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Noncommutative projective geometry

From Wikipedia, the free encyclopedia

In mathematics, noncommutative projective geometry is a noncommutative analog of projective geometry in the setting of noncommutative algebraic geometry.

Examples

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  • The quantum plane, the most basic example, is the quotient ring of the free ring:
  • More generally, the quantum polynomial ring is the quotient ring:

Proj construction

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By definition, the Proj of a graded ring R is the quotient category of the category of finitely generated graded modules over R by the subcategory of torsion modules. If R is a commutative Noetherian graded ring generated by degree-one elements, then the Proj of R in this sense is equivalent to the category of coherent sheaves on the usual Proj of R. Hence, the construction can be thought of as a generalization of the Proj construction for a commutative graded ring.

See also

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References

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  • Ajitabh, Kaushal (1994), Modules over regular algebras and quantum planes (PDF) (Ph.D. thesis)
  • Artin, Michael (1992), "Geometry of quantum planes", Contemporary Mathematics, 124: 1–15, MR 1144023
  • Rogalski, D (2014). "An introduction to Noncommutative Projective Geometry". arXiv:1403.3065 [math.RA].