  The goal of this paper is to establish a general rigidity statement for
abstract representations of elementary subgroups of Chevalley groups of rank at
least 2 over a class of commutative rings that includes the localizations of
1-generated rings and the coordinate rings of affine curves. This is achieved
by developing the approach introduced in our previous work, and in particular
by verifying condition (Z) over the class of rings at hand. Our main result
implies, for example, that any finite-dimensional representation of SL_n(Z[X])
(for n at least 3) over an algebraically closed field of characteristic 0 has a
standard description, yielding thereby the first unconditional rigidity
statement for finitely generated linear groups other than arithmetic
groups/lattices.
