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.
Categories: math.AC