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I am reading the chapter on Karoubi-Villamayor K-theory in Weibel's K-book. In particular he defines $\Omega R=(x^2-x)R[x]$ for a ring. This will eventually lead to a model for the loopspace

Corollary 11.13.1 For $n\geq 2$ we have, $$KV_n(R)\cong KV_{n-1}(\Omega R).$$

The proof uses the long exact sequence of the GL-fibration $$(x^2-x)R[x]\to xR[x]\to R$$

and uses the following fact to show that $KV_n(xR[x])$ vanishes:

Exercise 11.5 Let $R=R_0\oplus R_1\oplus\cdots$ be a graded ring. Then for every homotopy invariant functor $F$ on rings, possibly without unit, (i.e. the map $R\to R[x]$ induces homotopy equivalences; true for $KV$) then we have $F(R)\simeq F(R_0)$.

Now the question: Can't we just apply the exercise to $\Omega R$ as well? Or is the grading different? I am really confused here.

Edit: If anybody is put off by the word Karoubi-Villamayor, just assume $R$ regular, as we then have $KV=K$.

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# What is the grading of x(x−1)R[x]? Loopspace for Karoubi-Villamayor K-theory.

I am reading the chapter on Karoubi-Villamayor K-theory in Weibel's K-book. In particular he defines $\Omega R=(x^2-x)R[x]$ for a ring. This will eventually lead to a model for the loopspace

Corollary 11.13.1 For $n\geq 2$ we have, $$KV_n(R)\cong KV_{n-1}(\Omega R).$$

The proof uses the long exact sequence of the GL-fibration $$(x^2-x)R[x]\to xR[x]\to R$$

and uses the following fact to show that $KV_n(xR[x])$ vanishes:

Exercise 11.5 Let $R=R_0\oplus R_1\oplus\cdots$ be a graded ring. Then for every homotopy invariant functor $F$ on rings, possibly without unit, (i.e. the map $R\to R[x]$ induces homotopy equivalences; true for $KV$) then we have $F(R)\simeq F(R_0)$.

Now the question: Can't we just apply the exercise to $\Omega R$ as well? Or is the grading different? I am really confused here.