Corrigendum to “Maps between non-commutative spaces”

Abstract:

The statement of Lemma 3.1 in Maps between non-commutative spaces (Trans. Amer. Math. Soc. 356 (2004), no. 7, 2927–2944) 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 $\mathbb {N}$-graded $k$-algebra such that $\operatorname {dim}_k(A_n)< \infty$ for all $n$, and $J$ a graded two-sided ideal of $A$. If the non-commutative scheme $\operatorname {Proj}_{nc}(A)$ is isomorphic to a projective scheme $X$, then there is a closed subscheme $Z \subseteq X$ such that $\operatorname {Proj}_{nc}(A/J)$ is isomorphic to $Z$. This result is a geometric translation of what we actually prove: if the category $\operatorname {QGr}(A)$ is equivalent to $\operatorname {Qcoh}(X)$, then $\operatorname {QGr}(A/J)$ is equivalent to $\textsf {Qcoh}(Z)$ for some closed subscheme $Z \subseteq X$.