The notion of depth two is important in a certain noncommutative Galois theory, which generates Hopf algebroids in place of the more classical Galois groups, whereas the notion of depth greater than two measures the defect, or distance, from being depth two in a tower of iterated endomorphism rings above the subring.
A more recent definition of depth of any unital subring in any associative ring is proposed (see below) in a paper studying the depth of a subgroup of a finite group as group algebras over a commutative ring.
for some positive integer n; by switching to natural B-A-bimodules, there is a corresponding definition of left depth two.
(n times) as well as the common notion, p is a split epimorphism if there is a homomorphism q in the reverse direction such that pq = identity on the image of p. (Sometimes the subring B in A is referred to as the ring extension A over B; the theory works as well for a ring homomorphism B into A, which induces right and left B-modules structures on A.)
Equivalently, the condition for left or right depth two may be given in terms of a split monomorphism of bimodules where the domains and codomains above are reversed.
Let B be the group (sub)algebra of a normal subgroup H of index n in G with coset representatives
(and extended linearly to a mapping A into B, a B-B-module homomorphism since H is normal in G): the splitting condition pq = the identity on
Assume A is a finite projective B-module, so there are B-linear mapping
For a Frobenius algebra extension A | B (such as A and B group algebras of a subgroup pair of finite index) the two one-sided conditions of depth two are equivalent, and a notion of depth n > 2 makes sense via the right endomorphism ring extension iterated to generate a tower of rings (a technical procedure beyond the scope of this survey, although the first step, the endomorphism ring theorem, is described in the section on Frobenius extension under Frobenius algebra).
Other examples come from the fact that finite Hopf-Galois extensions are depth two in a strong sense (the split epimorphism in the definition may be replaced by a bimodule isomorphism).
The depth of Q as an R-module is defined in that paper to be the least positive integer n such that Q⊗⋅⋅⋅⊗Q (n times Q, tensor product of R-modules, diagonal action of R from the right) has the same constituent indecomposable modules as Q ⊗⋅⋅⋅⊗ Q (n+1 times Q) (not counting multiplicities, an entirely similar definition for depth of Q as an H-module with closely related results).
As a consequence, the depth of R in H is finite if and only if its "generalized quotient module" Q represents an algebraic element in the representation ring (or Green ring) of R. This is the case for example if Q is a projective module, a generator H-module or if Q is a permutation module over a group algebra R (i.e., Q has a basis that is a G-set).
In case H is a Hopf algebra that is a semisimple algebra, the depth of Q is the length of the descending chain of annihilator ideals in H of increasing tensor powers of Q, which stabilize on the maximal Hopf ideal within the annihilator ideal, Ann Q = { h in H such that Qh = 0 } (using a 1967 theorem of Rieffel).
and similarly for all powers of the inclusion matrix M, the condition of being depth
For example, a depth one subgroup H of a finite group G, viewed as group algebras CH in CG over the complex numbers C, satisfies the condition on the centralizer
, the order 2 and order 6 permutation groups on three letter a,b,c where the subgroup fixes c. The inclusion matrix may be computed in at least three ways via idempotents, via character tables or via Littlewood-Richardson rule coefficients and combinatorics of skew tableaux to be (up to permutation) the 2 by 3 matrix with top row 1,1,0 and bottom row 0,1,1, which has depth three after applying the definition.
In a 2011 article in the Journal of Algebra by R. Boltje, S. Danz and B. Kuelshammer, they provide a simplified and extended definition of the depth of any unital subring B of associative ring A to be 2n+1 if
(n times A) as B-B-bimodules for some positive integer m; similarly, B has depth 2n in A if the same condition is satisfied more strongly as A-B-bimodules (or equivalently for free Frobenius extensions, as B-A-bimodules).
(This definition is equivalent to an earlier notion of depth in case A is a Frobenius algebra extension of B with surjective Frobenius homomorphism, for example A and B are complex semisimple algebras.)
They then apply this to the group algebras of G and H over any commutative ring R. They define a minimum combinatorial depth
(i.e., G equals the product of H and its centralizer subgroup in G); in particular, H is normal in G. In general, the minimum depth
, which in turn is bounded by twice the index of the normalizer of H in G. Main classes of examples of depth two extensions are Galois extensions of algebras being acted upon by groups, Hopf algebras, weak Hopf algebras or Hopf algebroids; for example, suppose a finite group G acts by automorphisms on an algebra A, then A is a depth two extension of its subalgebra B of invariants if the action is G-Galois, explained in detail in the article on Frobenius algebra extension (briefly called Frobenius extensions).
Conversely, any depth two extension A | B has a Galois theory based on the natural action of
(often called a theory of duality of actions, which dates back in operator algebras to the 1970s).
If A | B is in addition to being depth two a Frobenius algebra extension, the right and left endomorphism rings are anti-isomorphic, which restricts to an antipode on the bialgebroid
There is the following relation with relative homological algebra: the relative Hochschild complex of A over B with coefficients in A, and cup product, is isomorphic as differential graded algebras to the Amitsur complex of the R-coring S (with group-like element the identity on A; see Brzezinski-Wisbauer for the definition of the Amitsur cochain complex with product).
For example, given a depth three Frobenius extension of ring A over subring B, one can show that the left multiplication monomorphism
The main theorem in this subject is the following based on algebraic arguments in two of the articles below, published in Advances in Mathematics, that are inspired from the field of operator algebras, subfactors: in particular, somewhat related to A. Ocneanu's definition of depth, his theory of paragroups, and the articles by W. Szymanski, Nikshych-Vainerman, R. Longo and others.
The proof of this theorem is a reconstruction theorem, requiring the construction of a Hopf algebra as a minimum, but in most papers done by construction of a nondegenerate pairing of two algebras in the iterated endomorphism algebra tower above B in A, and then a very delicate check that the resulting algebra-coalgebra structure is a Hopf algebra (see for example the article from 2001 below); the method of proof is considerably simplified by the 2003 article cited below (albeit packaged into the definition of Hopf algebroid).
The condition that the Frobenius homomorphism map A onto all of B is used to show that B is precisely the invariant subalgebra of the Hopf-Galois action (and not just contained within).