Subring Depth, Frobenius Extensions, and Towers

The minimum depth ð ‘‘ ( ð µ , ð ´ ) of a subring ð µ âŠ† ð ´ 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 ð ‘‘ ( ð µ , ð ´ ) < ∞ if ð ´ is a finite-dimensional algebra and ð µ ð ‘’ 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 ð ´ âŠ‡ ð µ is a QF extension, minimum left and right even subring depths are shown to coincide. If ð ´ âŠ‡ ð µ 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 ð ‘› 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.