Zoran Skoda
semicommutative scheme

It is often in important applications that a noncommutative algebra AA has a big center, e.g. that AA is of finite type over its center Z(A)Z(A).

Let SNAffSNAff be the category of pairs (Z,A)(Z,A) of a unital ring AA and a unital subring ZZ(A)Z\subset Z(A). One can equip the spectrum Spec(Z)Spec(Z) with a sheaf 𝒪 A\mathcal{O}_A of noncommutative algebras in the category of finite rank 𝒪 Z\mathcal{O}_Z-bimodules; as usual AA can be obtained by taking the global section functor. Gluing such spaces in Zariski topology one obtains the category SNSchSNSch of semicommutative schemes.

A prominent source of examples are quantum groups at root of unity, their homogeneous spaces etc.

For a generalization see semicommutative formal scheme.

Cf.

  • N. Reshetikhin, A. Voronov, A. Weinstein, Semiquantum geometry, Algebraic geometry, 5. J. Math. Sci. 82 (1996), no. 1, 3255-3267. q-alg/9606007.

Last revised on April 15, 2010 at 21:35:19. See the history of this page for a list of all contributions to it.