Noncommutative projective schemes

For a commutative graded ring \(A\), Serre’s theorem recovers the projective scheme \(\operatorname{Proj} A\) from the category of coherent sheaves, which is the quotient of finitely generated graded \(A\)-modules by those of finite length. Artin–Zhang took this as a definition in the noncommutative setting.

Given a connected graded algebra \(A\), let \(\operatorname{gr} A\) be the category of finitely generated graded right \(A\)-modules and \(\operatorname{tors} A\) the full subcategory of finite-dimensional ones. The noncommutative projective scheme is the quotient category

together with the image \(\mathcal{O}\) of \(A\) and its shift. One should think of \(\operatorname{qgr} A\) as “the noncommutative variety \(\operatorname{Proj} A\)”: for a four-dimensional quadratic AS-regular algebra it is a noncommutative \(\mathbb{P}^3\).

Invariants of the algebra and of its qgr can differ — most visibly their Hochschild cohomologies, which measure deformations of the variety rather than of the coordinate ring.