Gröbner deformations and F‐singularities

Abstract

For polynomial ideals in positive characteristic, defining F‐split rings and admitting a squarefree monomial initial ideal are different notions. In this note, we show that, however, there are strong interactions in both directions. Moreover, we provide an overview on which F‐singularities are Gröbner deforming. Also, we prove the following characteristic‐free statement: If p$\mathfrak {p}$ is a height h prime ideal such that infalse(pfalse(hfalse)false)$\mathrm{in}(\mathfrak {p}^{(h)})$ contains at least one squarefree monomial, then infalse(frakturpfalse)$\mathrm{in}(\mathfrak {p})$ is a squarefree monomial ideal.