Inverse limit theorem and expansion theorem for cohomological dimension

Abstract

For a given ground ring L, let dimL(X) denote the sheaf theoretic cohomological dimension of a space X. In this paper we prove an inverse limit theorem for this dimension function. Then we apply this theorem to show that for a large classs of rings the Freudenthal's expansion theorem, expressing a compact metric space X of dimL(X) as the inverse limit of an system of compact polyhedra Kα of dimL(Kα), is not valid.