Maps between non-commutative spaces

Abstract:

Let $J$ be a graded ideal in a not necessarily commutative graded $k$-algebra $A=A_0 \oplus A_1 \oplus \cdots$ in which $\dim _k A_i < \infty$ for all $i$. We show that the map $A \to A/J$ induces a closed immersion $i:\operatorname {Proj}_{nc} A/J \to \operatorname {Proj}_{nc}A$ between the non-commutative projective spaces with homogeneous coordinate rings $A$ and $A/J$. We also examine two other kinds of maps between non-commutative spaces. First, a homomorphism $\phi :A \to B$ between not necessarily commutative $\mathbb {N}$-graded rings induces an affine map $\operatorname {Proj}_{nc} B \supset U \to \operatorname {Proj}_{nc} A$ from a non-empty open subspace $U \subset \operatorname {Proj}_{nc} B$. Second, if $A$ is a right noetherian connected graded algebra (not necessarily generated in degree one), and $A^{(n)}$ is a Veronese subalgebra of $A$, there is a map $\operatorname {Proj}_{nc} A \to \operatorname {Proj}_{nc} A^{(n)}$; we identify open subspaces on which this map is an isomorphism. Applying these general results when $A$ is (a quotient of) a weighted polynomial ring produces a non-commutative resolution of (a closed subscheme of) a weighted projective space.