Let $X$ be the zero locus of $e\_1, \dots, e\_n$ sections of a vector bundle $\mathcal{E}$ of rank $r$ on $Y$. Assume that the codimension of $X$ is strictly less than $n$, then the Koszul complex associated to the sections is not exact. What can be said about the cohomologies of this complex in general? Are there nice cases in which, even though the complex is not exact, the cohomologies can be known?



Koszul resolution with wrong dimension
Assume $X$ is a locally complete intersection of codimension $m < n$. Then the natural morphism
$$
E^\vee|\_X \to I\_X \otimes O\_X = I\_X/I\_X^2 = N^\vee\_{X/Y}
$$
is surjective, let $F$ be its kernel (it is a vector bundle on $X$ of rank $n - m$). Then
$$
H\_i(Kosz(E,e)) \cong \wedge^iF.
$$


