Corollary

Let $A$ be a local Macaulay ring, and $M$ a nonzero finitely generated $A$-module, of dimension $e$. The $A$-module $H_A^e(M)$ is nonzero.

By Remark 2 of No. 1, one may suppose that the local ring $A$ is complete. In this case $A$ has a dualizing module $\Omega$ ($§ 9$, No. 3, Corollary 3 of Proposition 6); if $H_A^e(M)$ is zero, the same is true of its Matlis dual $\mathrm{Ext}_A^{d-e}(M, \Omega)$, which contradicts Proposition 3, b) of $§ 9$, No. 1.