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 /mω 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) /mω (−n) ) − 1, where p is the characteristic of the field K.

On the canonical ideal of the Ehrhart ring of the chain polytope of a poset

“…Note that if one replaces condition N' to condition N in (2) and (3) of Theorem 4.17, one obtains a criterion of level (resp. anticanonical level) property of K[O(P )] [Miy1,Miy2]. Since condition N' is a weaker condition than condition N, we see the following.…”

“…The Ehrhart ring K[O(P )] of the order polytope of P is identical with the ring considered by Hibi [Hib], which is nowadays called the Hibi ring. We studied in our previous papers [Miy1,Miy2] the canonical ideals of Hibi rings.…”

“…ω (−1) ) by "sequence with condition N' " in P . Recall that we [Miy1,Miy2] described generators of the canonical and anticanonical ideals of K[O(P )] by "sequence with condition N" in P . First we recall the definition of condition N and define condition N' of a sequence of elements of P .…”

“…In our previous papers, we studied many aspects of Hibi rings, especially described generators of the canonical ideals of Hibi rings and characterized level and anticanonical level properties of Hibi rings [Miy1,Miy2]. The key notion to investigate the generators of the canonical ideal of a Hibi ring is "sequence with condition N" (see below).…”

Non-Gorenstein loci of Ehrhart rings of chain and order polytopes

“…E K [C (P )]) is sequences with Condition N (resp. Condition N') [Miy1,Miy3,Miy2]. Conditions N and N' are quite similar but N' is a little weaker than N. A similar condition also showed up as the study of "mixed paths" in [Pag].…”