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I want to know the value $Ext^1({\mathcal O_{\mathbb{P}^1}}(n),{\mathcal O_{\mathbb{P}^1}}(m))$ for integer m, n.

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Have you considered short exact sequences $0\to\mathcal O_{\mathbb P^1}(m)\to U\to\mathcal O_{\mathbb P^1}(n)\to 0$? – Berci Oct 6 '12 at 8:55
Yes, I have. But I think that computation of $Ext^1% as extension is difficult. Are you agree? – aska langley Oct 6 '12 at 10:25
up vote 3 down vote accepted

a) The sheaf ext, $\mathcal {Ext}^1({\mathcal O_{\mathbb{P}^1}}(n),{\mathcal O_{\mathbb{P}^1}}(m))=0 $ , is the zero sheaf for all $n,m\in \mathbb Z$.
[In general, if $\mathcal F$ is locally free, the sheaf ext $\mathcal {Ext}^i(\mathcal F,\mathcal G)$ is zero for $i\gt0$ and for all coherent $\mathcal G$ because the functor $\mathcal {Hom}(\mathcal F,\bullet ) $ is exact].

b) What you probably want is the $k$-vector space ext, ${Ext}^1({\mathcal O_{\mathbb{P}^1}}(n),{\mathcal O_{\mathbb{P}^1}}(m))$.
It is isomorphic to $$H^1(\mathbb P^1_k,\mathcal {Hom}(\mathcal O_{\mathbb{P}^1}(n) ,\mathcal O_{\mathbb{P}^1}(m))=H^1(\mathbb{P}^1,\mathcal O_{\mathbb{P}^1}(m-n)) \quad (*)$$
The explicit calculation of your ext vector space then follows from Serre duality $dim_k H^1(\mathbb{P}^1,\mathcal O_{\mathbb{P}^1}(m-n))= dim_k H^0(\mathbb{P}^1,\mathcal O_{\mathbb{P}^1}(n-m-2))$ and the well known result $dim_k H^0(\mathbb{P}^1,\mathcal O_{\mathbb{P}^1}(r))=r+1$ for $r\geq 0$ and $=0$ else.

The displayed isomorphism $(*)$ follows from the general spectral sequence $$E_2^{i,j} = H^i(X,\mathcal {Ext}^j(\mathcal E,\mathcal F)) \implies Ext^{i+j}(\mathcal E,\mathcal F),$$ of which you take the low degree ensuing exact sequence .

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$\mathrm{Ext}^1(O(n),O(m)) = \mathrm{Ext}^1(O,O(m-n)) = H^1(O(m-n))$ and the cohomology groups are well-known.

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