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

TL;DR

This corrigendum corrects earlier proofs regarding non-commutative projective schemes and establishes a new geometric result linking algebraic quotients to closed subschemes in the commutative setting.

Contribution

It provides corrected proofs of key lemmas and theorems, and introduces a new result connecting non-commutative algebraic geometry with classical schemes.

Findings

Corrected proof of Theorem 3.2.

New result relating non-commutative schemes to closed subschemes.

Establishment of conditions under which non-commutative projective schemes correspond to classical subschemes.

Abstract

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 $\mathbb{N}$-graded $k$-algebra such that ${\rm dim}_k(A_n)< \infty$ for all $n$, and $J$ a graded two-sided ideal of $A$. If the non-commutative scheme ${\sf Proj}_{nc}(A)$ is isomorphic to a projective scheme $X$, then there is a closed subscheme $Z \subseteq X$ such that ${\sf Proj}_{nc}(A/J)$ is isomorphic to $Z$. This result is a geometric translation of what we actually prove: if the category ${\sf QGr}(A)$ is equivalent to ${\sf Qcoh}(X)$, then ${\sf QGr}(A/J)$ is equivalent to ${\rm Qcoh}(Z)$ for some closed subscheme $Z \subseteq X$.