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Non-commutative Castelnuovo–Mumford regularity

Published online by Cambridge University Press:  01 January 1999

PETER JØRGENSEN
Affiliation:
Matematisk Afdeling, Københavns Universitet, Universitetsparken 5, DK-2100 KøbenhavnØ, Danmark; e-mail: popjoerg@math.ku.dk

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.

Type
Research Article
Copyright
Cambridge Philosophical Society 1999

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