Classify all coherent sheaves on $\mathbb{A}^1_k$

algebraic-geometrycoherent-sheavesquasicoherent-sheavesschemessheaf-theory

Given a field $K$, I want to classify all coherent sheaves on $\mathbb{A}^1_k$, and moreover saying if there exist locally free sheaves that are not free on $\mathbb{A}^1_k$.

I am following Gathmann's notes on Algebraic Geometry, thus I only know the following:

  • The definition of quasi-coherent sheaf as sheaf of modules $\mathcal{F}$ on a scheme $X$ such that there is an affine open cover of $U_i=\operatorname{Spec} R_i$ with $\mathcal{F}$ restricted on the $U_i$ isomorphic to the sheaf $\tilde{M_i}$ associated to an $R_i$-module $M_i$;
  • The definition of coherent sheaf as above with the additional requirement that the $M_i$ are finitely generated $R_i$-modules.

To start, I know that $\mathbb{A}^1_k=\operatorname{Spec} K[x]$.

And I know that the structure sheaf is always quasi-coherent, so $O_{\operatorname{Spec} K[x]}$ is quasi-coherent, but I don't know why it is coherent, since this condition is not discussed on Gathmann's notes.

How can I find the other coherent sheaves? Any help is much appreciated. Thanks in advance.

Best Answer

First some generalities: As noted in the comments, every quasi-coherent sheaf on an affine scheme $\mathrm{Spec}(R)$ is of the form $\widetilde{M}$ for some $R$-module $M$. If $R$ is Noetherian, $\widetilde{M}$ is a coherent sheaf if and only if $M$ is finitely generated (the definition of coherent is a bit more technical when $R$ is not Noetherian). Further, if $\widetilde{M}$ is coherent, then $\widetilde{M}$ is locally free if and only if $M$ is a projective $R$-module.

Now let's try to answer your two questions:

  1. Classify all coherent sheaves on $\mathbb{A}^1_K$. Since $\mathbb{A}^1_K = \mathrm{Spec}(K[x])$, and $K[x]$ is Noetherian, the problem reduces to classifying all finitely generated $K[x]$-modules. Since $K[x]$ is a PID, these are classified by the structure theorem.

  2. Are there locally free sheaves that are not free on $\mathbb{A}^1_K$? I assume you mean to continue restricting to the coherent case here. Since a locally free coherent sheaf is of the form $\widetilde{M}$, where $M$ is a finitely generated projective $R$-module, the question reduces to asking whether there exist finitely generated projective $K[x]$-modules which are not free. The answer is no, since $K[x]$ is a PID.

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