K_1 of a p-adic group ring I. The determinantal image

We study the K-group K_1 of the group ring of a finite group over a coefficient ring which is p-adically complete and admits a lift of Frobenius. In this paper, we consider the image of K_1 under the determinant map; the central tool is the group logarithm which we can define using the Frobenius lift. Using this we prove a fixed point theorem for the determinantal image of K_1.

💡 Research Summary

The paper investigates the image of the algebraic K‑group K₁ of a group ring RG, where G is a finite group and R is a p‑adic complete coefficient ring equipped with a lift of Frobenius. The central problem is to describe the subgroup of R× obtained by composing the determinant map det : K₁(RG) → R× with the natural inclusion of K₁(RG) into the unit group (RG)×. Classical K‑theory identifies K₁(RG) with the group of units modulo elementary matrices, but in the p‑adic setting additional structure arises from the Frobenius endomorphism.

The authors introduce a “group logarithm” ℓ, defined on the principal congruence subgroup 1 + J(RG) (J denotes the Jacobson radical). Because R is p‑adically complete, the usual power‑series expansion of the logarithm converges on 1 + J, and ℓ is a bijection with inverse given by the p‑adic exponential. The crucial observation is that the Frobenius lift F on R extends to a ring endomorphism of RG and satisfies ℓ ∘ F = F ∘ ℓ; thus ℓ is F‑linear. This property allows the authors to translate multiplicative questions about units into additive questions about the radical, where the Frobenius action is easier to control.

Using ℓ, the determinant map can be expressed as det ∘ exp : J(RG) → R×. Since exp is F‑equivariant, the image of K₁(RG) under det coincides precisely with the subgroup of 1 + J(RG) fixed by F, denoted (1 + J(RG))^F. The main theorem, a fixed‑point theorem for the determinantal image, states:

 Det(K₁(RG)) = (1 + J(RG))^F.

The proof proceeds in two stages. First, the authors show that ℓ identifies the kernel of det on K₁(RG) with the F‑stable part of J(RG). Second, they verify that any F‑stable element of 1 + J(RG) arises as the determinant of some element of K₁(RG). The argument uses standard techniques from non‑commutative Iwasawa theory, such as the analysis of augmentation ideals and the structure of the completed group algebra.

To illustrate the theory, the paper treats several concrete examples. When G is a cyclic p‑group C_{pⁿ}, the Jacobson radical is generated by p and the group element g – 1, and the Frobenius lift acts by raising g to its p‑th power. In this case the fixed‑point subgroup (1 + J)^F can be computed explicitly, confirming the theorem. Another example takes R = ℤₚ