Sheaves with Finitely Generated Isomorphic Stalks and Homology Manifolds

Abstract

The setting is sheaves of modules over a commutative ring $L$. It is shown that on completely metrizable spaces certain sheaves having mutually isomorphic finitely generated stalks are locally constant over a dense open subset. This is used to show that a locally compact metrizable space $X$ that is homologically locally connected with respect to a principal ideal domain $L$ is a homology manifold over $L$ provided it has finite cohomological dimension with respect to $L$ and, for any two points $x,y \in X$, the modules ${H_k}(X,X - \{ x\} ;L)$ and ${H_k}(X,X - \{ y\} ;L)$ are isomorphic and finitely generated.