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Given a finite group $G$ and a commutative ring $R$, define the Swan $K$-theory $K_0(G, R)$ to be the Grothendieck group of the category finitely generated projective $R$-modules with $G$-action (with respect to all short exact sequences). When $R = \mathbb{C}$, this recovers the classical representation ring. When $R = \mathbb{F}_p$ and $G$ is a $p$-group, one gets $\mathbb{Z}$ for the Swan $K$-theory (everything is built up via short exact sequences from the trivial representation).

Are there computations in the literature when $R$ is non-regular in modular characteristic? I am interested in the example $G = C_2, R = \mathbb{Z}/4$ for starters. It may be difficult to classify representations of $C_2$ on free $\mathbb{Z}/4$-modules but I am hoping it is a little easier to understand the Swan $K$-theory.

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There seems to be a classification of representations for the example you mention (and more generally for representations of $C_p$ over $\mathbb{Z}/p^s\mathbb{Z}$) in

V. S. Drobotenko, E. S. Drobotenko, Z. P. Zhilinskaya and E. V. Pogorilyak, Representations of cyclic groups of prime order $p$ over rings of residue classes mod $p^s$, Ukrain. Mat. Z. 17 (1965), 28-42. MR0188304.

I've not been able to find a copy of the paper, which is in Russian, but if I understand correctly the review in Math Reviews, it seems as though the indecomposable representations, up to isomorphism, are as follows:

  1. For each monic irreducible polynomial $\varphi(x)$ over $\mathbb{Z}/2\mathbb{Z}$, and each $e\geq1$, a representation sending the generator of $C_2$ to $I+2X$, where $X$ is a companion matrix of $\varphi(x)^e$.
  2. For each monic irreducible polynomial $\varphi(x)$ over $\mathbb{Z}/2\mathbb{Z}$, and each $e\geq1$, a representation sending the generator of $C_2$ to $-I+2X$, where $X$ is a companion matrix of $\varphi(x)^e$.
  3. The regular representation.

These are all non-isomorphic except that the rank one representations given by (1) and (2) are the same.

The representations that are irreducible (in the sense of having no $\mathbb{Z}/4\mathbb{Z}$-free subrepresentations) are those that occur in (1) and (2) for $e=1$. Perhaps $K_0(G,R)$ will be $\mathbb{Z}$-free on the classes of these?

Edit: I don't think the statement of the classification can be quite right, as it seems to me that the representations given in (1) are isomorphic to those given in (2):

Let $X$ be the companion matrix, over $\mathbb{Z}/2\mathbb{Z}$, of a polynomial $p(t)$ and let $Y$ be the companion matrix of $p(t+1)$. Then $I+X$ has minimal and characteristic polynomial $p(t+1)$ and so is similar to $Y$. So $(I+X)A=AY$ for some invertible matrix $A$.

Lift $A$ to a matrix $\tilde{A}$ over $\mathbb{Z}/4\mathbb{Z}$. Then $\tilde{A}$ is automatically invertible, and $(I+2X)\tilde{A}=\tilde{A}(-I+2Y)$, so $\tilde{A}$ gives an isomorphism between the representation where a generator of $C_2$ acts as $I+2X$ and the one where it acts as $-I+2Y$.

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  • $\begingroup$ Thank you for your answer and reference! However, I don't quite follow the claim you made that $K_0(G, R)$ should be free on the classes of the irreducible modules. Does the analog of the Jordan-Holder theorem work here? $\endgroup$ May 6 '16 at 18:31
  • $\begingroup$ @AkhilMathew Sorry, that was meant to be more a speculation than a concrete claim. At the time I thought it was probably true, but I'm less sure now. I've edited to make that clearer. Also, I think the statement of the classification must be slightly wrong (see my edit). $\endgroup$ May 7 '16 at 8:55

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