Dzyaloshinsky-Moriya antisymmetric exchange coupling in cuprates: Oxygen   effects
  We revisit a problem of Dzyaloshinsky-Moriya antisymmetric exchange coupling for a single bond in cuprates specifying the local spin-orbital contributions to Dzyaloshinsky vector focusing on the oxygen term. The Dzyaloshinsky vector and respective weak ferromagnetic moment is shown to be a superposition of comparable and, sometimes, competing local Cu and O contributions. The intermediate oxygen $^{17}$O Knight shift is shown to be an effective tool to inspect the effects of Dzyaloshinsky-Moriya coupling in an external magnetic field. We predict the effect of $strong$ oxygen weak antiferromagnetism in edge-shared CuO$_2$ chains due to uncompensated oxygen Dzyaloshinsky vectors. Finally, we revisit the effects of symmetric spin anisotropy, in particular, those directly induced by Dzyaloshinsky-Moriya coupling. 
A. S. Moskvin