On a conjecture by Kalai

Abstract

We show that monomial ideals generated in degree two satisfy a conjecture by Eisenbud, Green and Harris. In particular we give a partial answer to a conjecture of Kalai by proving that h-vectors of flag Cohen-Macaulay simplicial complexes are h-vectors of Cohen-Macaulay balanced simplicial complexes.Date: October 22, 2018.2010 Mathematics Subject Classification. 13D40, 13F55, 05E45. Key words and phrases. h-vector and face vector and Cohen Macaulay simplicial complex and flag simplicial complex, EGH conjecture, regular sequences .

Square-free Gröbner degenerations

“…Furthermore, depth(S/ in(I)) = depth(S/I) by Corollary 2.7. So the conclusion follows using [CCV14], because S/I and S/ in(I) have the same Hilbert function: in fact [CCV14, Theorem 2.1] implies that the h-vector of S/ in(I) equals the h-vector of S/J where J is a homogeneous ideal containing (x 2 1 , . .…”

Hilbert functions of monomial ideals containing a regular sequence

“…We conclude that the EGH conjecture is true when I is a monomial ideal. A related result was proved in [3] when I is generated by monomials of degree 2 and in [6] when f 1 , . .…”

“…In [3], Caviglia, Constantinescu and Varbaro proved that h-vectors of Cohen-Macaulay flag simplicial complexes are h-vectors of Cohen-Macaulay balanced simplicial complexes. We generalize by proving that if ∆ is a Cohen-Macaulay simplicial complex such that I ∆ of height t and containing a regular sequence f 1 , .…”

On the Lex-plus-powers Conjecture

“…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…”

“…It is worth noticing that, although the two statements are apparently independent of each other, Conjecture 1.1 is actually equivalent to the special case i = 0 of Conjecture 1.2, see e.g. [ [1,3,8,10,17,20,27] for other special cases. On the other hand, much less is known about Conjecture 1.2, cf.…”