I show that a quasicompact morphism f : X → Y of schemes is an effective descent map for quasicoherent modules if and only if the map OY → f∗(OX) is injective, and remains injective after any base change. This generalizes Grothendieck’s result that faithfully flat quasicompact morphisms are effective descent maps.

We prove that every first-order formula that is invariant under quasi-injective bisimulations is equivalent to a formula of the hybrid logic H{↓}. Our proof uses a variation of the usual unravelling technique. We also briefly survey related results, and show in a standard way that it is undecidable whether a first-order formula is invariant under quasi-injective bisimulations.

In this paper we study some properties of GC -projective, injective and flat modules, where C is a semidualizing module and we discuss some connections between GC -projective, injective and flat modules , and we consider these properties under change of rings such that completions of rings, Morita equivalences and the localizations.

It is shown that certain natural maps between the ideal, Gromov, and end boundaries of a complete CAT(0) space can fail to be either injective or surjective. Additionally the natural map from the Gromov boundary to the end boundary of a complete CAT(−1) space can fail to be either injective or surjective.