In this paper we improve recent results dealing with cellular covers of R-modules. Cellular covers (sometimes called colocalizations) come up in the context of homotopical localization of topological spaces. They are related to idempotent cotriples, idempotent comonads or coreflectors in category theory. Recall that a homomorphism of R-modules π: G → H is called a cellular cover over H if π induces an isomorphism $\pi ⁎:Ho{m}_R(G,G)\cong Ho{m}_R(G,H)$, where π⁎(φ) = πφ for each $\phi \in Ho{m}_R(G,G)$ (where maps are acting on the left). On the one hand,...

Let $A$ be a finite subset of an abelian group $G$ and let $P$ be a closed half-plane of the complex plane, containing zero. We show that (unless $A$ possesses a special, explicitly indicated structure) there exists a non-trivial Fourier coefficient of the indicator function of $A$ which belongs to $P$. In other words, there exists a non-trivial character $\chi \in \widehat{G}$ such that ${\sum }_{a\in A}\chi \left(a\right)\in P$.

The discrete algebras $A$ over a commutative ring $R$ which can be realized as the full endomorphism algebra of a torsion-free $R$-module have been investigated by Dugas and Göbel under the additional set-theoretic axiom of constructibility, $V=L$. Many interesting results have been obtained for cotorsion-free algebras but the proofs involve rather elaborate calculations in linear algebra. Here these results are rederived in a more natural topological setting and substantial generalizations to topological...