An analytic Koszul complex in a Banach space

We show that the holomorphic ideal sheaf of a linear section of a pseudoconvex open subset $\mathrm{\Omega }$ of, say, a Hilbert space $X={\mathrm{\ell }}_2$ is acyclic. We also prove an analog of Hefer’s lemma, i.e., if $f:\mathrm{\Omega }\times \mathrm{\Omega }\to \text{CC}$ is holomorphic and $f(x,x)=0$ for $x\in \mathrm{\Omega }$, then there is a holomorphic $g:\mathrm{\Omega }\times \mathrm{\Omega }\to X^{\ast }$ with values in the dual space $X^{\ast }$ of $X$ such that $f(x,y)=g(x,y)(x-y)$