Homomorphisms of infinitely generated analytic sheaves
Abstract
Description
We prove that every homomorphism $\mathcal{O}^E_ζ\to\mathcal{O}^F_ζ$, with $E$ and $F$ Banach spaces and $ζ\in\mathbb{C}^m$, is induced by a $\mathop{\mathrm{Hom}}(E,F)$-valued holomorphic germ, provided that $1\leq m<\infty$. A similar structure theorem is obtained for the homomorphisms of type $\mathcal{O}^E_ζ\to\mathcal{S}_ζ$, where $\mathcal{S}_ζ$ is a stalk of a coherent sheaf of positive $\mathfrak{m}_ζ$-depth. We later extend these results to sheaf homomorphisms, obtaining a condition on coherent sheaves which guarantees the sheaf to be equipped with a unique analytic structure in the sense of Lempert-Patyi.