Differentially homogeneous algebras
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} \] {}.
scientific article; zbMATH DE number 1340593