Motivated by Smith's work \cite{ Smith2003, Smith2016} on maps between non-commu\-tative projective spaces of the form ${ \rm Proj}_{ nc} A$ in the setting of non-commutative projective geometry developed by Rosenberg and Van den Bergh, and the notion of schematicness introduced by Van Oystaeyen and Willaert \cite{ VanOystaeyenWillaert1995} to $\mathbb{ N}$-graded rings with the aim of formulating a non-commutative scheme theory \`a la Grothendieck \cite{ EGAII1961}, in this paper we consider a first approach to maps in the Smith's sense in the more general setting of non-commutative projective spaces over semi-graded rings defined by Lezama and Latorre \cite{ LezamaLatorre2017}. We extend Smith's key result \cite[Theorem 3.2]{ Smith2003}, \cite[Theorem 1.2]{ Smith2016} from the category of schematic $\mathbb{ N}$-graded rings to the category of schematic semi-graded rings.
Categories: math.AG math.QA