# In zero dimensional graded module the degree $1$ component is non zero.

I am reading the commutative algebra book by W. Bruns and H. Herzog. I am stuck at the Corollary 4.1.8 which comes from lemma 4.1.7. Actually in the corollary $d=0$ case does not come from the previous lemma. It requires an independent proof. So far it is clear to me that Hilbert polynomial is the $0$ polynomial. What I couldn't understand is $Q_M(1)$ is non zero in this case. I need some help. Many many thanks.

• Please include the full details of your question (the title does not make much sense to me as is. If the module is zero dimensional, then all graded components are also $0$). – Tobias Kildetoft Jan 2 '17 at 8:29
• When $d=0$, $Q_M(1)$ is the sum of the dimensions of graded components of $M$ which is not $0$. – user26857 Jan 2 '17 at 10:45