# Injective Resolutions in $\mathfrak{Ab}(X)$

Using right derived functors of the global sections functor, I'd like to calculate the first cohomology group of the constant sheaf $\mathbf{Z}$ on $S^1$ with its usual topology, $H^1(S^1,\mathbf{Z})$. I understand this can be done using Čech cohomology, but I'd like to compute it with derived functors.

To do so, I need to write down an injective resolution of the constant sheaf $\mathbf{Z}$. One such injective resolution involves products of skyscraper sheaves, but this seems fairly messy if all I want to show is $H^1(S^1,\mathbf{Z})=\mathbb{Z}$. Is there a cleaner injective resolution of $\mathbf{Z}$ in $\mathfrak{Ab}(X)$, the category of sheaves of abelian groups on $X$?

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Injective resolutions are not easy things to find; that is why we have Čech cohomology! –  Zhen Lin Apr 26 '13 at 6:42
This is exervise 3.2.7 in Hartshorne's "Algebraic Geometry". A solution can be found on page 103 in this PDF: math.northwestern.edu/~jcutrone/Work/… –  Fredrik Meyer Apr 26 '13 at 7:12
@FredrikMeyer: I've seen this solution, and it uses products of skyscraper sheaves. My question asks if there is a cleaner injective resolution. –  Jared Apr 26 '13 at 13:32