Multiplicity bounds in prime characteristic

Abstract

We extend a result by Huneke and Watanabe ([HW15]) bounding the multiplicity of F -pure local rings of prime characteristic in terms of their dimension and embedding dimensions to the case of F -injective, generalized Cohen-Macaulay rings. We then produce an upper bound for the multiplicity of any local Cohen-Macaulay ring of prime characteritic in terms of their dimensions, embedding dimensions and HSL numbers. Finally, we extend the upper bounds for the multiplicity of generalized Cohen-Macaulay rings in characteristic zero which have dense F -injective type.

Upper bound of multiplicity in prime characteristic

“…It is shown thatWe also improve the bound for F -nilpotent rings. Our result extends the main results of Huneke and Watanabe [6] and of Katzman and Zhang [9].…”

“…R is F -pure). If R is Cohen-Macaulay, Katzman and Zhang [9,Theorem 3.1] proved the following inequality…”

“…In fact Katzman and Zhang[9, Theorem 2.2] needed to assume that every c ∈ R • is a non-zero divisor, i.e. R has no embedded primes.…”