Noncommutative projective geometry

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:

k ⟨ x , y ⟩ / ( y x − q x y ) {\displaystyle k\langle x,y\rangle /(yx-qxy)}

* More generally, the quantum polynomial ring is the quotient ring:

k ⟨ x 1 , … , x n ⟩ / ( x i x j − q i j x j x i ) {\displaystyle k\langle x_{1},\dots ,x_{n}\rangle /(x_{i}x_{j}-q_{ij}x_{j}x_{i})}

## Proj construction

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See also: Proj construction

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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* Elliptic algebra
* Calabi–Yau algebra
* Sklyanin algebra

## 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 11440231-15&rft.date=1992&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=1144023#id-name=MR&rft.aulast=Artin&rft.aufirst=Michael&rfr_id=info:sid/en.wikipedia.org:Noncommutative+projective+geometry" class="Z3988">
* Rogalski, D (2014). "An introduction to Noncommutative Projective Geometry". arXiv:1403.3065 [math.RA].