Coherent sheaves of finite length over $\mathbb{P}^n_k$

Question

Let $k$ be an algebraically closed field. Are there any nonzero coherent sheaves on the projective space $\mathbb{P}^n_k$ that are supported at (only) finitely many closed points? If they don't exist, what is the reason?

Answer 1

So let me make my comment to an answer (just to take this question off the list of unanswered ones):

If $i : Z \to X$ is a closed immersion and $M$ is a quasi-coherent module on $Z$ of finite type, then also $i_* M$ is a quasi-coherent module on $X$ of finite type. For noetherian schemes, finite type = coherent. Now let $x \in X$ be a closed point and $i : \mathrm{Spec} k(x) \to X$ the corresponding inclusion. Then $i_* \mathcal{O}_{k(x)}$ is a quasi-coherent module of finite type on $X$, which is supported at $\{x\}$. This is a skyscraper sheaf.