Non-commutative Castelnuovo-Mumford regularity (1999)

Author(s): Jørgensen P

    Abstract: We define Castelnuovo–Mumford regularity for graded modules over non-commutative graded algebras. Two fundamental commutative results are generalized to the non-commmutative case: a vanishing-theorem by Mumford, and a theorem on linear resolutions and syzygies by Eisenbud and Goto. The generalizations deal with sufficiently well-behaved algebras (i.e. so-called quantum polynomial algebras). We go on to define Castelnuovo–Mumford regularity for sheaves on a non-commutative projective scheme, as defined by Artin. Again, a version of Mumford's vanishing-theorem is proved, and we use it to generalize a result by Martin, Migliore and Nollet, on degrees of generators of graded saturated ideals in polynomial algebras, to quantum polynomial algebras. Finally, we generalize a practical result of Schenzel which determines the regularity of a module in terms of certain Tor-modules.

      • Date: 01-01-1999
      • Journal: Mathematical Proceedings of the Cambridge Philosophical Society
      • Volume: 125
      • Issue: 2
      • Pages: 203-221
      • Publisher: Cambridge Philosophical Society
      • Publication type: Article
      • Bibliographic status: Published

      Professor Peter Jorgensen
      Professor of Mathematics