Recall that given a Riemann Surface $X$, the divisor sheaf is the sheaf ${\cal D}$ which assigns to each open set $U$ the collection of maps $\phi:U \to \mathbb{Z}$ such that $\phi(p)=0$ for all but finitely many $p$ in $U$ with the obvious restriction maps. A partition of unity for a sheaf ${\cal D}$ subordinate to a locally finite open cover $\{ U_\alpha \}$ is a family $h_\alpha \in \mathrm{Hom}({\cal D}, {\cal D})$ such that the support of $h_\alpha \subseteq U_\alpha$ and $\sum_\alpha h_\alpha = \mathrm{Id}$.

Gunning, in his text "Lectures on Riemann Surfaces", claims that the divisor sheaf of a Riemann surface has partitions of unity subordinate to any locally finite cover (in other words, it is a fine sheaf). Unfortunately, he leaves it as an exercise to the reader to verify this and I have been unable to construct one myself. Any help would be greatly appreciated.

  • $\begingroup$ Couldn't you do it arbitrarily? Unlike constructing smooth partitions of unity, it seems to me here that a discontinuous integer-valued one will work just as well... $\endgroup$ – Zhen Lin Dec 28 '12 at 2:21
  • $\begingroup$ As the maps $\phi$ have no restrictions on their topology I agree with you. The construction of such a partition of unity will be more "algebraic" than "topological". $\endgroup$ – Mark Penney Dec 28 '12 at 5:33

The trick is to introduce an ordering of the cover. So consider $U_{\alpha}$ a covering of $X$. Since we can assume it's countable, we have now a countable cover $\{ U_n \}_{n \in \mathbb Z}$. Define $\varphi_n(x) = 1$ if $n$ is the smallest number such that $x \in U_n$ and $0$ else. This is the desired partition of unity.


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