A generalization of Castelnuovo–Mumford regularity for representations of noncommutative algebras

We introduce and generalize the notion of Castelnuovo–Mumford regularity for representations of noncommutative algebras, effectively establishing a measure of complexity for such objects. The Gröbner–Shirshov basis theory for modules over noncommutative algebras is developed, by which a noncommutative analogue of Schreyer's Theorem is proved for computing syzygies. By a repeated application of this theorem, we construct free resolutions for representations of noncommutative algebras. Some interesting examples are included in which graded free resolutions and regularities are computed for representations of various algebras. In particular, using the Bernstein–Gelfand–Gelfand resolutions for integrable highest weight modules over Kac–Moody algebras, we compute the projective dimensions and regularities explicitly for the cases of finite type and affine type .