Maps between non-commutative spaces
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- by S. Paul Smith
- Trans. Amer. Math. Soc. 356 (2004), 2927-2944
- DOI: https://doi.org/10.1090/S0002-9947-03-03411-1
- Published electronically: November 18, 2003
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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.References
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Bibliographic Information
- S. Paul Smith
- Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
- MR Author ID: 190554
- Email: smith@math.washington.edu
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 356 (2004), 2927-2944
- MSC (2000): Primary 14A22; Secondary 16S38
- DOI: https://doi.org/10.1090/S0002-9947-03-03411-1
- MathSciNet review: 2052602