Unstable operations on K-theory for singular schemes

We study the algebraic structures, such as the lambda ring structure, that arise on K-theory seen as an object of some homotopy categories coming from model categories of simplicial presheaves. In particular, we show that if we take the Jardine local injective model category of simplicial presheaves over the category of divisorial, hence possibly singular, schemes with respect to the Zariski topology, these structures are in bijection with the ones we have on K0 seen as a presheaf of sets. This extends some results of Riou ([63]) from smooth schemes to singular ones and does not require A1-invariance. We also discuss similar results for symplectic K-theory.