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?



Noetherian local ring and the growth of $\dim_k \operatorname{Ext}^i(k,k)$
The commutative noetherian rings such that the Betti numbers of $k$ eventually grow polynomially are precisely the complete intersections. This is a theorem of Gulliksen, see Theorem 2.3  [here](https://www.jstor.org/stable/24491379) .


The degree of the polynomial giving the growth (which is called the complexity if you add 1) determines the codimension of the complete intersection. In particular, if the Betti numbers are eventually constant the ring is a hypersurface.


Both of your examples are complete intersections, as the theorem says, with $\mathbb{Z}/p^n\mathbb{Z}$ a hypersurface and $k[x,y]/(x^2,y^2)$ of codimension $2$.


