4. The reverse lexicographic order
We have reg(I) = reg(gI) for any ideal I and a coordinate transformation g ∈ GLn because the Betti tables of I and gI coincide. However, taking the initial ideal does not commute with coordinate transformation because syzygy calculations are not preserved under coordinate transformations. Where reg(I) ≤ reg(inτ (I)) for any order τ , the following theorem of Bayer and Stillman shows that the graded reverse lexicographic order gives the lowest possible regularity for generic initial ideals.
