We show that if a class of modules is closed under pure quotients, then it is precovering if and only if it is covering, and this happens if and only if it is closed under direct sums. This is inspired by a dual result by Rada and Saorín. We also show that if a class of modules contains the ground ring and is closed under extensions, direct sums, pure submodules, and pure quotients, then it for...

We prove a generalization of the Flat Cover Conjecture by showing for any ring R that (1) each (right R-) module has a Ker Ext(−, C)-cover, for any class of pure-injective modules C, and that (2) each module has a Ker Tor(−,B)-cover, for any class of left R-modules B. For Dedekind domains, we describe Ker Ext(−, C) explicitly for any class of cotorsion modules C; in particular, we prove that (1...

Consider a finite morphism f : X → Y of smooth, projective varieties over a finite field F. Suppose X is the vanishing locus in PN of r forms of degree at most d. We show that there is a constant C depending only on (N, r, d) and deg( f ) such that if |F| > C, then f (F) : X(F) → Y(F) is injective if and only if it is surjective.

A bidouble cover is a nite at Galois morphism with Galois group (Z=2) 2 ?. The structure theorem for smooth Galois (Z=2) 2 ? covers was given in Cat2] pag. 491-493] where bidouble covers of P 1 P 1 were introduced in order to nd interesting properties of the moduli spaces of surfaces of general type. In this paper we develop general formulae for the case of resolutions of singular bidouble cove...

The main purpose of this paper is to establish general conditions under which T2-spaces are compact-covering images of metric spaces by using the concept of cfpcovers. We generalize a series of results on compact-covering open images and sequencecovering quotient images of metric spaces, and correct some mapping characterizations of g-metrizable spaces by compact-covering σ-maps and mssc-maps.