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For a given $(d-1)$-dimensional simplicial complex $\Gamma$, we denote its
$h$-vector by $h(\Gamma)=(h_0(\Gamma),h_1(\Gamma),\ldots,h_d(\Gamma))$ and
set $h_{-1}(\Gamma)=0$. The known Swartz equality implies that if $\Delta$
is a $(d-1)$-dimensional Buchsbaum simplicial complex over a field, then
for every $0\leq i\leq d$, the inequality $ih_i(\Delta)+ (d-i+1)h_{i-1}(
\Delta)\geq0$ holds true. In this paper, by using these inequalities, we
give a simple proof for a result of Terai (N. Terai, On $h$-vectors of
Buchsbaum Stanley--Reisner rings, {\it Hokkaido Math. J.} {\bf 25} (1996),
no. 1, 137--148) on the $h$-vectors of Buchsbaum simplicial complexes. We
then generalize the Swartz equality (E. Swartz, Lower bounds for
$h$-vectors of $k$-CM, independence, and broken circuit complexes, {\it
SIAM J. Discrete Math.} {\bf 18} (2004/05), no. 3, 647--661), which in turn
leads to a generalization of the above-mentioned inequalities for
Cohen--Macaulay simplicial complexes in co-dimension $t$.
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