I am considering the following problem:
For the projective space $X=\mathbb P^n$ is there an exact sequence of the form
$0\rightarrow \binom{n+1}{n+1} \mathcal O(-n-1)\rightarrow \binom{n+1}{n}\mathcal O(-n)\rightarrow ... \rightarrow \binom{n+1}{1} \mathcal O(-1)\rightarrow \binom{n}{0}\mathcal O\rightarrow 0$
where for a sheaf $\mathcal F$ $m\mathcal F$ is shorthand for $\bigoplus_{i=1}^m \mathcal F$?
I tried proving it with Koszul complexes. For this I first tried to apply Koszul complex of locally free sheaves but I couldn't find a locally free sheaf of rank $n+1$ which gives me my answer.
I also tried proving it locally, but the usual Koszul complex works with exterior products, I couldn't "switch them out" to direct sums.
Background: I am trying to prove that in the Grothendieck group of $\mathbb P^n$ for $H=[\mathcal O]-[\mathcal O(-1)]$
$H^{n+1}=0$, where the product in the Grothendieck group is defined by
$[E][F]=\sum \limits _{i=0}^n (-1)^i[Tor_i(E,F)]$
Could anyone point out a hint to proving the existence of such an exact sequence (if it exists)?
sincerely, slinshady