In ring theory and Frobenius algebra extensions, areas of mathematics, there is a notion of depth two subring or depth of a Frobenius extension. 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.
Definition and first examples
A unital subring has (or is) right depth two if there is a split epimorphism of natural A-B-bimodules from for some positive integer n; by switching to natural B-A-bimodules, there is a corresponding definition of left depth two. Here we use the usual notation (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.
For example, let A be the group algebra of a finite group G (over any commutative base ring k; see the articles on group theory and group ring for the elementary definitions). Let B be the group (sub)algebra of a normal subgroup H of index n in G with coset representatives . Define a split A-B epimorphism p: by . It is split by the mapping defined by where for g in the coset (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 is satisfied. Thus B is right depth two in A.
As another example (perhaps more elementary than the first; see ring theory or module theory for some of the elementary notions), let A be an algebra over a commutative ring B, where B is taken to be in the center of A. Assume A is a finite projective B-module, so there are B-linear mapping and elements (i = 1,...,n) called a projective base for the B-module A if it satisfies for all a in A. It follows that B is left depth two in A by defining with splitting map as the reader may verify. A similar argument naturally shows that B is right depth two in A.
Depth in relation to Hopf algebras
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). For example, if B is a Hopf subalgebra of a finite-dimensional Hopf algebra, then B has depth two in A if and only if B is normal in A (i.e. invariant under the left and right adjoint actions of A). Since a group algebra is a Hopf algebra, the first example above illustrates the back implication of the theorem. 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).