Stabilization Methods

Abstract

In K-theory the “stabilization philosophy” that a problem posed in dimension n may become simpler if looked at in dimension n + m, is applied successfully in many places. For example for suitable rings R a projective R-module M which is not free may become free after stabilization, i.e. after replacing M by M ⊕ Rm. for suitable m, or a matrix A ∈ Gln(R) which is not a product of elementary matrices may become a product of such matrices after stabilization, i.e. after replacing A by $$\left( {\begin{array}{*{20}{c}}A&0 \\0&{{I_m}}\end{array}} \right) \in G{l_{n + m}}\left( R \right)$$, where Imdenotes the m × m identity matrix. The idea of using stabilization methods in the context of polynomial maps was first introduced by H. Bass, E. Connell and D. Wright in [29] and Yagzhev in [297]. Among other things they showed that it suffices to prove the Jacobian Conjecture for all polynomial maps of degree ≤ 3 in any dimension. For the precise formulation see 6.3.1. A little later this result was improved by Druikowski in [93] (see 6.3.2). Both results are now considered as classical. More recently J. Yu in his thesis, [299], pursues the stabilization ideas and shows that the Jacobian Conjecture is true for a special class of polynomial maps (see 6.3.4 and [301]). To explain the stabilization idea, we start in §1 with some instructive examples. Then in §2 we lay the foundations for the main results of this chapter which can be found in §3.