a$ to $va$. Since every element $v$ of $V^i$ can be written as $v=v'e_i$ for some $v'\in V$, we see $\varphi_{i,V}(v\otimes e_i)=v$. Hence the induced $K$-linear $\varphi_{i,V}^i$ is an isomorphism: $$(V^i\otimes_KP_i)^i\cong V^i\otimes_KP_i^i\cong V^i.$$ All $\varphi_{i,V}$ together give a canonical surjection $$\varphi_V:\bigoplus_{j}V^j\otimes_KP_j\to V.$$ Set $W=\operatorname{Ker}\varphi_V$, and consider the canonical surjection $$\varphi_W:\bigoplus_{i}W^i\otimes_KP_i\to W$$ for $W$. Here, for a non-zero $V$, define $d(V)$ by the following formula: $$d(V)=\max \{j\in \mathbb{Z}_{\geqq 0}|V^j\neq 0\}.$$ Since $\varphi_V^i$ is a surjective $K$-linear $\oplus_{i\leqq j}V^j\otimes_KP_j^i\cong (\oplus_{j\leqq m}V^j\otimes_KP_j)^i\to V^i$, we see that $$\dim W^i
=\sum_{i<j\leqq d(V)}\dim V^j\dim P_j^i
=\sum_{i<j\leqq d(V)}\dim V^j\dim \operatorname{Hom}(G_i,G_j).$$ In particular, $W^{i}=0$ for all $i\geqq d(V)$. These consideration leads to the following.

\[projresol\] Every finitely generated right $A$-module $V$ has a bounded projective resolution of the following form $$0\to P_0^{\oplus e_{m,0}}\to\dots\to \bigoplus_{j=0}^{m-l}P_{j}^{\oplus e_{l,j}}\to\dots\to
\bigoplus_{j=0}^{m}P_{j}^{\oplus e_{0,j}}\to V\to 0$$ where $e_{0,j}=\dim V^j$ for all $j=0,\dots,m$ and, for any $l\geqq 1$ and any $j\leqq m-l$, $e_{l,j}$ is determined inductively by the following formula: $e_{l,j}=\sum_{j<k}e_{l-1,k}\dim \operatorname{Hom}(G_j,G_k)$.

\[MinimalResol\] The resolution above is not minimal in general. Throughout this paper, we shall denote by $$0\to P_0^{\oplus e_{m,0}'}\to\dots\to \bigoplus_{j=0}^{m-l}P_{j}^{\oplus e_{l,j}'}\to\dots\to
\bigoplus_{j=0}^{m}P_{j}^{\oplus e_{0,j}'}\to V\to 0$$ a minimal resolution of $V$ with $0\leqq e_{l,j}'\leqq e_{l,j}$ for all $0\leqq j\leqq m-l\leqq m$.

\[general resolution\] Under the 