Fiber cones, analytic spreads of the canonical and anticanonical ideals and limit Frobenius complexity of Hibi rings

Abstract

Let ℛ_𝕂[𝐻] be the Hibi ring over a field 𝕂 on a finite distributive lattice 𝐻, 𝑃 the set of join-irreducible elements of 𝐻 and 𝜔 the canonical ideal of ℛ_𝕂[H]. We show the powers 𝜔^(𝑛) of 𝜔 in the group of divisors Div(ℛ_𝕂[𝐻]) is identical with the ordinary powers of 𝜔, describe the 𝕂-vector space basis of 𝜔^(𝑛) for 𝑛 ∈ ℤ. Further, we show that the fiber cones ⨁_{𝑛 ≥ 0} 𝜔^𝑛/ 𝔪 𝜔^𝑛 and ⨁_{𝑛 ≥ 0} (𝜔⁽⁻¹⁾)^𝑛/ 𝔪 (𝜔⁽⁻¹⁾)^𝑛 of 𝜔 and 𝜔⁽⁻¹⁾ are sum of the Ehrhart rings, defined by sequences of elements of 𝑃 with a certain condition, which are polytopal complex version of Stanley–Reisner rings. Moreover, we show that the analytic spread of 𝜔 and 𝜔⁽⁻¹⁾ are maximum of the dimensions of these Ehrhart rings. Using these facts, we show that the question of Page about Frobenius complexity is affirmative: lim_{𝑝 → ∞} cx_𝐹 (ℛ_𝕂[𝐻]) = dim(⨁_{𝑛 ≥ 0} 𝜔^(−𝑛)/ 𝔪 𝜔^(−𝑛)) −1, where 𝑝 is the characteristic of the field 𝕂.