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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?

Must
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    Sure, skyscraper sheaves ... – Martin Brandenburg Feb 23 '14 at 17:40
  • But are skyscraper sheaves coherent? – Must Feb 23 '14 at 17:43
  • In general, if $i : Z \to X$ is a closed immersion of noetherian schemes, and $F \in \mathrm{Coh}(Z)$, then $i_* F \in \mathrm{Coh}(X)$. One simply reduces to the observation that $\mathcal{O}_X/I$ is coherent if $I$ defines $i$. Or am I missing something? – Martin Brandenburg Feb 23 '14 at 17:45
  • I am confused by the answer provided to this question: http://math.stackexchange.com/questions/642262/is-skyscraper-sheaf-quasi-coherent – Must Feb 23 '14 at 17:55
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    There is no reason to be confused: there exist non-coherent skyscraper sheaves and I have described one in the answer you link to. But Martin gives and justifie a perfectly correct recipe for the construction of coherent skyscraper sheaves. – Georges Elencwajg Feb 23 '14 at 20:39

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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.

Martin Brandenburg
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