Diagonalizing the Frobenius

2007/05/09 by Esben Bistrup Halvorsen, Halvorsen, Esben Bistrup
Mathematics · #13A35 #13D22 #13H15 #14F17 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AC #msc:13A35 #msc:13D22 #msc:13H15 #msc:14F17

paper · pdf · doi:10.48550/arxiv.0705.1251

Revised (simplified) version. 12 pages

openalex publication_date 2007/05/09 · arxiv created 2007/09/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Over a Noetherian, local ring R of prime characteristic p, the Frobenius functor F induces a diagonalizable map on certain quotients of rational Grothendieck groups. This leads to an explicit formula for the Dutta multiplicity, and it is shown that a weaker version of Serre's vanishing conjecture holds if only chi(F(X)) = p^{dim R}chi(X) for all bounded complexes X of finitely generated, projective modules with finite length homology.

Related

• On the limit of Frobenius in the Grothendieck group
• Frobenius and Homological Dimensions of Complexes
• Frobenius Betti numbers and modules of finite projective dimension
• The Acyclicity of the Frobenius Functor for Modules of Finite Flat Dimension
• The Frobenius Complexity of a Local Ring of Prime Characteristic