous lemma: apply this to using the hom-tensor adjoint relation: there are -equivalences and isomorphisms of -bimodules,
for each and . Compare (3.3) and (3.4) with to get , which is the right depth 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 -Frobenius or even twisted QF extensions presents technical problems and is unknown to the author.

#### 4. Frobenius Extensions

As noted above a Frobenius extension is characterized by any of the following four conditions [23]. First, is finite projective and . Second, is finite projective and . Third, coinduction and induction of right (or left) -modules is naturally equivalent. Fourth, there is a Frobenius coordinate system ,, which satisfies These (dual bases) equations may be used to show the useful fact that .

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 is a generator is a generator there are elements and such that is surjective. *

*Proof. *The bimodule isomorphism is realized by (with inverse ). If is a generator, then there are elements of and mappings of such that . Let . Then, .

Another bimodule isomorphism is realized by . Then writing the last equation as exhibits 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 for some invertible .

A Frobenius (or QF) extension enjoys an *endomorphism ring theorem* [21, 32], which shows that is a Frobenius (resp., QF) extension, where the default ring homomorphism is understood to be the left multiplication mapping where . It is worth noting that is a left split -monomorphism (by evaluation at ) so is a generator.

The *tower* of a Frobenius (resp., QF) extension is obtained by iteration of the endomorphism ring and , obtaining a tower of Frobenius (resp. QF) extensions where occasionally we need the notation and
so , and so forth. By transitivity of Frobenius extension or QF extension [21, 22], all subextensions in the tower are also Frobenius (resp. QF) extensions.

The rings are -equivalent to as -bimodules in case is a QF extension. This follows from noting the also holding as natural --bimodules, obtained by substitution of . This observation is then iterated followed by cancellations of the type .

##### 4.1. Tower above Frobenius Extension

Specialize now to a Frobenius extension with Frobenius coordinate system and . Then the -equivalences above are replaced by isomorphisms, and for each as ring isomorphisms with respect to a certain induced β-multiplication.β The -multiplication on is induced from the endomorphism ring given by with inverse . The outcome of -multiplication on is given by with unity element . Note that the -bimodule structure on induced by corresponds to the natural -bimodule .

The -multiplication is defined inductively on using the Frobenius homomorphism obtained by iterating the following natural Frobenius coordinate system on , given by and , [23] as one checks.

The iterative -multiplication on clearly exists as an associative algebra, but it seems worthwhile (and not available in the literature) to compute it explicitly. The multiplication on is given by () The identity on is in terms of the dual bases,

The multiplication on is given by with identity Denote in brief notation the rings and distinguish them from the isomorphic rings ().

The inclusions are given by , which works out in the odd and even cases to

The bimodule structure on over a subalgebra (with via composition of left multiplication mappings ) is just given in terms of the multiplication in as follows: with a similar formula for the right module structure.

The formulas for the successive Frobenius homomorphisms are given in even degrees by for . The formula in the odd case is for .

The dual bases of denoted by and are given by all-in-one formulas for (where ). Note that .

With another choice of Frobenius coordinate system for , there is in fact an invertible element in the centralizer subring of such that and [22, 23]; whence an isomorphism of the -multiplication onto the -multiplication, both on , is given by . If the tower with -multiplication is denoted by and the tower with -multiplication is denoted by , there is a sequence of ring isomorphisms which commute with the inclusions .

Theorem 4.2. *The multiplication, module, and Frobenius structures for the tower ( times ) above a Frobenius extension are given by formulas (4.4) to (4.16). *

*Proof. *First define Temperley-Lieb generators iteratively by for , which results in the explicit formulas
These satisfy braid-like relations [4, page 106], namely,
(The generators above fail to be idempotents to the extent that differs from .) 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
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 and and uniqueness of bases with respect to the same Frobenius homomorphism. In fact for any (a symmetrical formula holds as well) and .

Since the inductive definitions of the ring and module structures on the 's also satisfy the relations listed above and agree on and below , the proof is finished with an induction argument based on expressing tensors as words in Temperley-Lieb generators and elements of .

We note that
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 's satisfying the hypotheses below (for one compares with (4.11)). This is true as well if is a ring with in its center, since the ordinary tensor algebra on may be extended to an -fold tensor product algebra .

Proposition 4.3. * Let be a ring extension. Suppose that there is a ring structure on each for each , a ring homomorphism for each , and that the composite induces the natural bimodule given by . Then, has depth if and only if has depth . *

*Proof. * If has depth , then as -bimodules. By tensoring repeatedly by , also as -bimodules. But . Then, has depth three.

Conversely, if has depth , then as -bimodules. But via the split -bimodule epi . Then, for some . It follows that has depth .

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 is a QF extension. If has depth , then has depth . Conversely, if has depth and is a generator, then has depth . *

*Proof. *Since is a QF extension of , we have as --bimodules. Then, as --bimodules. Given the depth condition, as -bimodules, it follows by two substitutions that as --bimodules. Consequently, as --bimodules. Hence, has right depth , and since it is a QF extension by the endomorphism ring theorem and transitivity, also has left depth .

Conversely, we are given a progenerator, so that and are Morita equivalent rings, where and are the context bimodules. If has depth two, then as --bimodules. Then as --bimodules. Since as -bimodules, a cancellation of the bimodules follows, so as -bimodules. Since , it follows that for some . Then has depth .

*Example 4.5. *To illustrate that the theorem does not extend to when is not a QF extension, consider , (a hereditary algebra) and (a semisimple algebra), and left be an algebraically closed field of characteristic zero. (Since is, is not a QF-algebra it follows by transitivity that is not a QF extension.) It was computed that in Example 1.7. Thinking of the columns of as , it is quite easy to see that End and that the inclusion of is given by
Its restriction to is given by
with inclusion matrix