  Let $R=\mathbb{K}[X_1, \ldots , X_n ]$ be a polynomial ring over a field
$\mathbb{K}$. We introduce an endomorphism $\mathcal{F}^{[m]}: R \rightarrow R
$ and denote the image of an ideal $I$ of $R$ via this endomorphism as
$I^{[m]}$ and call it to be the $m$ \textit{-th square power} of $I$. In this
article, we study some homological invariants of $I^{[m]}$ such as regularity,
projective dimension, associated primes and depth for some families of ideals
e.g. monomial ideals.
