Extention of vector bundles on projective line: $Ext^1({\mathcal O_{\mathbb{P}^1}}(n),{\mathcal O_{\mathbb{P}^1}}(m))=$??

Question

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

Answer 1

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 .

Answer 2

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