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Corrigendum to ``Maps between non-commutative spaces''


Author: S. Paul Smith
Journal: Trans. Amer. Math. Soc. 368 (2016), 8295-8302
MSC (2010): Primary 14A22
DOI: https://doi.org/10.1090/tran/6908
Published electronically: February 29, 2016
MathSciNet review: 3546801
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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 $ \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 {\sf QGr}(A)$ is equivalent to $ {\sf Qcoh}(X)$, then $ \operatorname {\sf QGr}(A/J)$ is equivalent to $ {\sf Qcoh}(Z)$ for some closed subscheme $ Z \subseteq X$.


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Additional Information

S. Paul Smith
Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195
Email: smith@math.washington.edu

DOI: https://doi.org/10.1090/tran/6908
Keywords: Closed subspaces, non-commutative algebraic geometry
Received by editor(s): July 4, 2015
Received by editor(s) in revised form: January 22, 2016
Published electronically: February 29, 2016
Article copyright: © Copyright 2016 American Mathematical Society