The asymptotic behavior of Frobenius direct images of rings of invariants

2015/09/09 by Mitsuyasu Hashimoto, Hashimoto, Mitsuyasu, Peter Symonds +1
Mathematics · #13A35 #Commutative Algebra (math.AC) #FOS: Mathematics #Primary 13A50 #Representation Theory (math.RT) #math.AC #math.RT #msc:13A35 #msc:13A50

paper · pdf · doi:10.48550/arxiv.1509.02592

25 pages

arxiv created 2015/09/09 · arxiv updated 2015/09/10

Abstract

We define the Frobenius limit of a module over a ring of prime characteristic to be the limit of the normalized Frobenius direct images in a certain Grothendieck group. When a finite group acts on a polynomial ring, we calculate this limit for all the modules over the twisted group algebra that are free over the polynomial ring; we also calculate the Frobenius limit for the restriction of these to the ring of invariants. As an application, we generalize the description of the generalized F-signature of a ring of invariants by the second author and Nakajima to the modular case.

Related

• F-rationality of the ring of modular invariants
• On the limit of Frobenius in the Grothendieck group
• Generalized F-signatures of the rings of invariants of finite group schemes
• Densities of eigenspaces of Frobenius and distributions of R-modules
• Frobenius Betti numbers and modules of finite projective dimension