Suppose $\mathcal E$ is a bundle over a smooth variety $X\subset \mathbb P^n$ such that the zero-locus $Z_s$ of a generic global section $s$ is a reduced zero-dimensional scheme. Then this bundle admits a resolution given by the Koszul complex $$ 0\to \bigwedge^r\mathcal E^\ast\xrightarrow{\phi_r}\bigwedge^{r-1}\mathcal E^\ast\xrightarrow{\phi_{r-1}}\dots\xrightarrow{\phi_1}\mathcal E^\ast\to I_{Z_s}\to 0. $$

I am interested if there are standard techniques to describe the induced map $(\phi_i)_*:H^{m}(\bigwedge^i \mathcal E^\ast)\to H^m(\bigwedge^{i-1} \mathcal E^\ast) $ from the explicit knowledge of the global sections, $H^m(\bigwedge^{i} \mathcal E^\ast), H^m(\bigwedge^{i-1} \mathcal E^\ast)$, and the Koszul maps.

Many thanks in advance.

Suppose $\mathcal E$ is a bundle over a rational homogeneous smooth variety $X\subset \mathbb P^n$ such that the zero-locus $Z_s$ of a generic global section $s$ is a reduced zero-dimensional scheme. Then this bundle admits a resolution given by the Koszul complex $$ 0\to \bigwedge^r\mathcal E^\ast\xrightarrow{\phi_r}\bigwedge^{r-1}\mathcal E^\ast\xrightarrow{\phi_{r-1}}\dots\xrightarrow{\phi_1}\mathcal E^\ast\to I_{Z_s}\to 0. $$

I am interested if there are standard techniques to describe the induced map $(\phi_i)_*:H^{m}(\bigwedge^i \mathcal E^\ast)\to H^m(\bigwedge^{i-1} \mathcal E^\ast) $ from the explicit knowledge of the global sections, $H^m(\bigwedge^{i} \mathcal E^\ast), H^m(\bigwedge^{i-1} \mathcal E^\ast)$, and the Koszul maps.

Many thanks in advance.