Differentially homogeneous algebras
type: Item
instance of: scholarly article
title: Differentially homogeneous algebras
zbMATH Open document ID: 0944.13019
author: Pedro J. Sancho de Salas
published in: Journal of Algebra
publication date: 2000-05-04T00:00:00Z
review text: Let \(k\) be a ring and let \(A\) be a flat finitely generated \(k\)-algebra. The \(k\)-algebra \(A\) is said to be differentially homogeneous when the \(A\)-modules of jets \(J_{A/k}^r=(A\otimes_kA)/\Delta^{r+1}\) are projective for any \(r\geq 0\), where \(\Delta\) stands for the diagonal ideal. The following result concerning the local structure of differentially homogeneous algebras is obtained. Theorem. Let \(k\) be a Noetherian ring with connected spectrum and let \({\mathcal O}=k\xi_1,\dots,\xi_n\) be a complete separated \(k\)-algebra with respect to the ideal \(\mathfrak m=(\xi_1,\dots,\xi_n)\) where \({\mathcal O}/{\mathfrak m}=k\). Then \(\mathcal O\) is (formally) differentially homogeneous if and only if there exists a faithfully flat morphism \(k\to K\) such that \[ {\mathcal O}\widehat\otimes_kK=\begin{cases} Kx_1,\dots,x_n &\text{when the characteristic of \(k\) is not prime,}\\ Kx_1,\dots,x_n/(x_1^{p^{e_1}},\dots,x_r^{p^{e_r}}) &\text{when \(\text{char}(k)=p\) is prime.}\end{cases} \] {}.
Mathematics Subject Classification ID: 13J10
Mathematics Subject Classification ID: 13F25
zbMATH DE Number: 1340593
zbMATH Keywords: differentially homogeneous algebra
zbMATH Keywords: affine rings of algebraic groups
zbMATH Keywords: Noetherian ring
zbMATH Keywords: complete algebra
zbMATH Keywords: faithfully flat morphism
zbMATH Keywords: formal power series ring
reviewed by: E. V. Pankrat'ev
MaRDI profile type: Publication
full work available at URL: jabr.1998.7762
OpenAlex ID: W2006197913
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DOI: 10.1006/JABR.1998.7762
description: scientific article; zbMATH DE number 1340593