Counter-examples for quasi-coherent, coherent, locally free and invertible sheaves

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I'm trying to find at least one counter-example for each of these concepts to feel more comfortable with understanding the ideas behind them but I cannot even get started 🙁 Please help me find counter-examples for the following concepts:

  1. A quasi-coherent sheaf that is not coherent
  2. A coherent sheaf that is not locally free
  3. A locally free sheaf that is not globally free
  4. A locally free sheaf that is not invertible

I'm studying sheaves from Kempf's Algebraic Varieties. My main problem that prevents me from attacking the above questions is that I do not know how I can create new sheaves or modify old sheaves to make them have interesting properties. The only example of a sheaf I know is an algebraic variety with its structure sheaf (i.e. the $k$-algebra of regular functions over its open sets when it's considered as a space with functions). Kempf's definitions are so abstract for me and I would highly appreciate any glimpse of intuition or information that answers the above questions.

Best Answer

Here are the examples, I leave you as an exercise to check everything :-)

1) Take an uncoutable sum of $\mathcal O_X$.

2) Take a skyscraper sheaf, i.e $i_* \mathcal O_{p}$ where $i : p \to X$ is the inclusion.

3) This is less easy, the simplest example is the sheaf $\mathcal O(1)$ on $\Bbb P^1$.

4) Hint : a locally free sheaf of rank $r > 1$ can't be invertible.

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