On the Dimension of Modules and Algebras, VIII. Dimension of Tensor Products

Abstract

The questions concerning the dimension of the tensor product of two K-algebras have turned out to be surprisingly difficult. In this paper we follow a method using spectral sequences (§§1-3) which in some concrete cases yields complete results (§§4-5). In particular, complete results are obtained when r is a ring of matrices, triangular matrices, polynomials or rational functions, so that in the first three cases is respectively the ring of matrices, triangular matrices or polynomials with coefficients in the arbitrary algebra A.