### 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.

Original language | English |
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Pages (from-to) | 144-164 |

Number of pages | 21 |

Journal | Advances in Mathematics |

Volume | 305 |

DOIs | |

Publication status | Published - Jan 10 2017 |

### Keywords

- F-signature
- Frobenius direct image
- Frobenius limit
- Hilbert–Kunz multiplicity

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Hashimoto, M., & Symonds, P. (2017). The asymptotic behavior of Frobenius direct images of rings of invariants.

*Advances in Mathematics*,*305*, 144-164. https://doi.org/10.1016/j.aim.2016.09.020