2004/12/01 by Holger Brenner, Brenner, Holger, Mordechai Katzman +1
Computer Science · Engineering · Mathematics · #11A41 #13A35 #14H60 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #Computational Geometry and Mesh Generation #FOS: Mathematics #math.AC #math.AG #msc:11A41 #msc:13A35 #msc:14H60
paper · pdf · doi:10.48550/arxiv.math/0412033
openalex publication_date 2004/12/01 · arxiv created 2004/12/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We provide a negative answer to an old question in tight closure theory by showing that the containment x3y3 ∈ (x4,y4,z4)^* in K[x,y,z]/(x7+y7-z7) holds for infinitely many but not for almost all prime characteristics of the field K. This proves that tight closure exhibits a strong dependence on the arithmetic of the prime characteristic. The ideal (x,y,z) ⊂ K[x,y,z,u,v,w]/(x7+y7-z7, ux4+vy4+wz4+x3y3) has then the property that the cohomological dimension fluctuates arithmetically between 0 and 1.