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

摘要

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 T_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/mω~n and ⊕_n≥0~ω~(-1)/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→∞ cxF (R_k[H]) = dim(⊕_n≥0~ω~(-n))/ω~(-n)-1, where p is the characteristic of the field K.