Poset embeddings of Hilbert functions

Abstract

Abstract. For a standard graded algebra R, we consider embeddings of the poset of Hilbert functions of R-ideals into the poset of R-ideals, as a way of classification of Hilbert functions. There are examples of rings for which such embeddings do not exist. We describe how the embedding can be lifted to certain ring extensions, which is then used in the case of polarization and distraction. A version of a theorem of Clements-Lindström is proved. We exhibit a condition on the embedding that ensures that the classification of Hilbert functions is obtained with images of lexicographic segment ideals.

On the Lex-plus-powers Conjecture

“…To summarize, we have HF(I + (x i n )) = HF(K + (x i n )) = HF(H + (x i n )) for all i ≥ 1, and H is a d-SPP ideal of S with HF(I) = HF(H), so in particular L d (I) = L d (H). Now we may apply [4,Theorem 3.9]: in the language of that paper, the ring S ℘ has the embedding I ℘ → L d (I) ℘ , and this is precisely the embedding produced by [4, Proof of Theorem 3.3] starting from the embedding of S ℘ given by J ℘ → L d (J ) ℘ . We obtain that HF(H + (x i n )) HF(L d (H) + (x i n )), and the desired conclusion follows.…”

“…where A = k[x 0 , x 1 , x 2 , x 3 ] is the homogeneous coordinate ring of P 3 . By going modulo a general linear form in A/I Γ , we reduce to considering Artinian algebras R = S/I where S = k[x 1 , x 2 , x 3 ] with HF(R) = (1,3,6,10,12,12,12,12,11,9,6,2) and I contains a regular sequence of degrees (4,4,8…”

“….) and I contains a regular sequence of degrees (4,4). The upper bounds for the Betti table of I Γ obtained via the Bigatti-Hulett-Pardue Theorem and Theorem 3.8 are respectively 0 1 2 4 4 4 1 5 1 2 1 6 3 5 2 7 1 2 1 8 1 2 1 9 1 2 1 10 1 2 1 11 1 1 − 0 1 2 4 4 3 1 5 − − − 6 1 3 1…”

“…Let Ω ⊆ P 3 be a 0-dimensional complete intersection of degrees(4,4,8). Let Γ ⊆ Ω be a closed subscheme with Hilbert function HF(A/I Γ ) =(1, 4, 10, 20, 32, 44, 56, 68, 79, 88, 94, 96, 96, 96, .…”

Distractions of Shakin rings

“…Our first result, Theorem 3.4, which we derive as a direct consequence of all the available results on embeddings of Hilbert functions [CaKu1,CaKu2,CaSb], states that Shakin rings are Macaulaylex and that they satisfy the properties (1),(2) and (3) mentioned above (with the exception that, to prove (2) when P = (0), we assume char(K) = 0).…”