A linear bound for Frobenius powers and an inclusion bound for tight closure

2004/05/24 by Holger Brenner, Brenner, Holger
Mathematics · #13A35 #13D02 #14J60 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #math.AC #math.AG #msc:13A35 #msc:13D02 #msc:14J60

paper · pdf · doi:10.48550/arxiv.math/0405463

Some minor changes and some examples added; to appear in Mich. Math. Journal

openalex publication_date 2004/05/24 · arxiv created 2005/01/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let I denote an R₊ -primary homogeneous ideal in a normal standard-graded Cohen-Macaulay domain over a field of positive characteristic p. We give a linear degree bound for the Frobenius powers I^[q] of I, q=p^e, in terms of the minimal slope of the top-dimensional syzygy bundle on the projective variety Proj R. This provides an inclusion bound for tight closure. In the same manner we give a linear bound for the Castelnuovo-Mumford regularity of the Frobenius powers I^[q].

Related

• A note on the growth of regularity with respect to Frobenius
• Computing the tight closure in dimension two
• Growth of primary decompositions of Frobenius powers of ideals
• Betti numbers of the Frobenius powers of the maximal ideal over a\n general hypersurface
• Uniform bounds on projective dimension and Castelnuovo-Mumford regularity