Depth of noncommutative subrings

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

B

⊆

A

{\displaystyle B\subseteq A}

has (or is) right depth two if there is a split epimorphism of natural A-B-bimodules from

A

n

→

A

⊗

B

A

{\displaystyle A^{n}\rightarrow A\otimes _{B}A}

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

A

n

=

A

×

…

×

A

{\displaystyle A^{n}=A\times \ldots \times A}

(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

g

1

,

⋯

,

g

n

{\displaystyle g_{1},\cdots ,g_{n}}

. Define a split A-B epimorphism p:

A

n

→

A

⊗

B

A

{\displaystyle A^{n}\rightarrow A\otimes _{B}A}

by

p

(

(

a

1

,

⋯

,

a

n

)

)

=

∑

i

=

1

n

a

i

g

i

−

1

⊗

B

g

i

{\displaystyle p((a_{1},\cdots ,a_{n}))=\sum _{i=1}^{n}a_{i}g_{i}^{-1}\otimes _{B}g_{i}}

. It is split by the mapping

q

:

A

⊗

B

A

→

A

n

{\displaystyle q:A\otimes _{B}A\rightarrow A^{n}}

defined by

q

(

a

⊗

B

a

′

)

=

(

a

γ

1

(

a

′

)

,

⋯

,

a

γ

n

(

a

′

)

)

{\displaystyle q(a\otimes _{B}a')=(a\gamma _{1}(a'),\cdots ,a\gamma _{n}(a'))}

where

γ

i

(

g

)

=

δ

i

j

g

{\displaystyle \gamma _{i}(g)=\delta _{ij}g}

for g in the coset

g

j

H

{\displaystyle g_{j}H}

(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

A

⊗

B

A

{\displaystyle A\otimes _{B}A}

is satisfied.