Let $A$ be a noetherian local ring with residue field $k$, one can consider $Ext^i(k,k)$ for every natural number $i$. If it is zero for large $i$, then $A$ is regular and the converse is also true. Then, for which rings the dimension of $Ext^i(k,k)$ is bounded with respect to $i$ ?

Example: $\mathbb Z /p^n$

Non-example: $k[x,y]/(x^2,y^2)$

What about polynomial growth?

Let $A$ be a noetherian local ring with residue field $k$, one can consider $\operatorname{Ext}^i(k,k)$ for every natural number $i$. If it is zero for large $i$, then $A$ is regular and the converse is also true. Then, for which rings the dimension of $\operatorname{Ext}^i(k,k)$ is bounded with respect to $i$ ?

Example: $\mathbb Z /p^n$

Non-example: $k[x,y]/(x^2,y^2)$

What about polynomial growth?