Global sections of the structure sheaf on $\mathbb{P}_{k}^{n}$ in negative degree

Question

This question is about something written on the first page of the lecture notes here. As usual, let $\mathbb{P}_{k}^{n}$ denote the projective space over a field (or indeed any noetherian ring) $k$. There, Ravi gives the formula for dimension of the zeroth cohomology group in degree $m \geq 0$, $$ \dim H^{0}(\mathbb{P}_{k}^{n}, \mathcal{O}(m)) = {n + m \choose m}, $$ which just comes from a stars and bars combinatorics argument. This is all well and good. My confusion is in the next sentence. There is claims that this breaks down for the case that $m \leq -n-1$. But isn't $H^{0}(\mathbb{P}_{k}^{n}, \mathcal{O}(m))$ just $0$ as soon as $m < 0$? So what is meant by the claim that it "breaks down" for $m \leq -n-1$? If $m < 0$ then since $\mathbb{P}_{k}^{n} = \text{Proj }k[x_{0}, x_{1}, \ldots , x_{n}]$, surely there are no global sections in negative degree, right?

Answer 1

If you think of $$ \binom{n+m}{m} = \frac{(m+1)(m+2)\cdots(m+n)}{n!} $$ as of polynomial in $m$ (with $n$ considered as a parameter), it is nonzero for $m \le -n-1$, while $H^0 = 0$.