Let
X be a
projective curve over an
algebraically closed field k. A vector bundle on
X can be considered as a locally free
sheaf. Every semistable locally free
E on
X admits a Jordan-Hölder
filtration with stable
subquotients, i.e. : 0 = E_0 \subseteq E_1 \subseteq \ldots \subseteq E_n = E where E_i are locally free sheaves on
X and E_i/E_{i-1} are stable. Although the Jordan-Hölder filtration is not unique, the subquotients are, which means that gr E = \bigoplus_i E_i/E_{i-1} is unique up to isomorphism. Two semistable locally free sheaves
E and
F on
X are
S-equivalent if
gr E ≅
gr F.