Explicit description of transfer for $K_1$

Question

Let $R$ be a commutative regular ring and let $s \in R$ be an element such that $R / s$ is also regular. Then we have a long exact localization sequence $$ \ldots \rightarrow K_i(R/s) \rightarrow K_i(R) \rightarrow K_i(R_s) \rightarrow \ldots $$ for the algebraic K-groups of these rings. The first few $K$-groups have very explicit descriptions, specifically $K_1$ in terms of matrices: we have $K_1(R)=\mathrm{GL}(R) / \mathrm{E}(R)$, the quotient of the general linear group modulo the elementary matrices.

Is there an explicit description of the transfer homomorphism $K_1(R/s) \rightarrow K_1(R)$ in terms of matrices somewhere in the literature? In Bass' book on algebraic K-theory, the localization sequence is only stated starting at $K_1(R)$, so this suggests that the description is not completely obvious.

Answer 1

As explained by John Klein, for commutative regular rings any automorphism of a finitely generated module produces a $K_1$-class. Given an automorphism $(R/s)^n \to (R/s)^n$, one may view these as finitely generated $R$-modules and obtain a $K_1$-class.

Let's see how this works explicitly and try to obtain an actual element in $\rm{GL}(R)$ from an element in $\rm{GL}_n(R/s)$. The key is to produce a short exact sequence relating the automorphism of $(R/s)^n$ to automorphisms of finitely generated free $R$-modules.

Given an element of $\rm{GL}_n(R/s)$, we may lift it to an $n\times n$-matrix $U\in M_n(R)$. This is only invertible mod $s$, so we find matrices $A$ and $B$ with $$ AU + sB = I_n $$ where $I_n$ is the $n\times n$ identity matrix. Observe that this makes the block matrix $$ \left(\begin{array}{cc}A & B \\ -sI_n & U\end{array}\right) $$ invertible (one can write down the inverse explicitly, in terms of some $C$ satisfying $UA + sC = I_n$).

Now we have an exact sequence $$ 0 \to R^n\oplus (sR)^n \to R^n\oplus R^n \to (R/s)^n\to 0, $$ where we act by automorphisms $$ \left(\begin{array}{cc}A & sB \\ -I_n & U\end{array}\right), \left(\begin{array}{cc}A & B \\ -sI_n & U\end{array}\right), U $$ which tells you that in $K_1(R)$, we have a relation $$ \left[\begin{array}{cc}A & B \\ -sI_n & U\end{array}\right] = \left[\begin{array}{cc}A & sB \\ -I_n & U\end{array}\right] + [U], $$ where the right hand $[U]$ requires the more general description of $K_1=G_1$ in terms of automorphisms of finitely generated modules mentioned at the beginning, but the other terms also work in the usual description via $\rm{GL}(R)$. So the transfer of $U$ isgiven by $$ \left[\begin{array}{cc}A & B \\ -sI_n & U\end{array}\right] -\left[\begin{array}{cc}A & sB \\ -I_n & U\end{array}\right], $$ or represented by the single $2n\times 2n$-matrix $$\left(\begin{array}{cc}A & B \\ -sI_n & U\end{array}\right) \cdot \left(\begin{array}{cc}A & sB \\ -I_n & U\end{array}\right)^{-1}. $$

Answer 2

For regular Noetherian rings $R$, $K_1(R) \cong G_1(R)$, where $G_1(R)$ is the Grothendieck group of the category of finitely generated $R$-modules equipped with an automorphism. The transfer $$ G_1(R/s) \to G_1(R) $$ sends a pair $(M,f)$, consisting of a finitely generated $R/s$-module $M$ and an automorphism $f\: M \to M$, to $(M,f)$ in which $M$ is now considered as a finitely generated $R$-module.