Properties of analogues of Frobenius powers of ideals

Abstract

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-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.