Maps between non-commutative spaces

Let J be a graded ideal in a not necessarily commutative graded k-algebra A = A 0 ○+ A 1 ○+... in which dim k A i < ∞ for all i. We show that the map A → A/J induces a closed immersion i: Proj nc A/J → 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 Φ: A → B between not necessarily commutative N-graded rings induces an affine map Proj nc B ⊃ U → Proj nc A from a non-empty open subspace U C 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 Proj nc A → 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.