noncommutative projective geometry

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

::$k\; \backslash langle\; x,\; y\; \backslash rangle\; /\; (yx\; -\; q\; xy)$

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

::$k\; \backslash langle\; x\_1,\; \backslash dots,\; x\_n\; \backslash rangle\; /\; (x\_i\; x\_j\; -\; q\_\{ij\}\; x\_j\; x\_i)$

{{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.

• Elliptic algebra
• Calabi–Yau algebra
• Sklyanin algebra

• {{citation |url=http://dspace.mit.edu/bitstream/handle/1721.1/28088/31369741.pdf?sequence=1 |title=Modules over regular algebras and quantum planes |first=Kaushal |last=Ajitabh |year=1994 |type=Ph.D. thesis}}
• {{citation | last1=Artin | first1=Michael | authorlink1=Michael Artin | title=Geometry of quantum planes | journal=Contemporary Mathematics| volume=124 | pages=1–15 | date=1992 | mr=1144023}}
• {{cite arXiv |first1=D |last1=Rogalski |title=An introduction to Noncommutative Projective Geometry |year=2014 |eprint=1403.3065|class=math.RA }}

Category:Fields of geometry