Let $R$ be a local commutative Noetherian ring with maximal ideal $m$.
My questions concern ideals $I \subseteq m$ of $R$ such that for any non-zero number $n \in \mathbb{N}$ the $R/I$-module $I^n/I^{n+1}$ is free.
There are natural examples:
- The maximal ideal $m$ of $R$ is such an ideal.
- For any regular element $x \in m$ the cyclic ideal $(x)$ is such an ideal. Actually, any regular sequence $(x_1,\ldots,x_n)$ of elements in $m$ generates such an ideal.
A Theorem by Ferrand and Vasconcelos states that an ideal $I$ of $R$ is generated by a regular sequence if and only if it has finite projective dimension and $I/I^2$ is a free $R/I$-module.
So the remaining examples of ideals of the type above have infinite projective dimension.
Here are my questions:
- Do such ideals have a name in commutative algebra?
- Do they have an interpretation in algebraic geometry?
- What is the geometric interpretation of ideals $I \subseteq m$ such that the conormal module $I/I^2$ is a free $R/I$-module?
Any comments or feedback will be appreciated!
