Corrigendum to "Maps between non-commutative spaces" [Trans. Amer. Math. Soc., 356(7) (2004) 2927-2944]

Original · EN

The statement of Lemma 3.1 in the published paper is not correct. Lemma 3.1 is needed for the proof of Theorem 3.2. Theorem 3.2 as originally stated is true but its "proof" is not correct. Here we change the statements and proofs of Lemma 3.1 and Theorem 3.2. We also prove a new result. Let k be a field, A a left and right noetherian N-graded k-algebra such that dimₖ(Aₙ)< ∞ for all n, and J a graded two-sided ideal of A. If the non-commutative scheme Projnc(A) is isomorphic to a projective scheme X, then there is a closed subscheme Z X such that Projnc(A/J) is isomorphic to Z. This result is a geometric translation of what we actually prove: if the category QGr(A) is equivalent to Qcoh(X), then QGr(A/J) is equivalent to Qcoh(Z) for some closed subscheme Z X.