On the asymptotic linearity of Castelnuovo–Mumford regularity
By Ngô Viêt Trung, Hsin-Ju Wang. 2002
79 citations recorded by the source corpus.
Abstract Let R be a standard graded ring over a commutative Noetherian ring with unity. Let I be an arbitrary graded ideal of R and M an arbitrary finitely generated graded R -module. We prove that there exist integers e and ρ M ( I ) such that reg ( I n M ) = ρ M ( I ) n + e for all large n . This generalizes earlier results in the case R is a polynomial ring. Note that standard techniques in the polynomial case cannot be used here.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/1ed760bbd04c4408d652aac26d3c1785b556f54c
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Citations in the corpus
- Powers of sums and their homological invariants (2016, background): “In another direction, by results due to Cutkosky, Herzog, Kodiyalam, N.V. Trung and Wang [16, 38, 49], the Castelnuovo–Mumford regularity reg Is is a linear function of s for s ≫ 0.”
- Rational normal scrolls and the defining equations of Rees algebras (2008, unknown): “regðF1;F2Þ 1⁄4 sðn þ sÞ þ n þ t 1 1⁄4 sd ðs 1Þtþ n 1: r It is shown in [10], [19], [25] that the regularity of the s power of any homogeneous ideal is a linear function of s for all sg 0.”
- Koszul Algebras and Regularity (2012, background): “egree 1, [CH] where M,N are S-modules andI,Ii are ideals of S. What happens if we replace in 5.1 the polynomial ring S with a Koszul algebra R and consider regularity over R? Trung and Wang proved in [TW] that regS(IuM)is asymptotically a linear function in u when I is an ideal of R and M is a R-mod”
- Powers of Ideals and Fibers of Morphisms (2008, background): “astelnuovo-Mumford regularity of It or the corresponding ideal sheaf (see §1). Thus Theorem 0.1 clarifies the following beautiful result of CutkoskyHerzog-Trung [1999], Kodiyalam[2000], and Trung-Wang [2005], at least in a special case. Theorem 0.2. If I is a homogeneous ideal in the polynomial ring ”
- Depth and regularity of powers of sums of ideals (2015, background): “For instance, see [14, 15, 17, 20, 22, 23, 26] for studies on depth of powers of ideals, and see [1, 3, 6, 7, 8, 11, 10, 13, 19, 21, 24] for studies on regularity of powers of ideals.”