Let B and Q be algebras over a commutative ring R and \(\pi\) : \(B\to Q\) a morphism of algebras. Given a right Q-module U and a left (right) B- module V, there is a change of rings spectral sequence \[ E^ 2=Tor^ Q_*(U,Tor^ B_*(Q,V))\quad \Rightarrow \quad Tor^ B_*(U,V), \]
\[ (E_ 2=Ext^*_ Q(U,Ext^*_ B(Q,V))\quad \Rightarrow \quad Ext^*_ B(U,V),\quad respectively). \] It is known that under additional hypotheses the differentials \(d^ 2\) \((d_ 2\), resp.) can be described as cup (cap) products with certain characteristic classes. The author develops a theory of characteristic classes for the change of rings spectral sequence without any additional hypotheses. More specifically, he shows that there are classes \[ V^ q_ 2\in E=Ext^ 2_ Q(Ext^ q_ B(Q,V),Ext_ B^{q-1}(Q,V)), \]
\[ (W^ q_ 2\in T=Ext^ 2_ Q(Tor^ B_ q(QV),Tor^ B_{q+1}(Q,V)),\quad resp.), \] such that up to a sign \(d_ 2=\mu (-\otimes V^ q_ 2): E_ 2^{p,q}\to E_ 2^{p+2,q-1},\) where \(\mu\) : \(E_ 2^{p,q}\otimes E\to E_ 2^{p+2,q-1}\) is a pairing \((d^ 2=\nu (W^ q_ 2\otimes - ): E^ 2_{p,q}\to E^ 2_{p-2,q+1},\) where \(\nu\) : \(T\otimes E^ 2_{p,q}\to E^ 2_{p-2,q+1}\) is a pairing, resp.).