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As $AG$ is noetherian of dimension 1*, finitely-generated projective $AG$-modules are classified by their rank and determinant, and so $$K_0(AG) = \mathbb{Z} \oplus Pic(AG).$$ Proposition 10.3.8 of to …

The algebraic K-groups of Z are to the homology of SLn(Z) as the Hermitian K-groups of Z are to the homology of Sp2g(Z). There is a paper by Berrick and Karoubi here in which they discuss, and make so …

Warning: I know nothing about this subject, but found the question interesting so decided to learn something about it. Approach the following with caution. Consider the ring of integers $\mathbb{Z}[\ …

I understand that this is because GLn(F1) is supposed to be Sigman, the symmetric group on n letters. Thus K(F1) = K(finite sets) which is the sphere spectrum by the Barratt-Priddy-Quillen-Segal theor …

Let $G_n := W(\tilde{A}_{n-1})$. If I understand your description correctly, there is an extension $$1 \to G_n \to S_{n} \wr \mathbb{Z} \overset{sum}\to \mathbb{Z} \to 1$$ and so a $\mathbb{Z}$-Galois …

I don't think that you can deduce homological stability of the coinvariants from the Serre spectral sequence as you suggest. But this precise situation was studied in my paper "An upper bound for the …

Suppose first that $F$ is a finite field of characteristic $p$. Then $U_n(F)$ is a Sylow $p$-subgroup of $GL_n(F)$, and so using the transfer in group homology one sees that the image of $f_k$ (for $k …

I think I have been able to reproduce the "argument by Wagoner" (perhaps it was removed from the published version?). It certainly holds in more generality that what I have written below, using the no …

This is explained in my paper "Group-Completion", local coefficient systems, and perfection, Q. J. Math. (Quillen Memorial Issue) 64 (3) (2013) 795-803. A sufficient condition is for the monoid to be …