Subring Depth, Frobenius Extensions, and Towers

الملخص EN

The minimum depth d(B,A) of a subring B⊆A introduced in the work of Boltje, Danz and Külshammer (2011) is studied and compared with the tower depth of a Frobenius extension.

We show that d(B,A) < ∞ if A is a finite-dimensional algebra and Be 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 A⊇B is a QF extension, minimum left and right even subring depths are shown to coincide.

If A⊇B 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.