Some adjoints in homotopy categories


Let $R$ be a ring. In a previous paper [11] we found a new description for the category $\mathbf{K}(R\text{-Proj})$; it is equivalent to the Verdier quotient $\mathbf{K}(R\text{-Flat})/{\mathscr S}$, for some suitable $\mathscr{S}\subset\mathbf{K}(R\text{-Flat})$. In this article we show that the quotient map from $\mathbf{K}(R\text{-Flat})$ to $\mathbf{K}(R\text{-Flat})/\mathscr{S}$ always has a right adjoint. This gives a new, fully faithful embedding of $\mathbf{K}(R\text{-Proj})$ into $\mathbf{K}(R\text{-Flat})$. Its virtue is that it generalizes to nonaffine schemes.