In Vasconcelos' paper (Ideals generated by R-sequences), he proved

If $R$ is a local ring, $I$ an ideal of finite projective dimension, and $I/I^2$ is a free $R/I$ module, then $I$ can be generated by a regular sequence.

This is a theorem for local ring.

In Kac's paper, (Torsion in cohomology of compact Lie groups and Chow rings of reductive algebraic groups), he referred this result (in appendix, Proof of Theorem 1), but used for non-local ring.

More precisely, he constructed a map for a compact lie group $K$, and a field $k$, $S(M)\stackrel{\psi}\to H^\bullet(K/T;k)$, where $M=L\otimes k$ with $L$ the weight lattice, and $T$ the maximal torus. He claim $\ker \psi$ is generated by a homogenous regular sequence.

Furthermore, what I believe to be right, is the following

For polynomial ring $R$ over field, and a graded ideal $I$ such that $I/I^2$ is free over $R/I$ (as graded module), then $I$ is generated by a homogenous regular sequences.

**My question is, how to prove this if it is true? If not, is the $\ker \psi$ in the paper generated by a regular sequence?**

Maybe some useful remarks,

- This is not true for example $k[x]/x^2$, and $I=(x)$. Since $k[x]/x=k$ never admits a finite projective(=free since local) $k[x]/x^2$ resolution by dimension argument.
- When the ring is local and $I$ is the maximal ideal, this is exactly the theorem of regular local ring. I tried to move the proof, but fails, because of the above example.
- The main step of Vasconcelos' paper, is to a result due to auslander and buchbaum. It discussed local ring specifically.
- Generally, there is a concept called regular ideal, but it is local.
- I do not even know whether we can pick the sequence to be arbitary choice of representative of basis.
- I also wounder whether it is true for all graded ring with $I$ of finite projective dimension.
- For $\psi$, it is more crutial when the field of positive characteristic. When it is of characteristic zero, the $\psi$ is nothing but the classical thing, the projection to coinvariant algebra.