Separable ring extensions and cohomological dimension (Q1104394)

The author introduces the notion of separable K-rings to generalize several results on cohomological dimension of algebras due to \textit{M. Auslander} [Nagoya Math. J. 9, 67-77 (1955; Zbl 0067.271) and ibid. 11, 61-65 (1957; Zbl 0212.058)], \textit{S. Eilenberg}, \textit{M. Ikeda} and \textit{T. Nakayama} [Nagoya Math. J. 8, 49-57 (1955; Zbl 0066.288) and Comment. Math. Helv. 28, 310-319 (1954; Zbl 0058.028)] and others. It is proved that the cohomological dimension of a semiprimary purely inseparable algebra over a field is infinite. Furthermore, a relative version of one inequality for projective and cohomological dimensions by \textit{G. Hochschild} [Nagoya Math. J. 13, 89-94 (1958; Zbl 0085.266)] is given.