2010/12/08 by Lars Kadison, Kadison, Lars
Computer Science · Mathematics · #Advanced Algebra and Logic #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #math.RA #math.RT
paper · pdf · doi:10.48550/arxiv.1012.1754
18 pages
arxiv created 2010/12/08 · openalex publication_date 2010/12/08 · arxiv updated 2010/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A minimum depth dI(S --> R) is assigned to a ring homomorphism S --> R and a R-R-bimodule I. The recent notion of depth of a subring d(S,R)in a paper by Boltje-Danz-Kuelshammer is recovered when I = R and S --> R is the inclusion mapping. Ideal depth gives lower bounds for d(S,R) in case of group C-algebra pair or semisimple complex algebra extensions. If R | S is a QF extension of finite depth, minimum left and right even depth are shown to coincide. If R < S is moreover a Frobenius extension with R a right S-generator, its subring depth is shown to coincide with its tower depth. In the process formulas for the ring, module, Frobenius and Temperley-Lieb structures are provided for the tensor product tower above a Frobenius extension. 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.