# Embedding dimension and codimension of tensor products of algebras over a field

## TL;DR

This paper studies the embedding dimension and codimension of tensor product localizations of k-algebras, extending the 'special chain theorem' to understand regularity transfer in algebraic structures.

## Contribution

It establishes an analogue of the 'special chain theorem' for embedding dimension in tensor products of k-algebras, linking prime spectra and dimension theory.

## Key findings

- Derived an analogue of the 'special chain theorem' for embedding dimension.
- Analyzed how regularity properties transfer or defect in tensor product localizations.
- Provided new insights into the structure of Noetherian local rings from tensor products.

## 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.