h ) \in (C+ \overline \Theta R)_{d-j}.$$ By Lemma \[2.3.2\], we have $(u_{d-j}/u_{\ell-s}) (\alpha_s u_{\ell -s} + \theta_d h) \in C+\overline \Theta R$. Hence $$\alpha_s u_{d-j} \in (C+ \overline \Theta R+ \theta_d R)_{d-j} \subset \Theta R,$$ where the last inclusion follows from Lemma \[E2.5\]. Since $\alpha_s \ne 0$, $u_{d-j}$ is contained in $\Theta R$. This fact shows $u_{d-j}=x_{v_d} x_{v_{d-1}}\cdots x_{v_{j+1}} \in J + (\Theta)$ when we regard $u_{d-j}$ as an element of $A$.

Proof of sufficiency
====================

In this section, we prove the sufficiency of Theorem \[1.2\].

Since the proof is a bit long, we first give a short outline of the proof. The key tools of the proof are the two main lemmas outlined below.

-   In Lemma \[3.A\], we introduce a simple way to make a new simplicial cell ball from a given simplicial cell ball by gluing a pair of simplexes along their boundaries.

-   In Lemma \[3.B\], we construct a special simplicial cell ball whose boundary has a nice property.

Then we construct a simplicial cell ball with the desired $h$-vector as follows: First, starting from a simplex, we gradually make a larger simplicial cell ball by using (I) repeatedly; Then, at a certain point, we 