Quillen-Suslin theory for the special linear group.

We prove that for any valuation ring <italic>R</italic> of Krull dimension ≤1 or any commutative local ring <italic>R</italic> with Krull dimension 0 and <italic>n</italic>≥3, the special linear group <italic>S</italic><italic>L</italic>_{<italic>n</italic>}(<italic>R</italic>[<italic>x</italic>]) coincides with the group of elementary matrices <italic>E</italic>_{<italic>n</italic>}(<italic>R</italic>[<italic>x</italic>]), and for an arbitrary arithmetical ring <italic>R</italic> of Krull dimension ≤1 and <italic>n</italic>≥3, the group <inline-graphic></inline-graphic>. These give partial solutions to two conjectures of Yengui. [ABSTRACT FROM AUTHOR]