Noncommutative projective geometry

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

## Contents

* 1 Examples
* 2 Proj construction
* 3 See also
* 4 References

## Examples

* The quantum plane, the most basic example, is the quotient ring of the free ring:

k⟨x,y⟩/(yx−qxy)

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

k⟨x1,…,xn⟩/(xixj−qijxjxi)

## 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

* Elliptic algebra
* Calabi–Yau algebra

## References

* Ajitabh, Kaushal (1994), Modules over regular algebras and quantum planes, http://dspace.mit.edu/bitstream/handle/1721.1/28088/31369741.pdf?sequence=1
* Artin M.: Geometry of quantum planes, Contemporary Mathematicsv. 124 (1992).
* Rogalski, D (2014). "An introduction to Noncommutative Projective Geometry". arXiv:1403.3065 [math.RA].

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