%0 Journal Article
%T Subring Depth, Frobenius Extensions, and Towers
%A Lars Kadison
%J International Journal of Mathematics and Mathematical Sciences
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/254791
%X The minimum depth (,) of a subring £¿ 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 (,) < ∞ if  is a finite-dimensional algebra and  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 £¿ is a QF extension, minimum left and right even subring depths are shown to coincide. If £¿ 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  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.
%U http://www.hindawi.com/journals/ijmms/2012/254791/