a key role in this paper; let us recall here briefly a projective resolution of a right $A$-module.

Let $p_i:G\to G_i$ be the projection, and $\iota_i:G_i\hookrightarrow G$ the inclusion. Set $e_i=\iota_i\circ p_i$ in $A$. Denote $e_iA$ by $P_i$. Then $P_i\cong \operatorname{Hom}(G,G_i)$ as right $A$-modules, and $A=\bigoplus_{i=0}^m P_i$; $P_i$ is projective and $P_i\otimes_AG\cong G_i$.

For a finitely generated right $A$-module $V$, a right $A$-submodule $V^{\leqq i}$ of $V$ is defined by the formula $V^{\leqq i}=\bigoplus_{j\leqq i}Ve_j$. We have a natural isomorphism $V^{\leqq i}\cong V\otimes_A A^{\leqq i}$, and associated to every module $V$ is an ascending filtration $$0=V^{\leqq -1}\subset V^{\leqq 0}\subset V^{\leqq 1}\subset \dots\subset V^{\leqq m}=V$$ by right $A$-submodules. Set $\operatorname{Gr}^iV=V^{\leqq i}/V^{\leqq i-1}$; $\operatorname{Gr}^iV$ is a right $A$-module. Denote by $V^i$ the $K$-vector subspace $Ve_i$ of $V$. Note that $V^i$ is not a $A$-submodule of $V$, but we have an isomorphism $\operatorname{Gr}^iV\cong V^i$ of $K$-vector spaces. For example, we have $$P_k^{\leqq i}\cong \bigoplus_{j\leqq i}\operatorname{Hom}(G_j, G_k)
\textrm{ and }P_k^j\cong \operatorname{Hom}(G_j, G_k).$$ Note in particular that $P_k^k\cong K$ and that $P_k^j=0$ if $j>k$. For a homomorphism $\varphi:V\to W$ of right $A$-modules, we denote by $\varphi^i$ the induced homomorphism $V^i\to W^i$ of $K$-vector spaces.

We have a natural right $A$-linear map $$\varphi_{i,V}:V^i\otimes_KP_i\to V$$ sending $v\otimes 