Closedness of coboundary modules of analytic sheaves

Abstract:

Suppose $A$ is a subvariety of a complex space $X$ and $\mathcal {J}$ is a coherent analytic sheaf on $X$. It is shown that, if the analytic sheaf $\mathcal {H}_{A}^{\nu }(\mathcal {J})$ of local cohomology is coherent for $0 \leqq \nu \leqq q$, then for $0 \leqq \nu \leqq q$ the local cohomology group $\mathcal {H}_{A}^{\nu }(X, \mathcal {J})$ with its natural topology is Hausdorff and hence is a Fréchet space.