Equi-dimensionalization via subdivision of simplices

We give an alternative proof of Suslin's equi-dimensionalization moving lemma using a different geometric construction. The new construction provides better control of the degrees of the polynomials describing the geometric procedure. The new degree bound can be used to improve an earlier result of Hiroyasu Miyazaki and the present author on algebraic cycles with modulus: the isomorphism in question is now valid for any fixed divisor, without the need to take the limit over thickenings.