Zoran Skoda semicommutative scheme

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

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