Maps between Non-commutative Spaces

Let J be a graded ideal in a not necessarily commutative graded k-algebra A = A0⊕A1⊕· · · in which dimk Ai <∞ for all i. We show that the map A→ A/J induces a closed immersion i : Projnc A/J → Projnc 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 Projnc B ⊃ U → Projnc A from a non-empty open subspace U ⊂ Projnc 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 Projnc A → Projnc 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.