Let R be a right GF-closed ring with finite left and right Gorenstein global dimension. We prove that if I is an ideal of R such that R/I is a semi-simple ring, then the Gorensntein flat dimensnion of R/I as a right R-module and the Gorensntein injective dimensnnion of R/I as a left R-module are identical. In particular, we show that for a simple module S over a commutative Gorensntein ring R, ...

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.