Properties of analogues of Frobenius powers of ideals

Let R = K[X-1, ..., X-n] be a polynomial ring over a field K. We introduce an endomorphism F-[m] : R -> R and denote the image of an ideal I of R via this endomorphism as I([m] )and call it to be the m-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.