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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Maps between non-commutative spaces

Author: S. Paul Smith
Journal: Trans. Amer. Math. Soc. 356 (2004), 2927-2944
MSC (2000): Primary 14A22; Secondary 16S38
Published electronically: November 18, 2003
MathSciNet review: 2052602
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Abstract: Let $J$ be a graded ideal in a not necessarily commutative graded $k$-algebra $A=A_0 \oplus A_1 \oplus \cdots$ in which $\dim_k A_i < \infty$ for all $i$. We show that the map $A \to A/J$ induces a closed immersion $i:\operatorname{Proj}_{nc} A/J \to \operatorname{Proj}_{nc}A$ 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 $\phi:A \to B$ between not necessarily commutative $\mathbb{N}$-graded rings induces an affine map $\operatorname{Proj}_{nc} B \supset U \to \operatorname{Proj}_{nc} A $from a non-empty open subspace $U \subset \operatorname{Proj}_{nc} B$. 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 $\operatorname{Proj}_{nc} A \to \operatorname{Proj}_{nc} A^{(n)}$; 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.

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Additional Information

S. Paul Smith
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195

Received by editor(s): September 18, 2002
Received by editor(s) in revised form: April 29, 2003
Published electronically: November 18, 2003
Additional Notes: The author was supported by NSF grant DMS-0070560
Article copyright: © Copyright 2003 American Mathematical Society