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I am using the Mayer Vietoris sequence to compute the de Rham cohomology of a twice punctured plane.

I computed it first by choosing my open cover to be $U=R^2-{pt}$ and $V=R^2-{pt}$ (the points are different) with $U\cup V = R^2$ and $U \cap V$ the double punctured plane. Everything worked as expected.

Then, I tried computing it another way. Assume the missing points are (0,1) and (0,-1). I take $U = [(x,y): y > -1/2] $ with (0,1) missing. Similarly, I take $V = [(x,y): y < 1/2] $ with (0,-1) missing. Then, $ U \cup V $ is the double punctured plane, and $ U \cap V $ is the convex strip $ [-1/2 < y < 1/2] $ . But $H^{0}(U) \oplus H^{0}(V) = H^{1}(U) \oplus H^{1}(V) = R^2$ while $H^{0}(U \cap V) = R$ and $H^{1}(U \cap V) = 0$, so that the sequence is not exact: $ H^{0}(U \cap V) = R$ $\rightarrow$ $H^{1}(U \cup V)$ $\rightarrow$ $H^{1}(U) \oplus H^{1}(V) = R^2$ $\rightarrow$ $H^1(U \cap V) = 0$
What blunder am I making here?


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Where in the sequence is the failure of exactness? – Nick D. Jan 23 '14 at 20:08
You're right. There's no problem – Guest Jan 23 '14 at 20:22
up vote 3 down vote accepted

There's no blunder. The MV sequence says this (where $M = U \cup V$), and I'm gonna be sloppy about formatting: $$ 0 \to H^0M \to H^0 U + H^0 V \to H^0(U \cap V) \to \\ H^1 (M) \to H^1 U + H^1 V \to H^1(U \cap V) \to \ldots $$ Let's fill in those groups: $$ 0 \to R \to R + R\to R\to \\ R+R \to R+R \to 0 \to \ldots $$ The maps themselves look like this: $$ 0 \to R : 0 \mapsto 0\\ R \to R + R :t \mapsto (t, t)\\ R + R\to R : (u,v) \mapsto u - v \\ R \to R + R :t \mapsto 0\\ R+R \to R+R : (u, v) \mapsto (u, v) \\ R+R \to 0 : (u, v) \mapsto 0. $$ That's exact at every step.

You might want to look at as well.

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