Suslin's lemma for rings containing an infinite field

Samiha Monceur¹ ; Ihsen Yengui²

¹Department of Mathematics Faculty of Sciences University of Sfax 3000 Sfax, Tunisia
²Department of Mathematics Faculty of Sciences of Sfax University of Sfax 3000 Sfax, Tunisia

Colloquium Mathematicum, Tome 146 (2017) no. 1, pp. 111-122

Résumé

A well-known lemma of Suslin says that for a commutative ring ${\bf A}$, if $^{\rm t}(v_1,\ldots ,v_n) \in {\bf A}[X]^{n\times 1}$ is unimodular where $v_1$ is monic of degree $d$ and $n\geq 3$, then there exist $\gamma _1,\ldots ,\gamma _{\ell } \in {\rm E}_{n-1}({\bf A}[X])$ such that, denoting by $w_i$ the first coordinate of $\gamma _i {^{\rm t}(}v_2,\ldots ,v_n)$, we have $$\langle {{\rm Res}_X(v_1, w_1), \ldots , {\rm Res}_X (v_1, w_{\ell })}\rangle = {\bf A}.$$ This lemma played a central role in Suslin’s resolution of Serre’s conjecture. In case ${\bf A}$ contains a set $E = \{y_0,\ldots ,y_{(n-2)d}\}$ such that $y_i -y_j \in {\bf A}^{\times }$ for $i \not =j$, we prove that the $\gamma _i$’s can simply correspond to the elementary operations $L_1 \rightarrow L_1 + \sum _{j=2}^{n-1} y_i^{j-2} L_j$, $0 \leq i \leq (n -2) d$. These efficient elementary operations enable us to give a new and simple algorithm (for the Quillen–Suslin theorem) for reducing unimodular rows with entries in ${\bf K}[X_1,\ldots ,X_k]$ to $^{\rm t}(1,0,\ldots ,0)$, using elementary operations in case ${\bf K}$ is an infinite field. This work generalizes a previous paper by Lombardi and the second author which corresponds to the particular case $n = 3$.

Détail

DOI : 10.4064/cm5969-2-2015

Keywords: well known lemma suslin says commutative ring ldots times unimodular where monic degree geq there exist gamma ldots gamma ell n denoting first coordinate gamma ldots have langle res ldots res ell rangle lemma played central role suslin resolution serre conjecture contains set ldots n y times prove gamma simply correspond elementary operations rightarrow sum n j leq leq these efficient elementary operations enable simple algorithm quillen suslin theorem reducing unimodular rows entries ldots ldots using elementary operations infinite field work generalizes previous paper lombardi second author which corresponds particular