# Stability theorems in algebraic K-theory

Assertions on the invariance of the groups $K _{i} (R)$ or their subgroups, given certain special extensions of the ground ring $R$( see Algebraic $K$- theory).
The following are the best-known stability theorems. Let $R$ be a regular ring (cf. Regular ring (in commutative algebra)) and let $R[t _{1} \dots t _{n} ]$ be the ring of polynomials in the variables $t _{1} \dots t _{n}$ over $R$. The stability theorem for Whitehead groups under the transfer from $R$ to $R[t _{1} \dots t _{n} ]$, , states that the natural homomorphism imbedding $R$ in $R[t _{1} \dots t _{n} ]$ induces an isomorphism between $K _{1} (R)$ and $K _{1} (R[t _{1} \dots t _{n} ])$( cf. also Whitehead group).
In the case of a skew-field $R$ that is finite-dimensional over its centre $Z(R)$, one can define a reduced-norm homomorphism $\mathop{\rm Nrd}\nolimits _{R} : \ R ^ \star \rightarrow Z(R) ^ \star$ of the multiplicative group $R ^ \star$ of $R$ into the multiplicative group $Z(R) ^ \star$ of its centre. The kernel of this homomorphism, usually written as $\mathop{\rm SL}\nolimits (1,\ R)$, determines the reduced Whitehead group $SK _{1} (R)$ of $R$: $$SK _{1} (R) \simeq { \mathop{\rm SL}\nolimits (1,\ R)} / {[R ^ \star ,\ R ^ \star ]}$$( see Special linear group), which is a subgroup in $K _{1} (R)$. If $Z(R)(t _{1} \dots t _{n} )$ is the field of rational functions in $t _{1} \dots t _{n}$ over $Z(R)$, then the algebra $$R(t _{1} \dots t _{n} ) = R \otimes _{Z(R)} Z(R)(t _{1} \dots t _{n} )$$ is a skew-field, and the natural imbedding $\phi _ {t _{1} \dots t _ n}$ of $R$ in $R(t _{1} \dots t _{n} )$ induces a homomorphism $$\psi _ {t _{1} \dots t _ n} ^ \prime : \ SK _{1} (R) \rightarrow SK _{1} (R(t _{1} \dots t _{n} )).$$ The stability theorem for reduced Whitehead groups states that the homomorphism $\psi _ {t _{1} \dots t _ n} ^ \prime$ is bijective (, see also ). Similar statements are also true in unitary and spinor algebraic $K$- theories , .
Theorems on stabilization for $K _{i}$- functors under the transfer from the stable objects $K _{i} (R)$ to unstable ones are also called stability theorems (see ).