Cohomology of projective space seen by residual complex (Q2716131)

The author constructs a residual complex \(J^{\bullet}\) on \({\mathbb{P}}^n\) which is an injective resolution of \({\mathcal O}(-n-1)\). For a locally free sheaf \(F\) on \({\mathbb{P}}^n\), a subcomplex \(F^{\bullet}\) of \(J^{\bullet} \otimes F(n+1)\) is constructed which gives rise to the cohomology modules of \(F\). As an application, for \(F= \Omega^p(m)\), a basis of \(H^i(F)\) is constructed explicitly for each \(i\). Bott's formula is proved by counting the cardinality of these bases.