2003/05/15 by Diane Maclagan, Gregory G. Smith
Computer Science · Mathematics · #Abelian group #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Connection (principal bundle) #Finitely-generated abelian group #Ideal (ethics) #Polynomial #Polynomial and algebraic computation #Property (philosophy) #Sheaf #Variety (cybernetics) #math.AC #math.AG #msc:13D45 #msc:14M25 #msc:14Q20
paper · pdf · doi:10.1515/crll.2004.040
published as Journal fur die reine und angewandte Mathematik (Crelle's Journal) 571 (2004) 179-212 · 30 pages, 5 figures
arxiv created 2003/05/15 · openalex publication_date 2004/01/07 · arxiv updated 2010/03/15 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated by toric geometry, we work with modules over a polynomial ring graded by a finitely generated abelian group. As in the standard graded case, our definition of multigraded regularity involves the vanishing of graded components of local cohomology. We establish the key properties of regularity: its connection with the minimal generators of a module and its behavior in exact sequences. For an ideal sheaf on a simplicial toric variety X, we prove that its multigraded regularity bounds the equations that cut out the associated subvariety. We also provide a criterion for testing if an ample line bundle on X gives a projectively normal embedding.