Embedding Dimension and Codimension of Tensor Products of Algebras over a Field

Abstract

Abstract. Let k be a field. This paper investigates the embedding dimension and codimension of Noetherian local rings arising as localizations of tensor products of k-algebras. We use results and techniques from prime spectra and dimension theory to establish an analogue of the "special chain theorem" for the embedding dimension of tensor products, with effective consequence on the transfer or defect of regularity as exhibited by the (embedding) codimension given by codim(R) := embdim(R) − dim(R).

Basically regular local homomorphisms

“…. Also, note that in [5], the quantities δ A (I) and δ ϕ B (I) are denoted by µ A (I) and µ ϕ B (I), respectively. The second deviation of A is the vector space dimension of the second André-Quillen homology group H 2 (A, K, K), i.e., ε 2 (A) := dim K (H 2 (A, K, K)).…”