Symbolic power containments in singular rings in positive characteristic

2019/11/14 by Grifo, Eloísa, Ma, Linquan, Schwede, Karl
#13A35 #13D99 #14B05 #14H20 #14M05 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1911.06307

Abstract

The containment problem for symbolic and ordinary powers of ideals asks for what values of a and b we have I^(a) ⊆ I^b. Over a regular ring, a result by Ein-Lazarsfeld-Smith, Hochster-Huneke, and Ma-Schwede partially answers this question, but the containments it provides are not always best possible. In particular, a tighter containment conjectured by Harbourne has been shown to hold for interesting classes of ideals - although it does not hold in general. In this paper, we develop a Fedder (respectively, Glassbrenner) type criterion for F-purity (respectively, strong F-regularity) for ideals of finite projective dimension over F-finite Gorenstein rings and use our criteria to extend the prime characteristic results of Grifo-Huneke to singular ambient rings. For ideals of infinite projective dimension, we prove that a variation of the containment still holds, in the spirit of work by Hochster-Huneke and Takagi.

Related

• Symbolic powers of ideals defining F-pure and strongly F-regular rings
• Comparing powers and symbolic powers of ideals
• Strong F-regularity and the Uniform Symbolic Topology Property
• Diagonal F-splitting and Symbolic Powers of Ideals
• The resurgence of ideals of points and the containment problem