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\newtheorem{thm}[df]{$BDjM}(B}
\newtheorem{prop}[df]{$BL?Bj(B}
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\begin{document}
\title{Calculation of Projective Resolutions}
\author{$BOF(B $B9n;V(B \\
$B90A0Bg3X(B $BM}9)3XIt(B $B?tM}%7%9%F%`2J3X2J(B}
%\date{}
\maketitle

\section{Introduction}
$BM-8BBN(B$k$$B$HM-8B72(B$G$$B$,M?$($i$l$?$H$-!"7W;;5!$r;H$C$F(B
$B724D(B$A=kG$$B>e$N(B$A$--$B2C72(B $M$ $B$N(B minimal projective resolutions $B$r(B
$B5a$a$k$3$H$r9M$($^$9!#(B
$A$--$B2C72(B $M$ $B$N(B minimal projective resolution$B$H$O!"(B
$B2C72(B $M$ $B$H(B $B<M1F2C72(B $P_i$ $(i=0,1,2,\ldots )$ $B$K$h$C$F9=@.$5$l$k(B
$B<!$N$h$&$J2C72(B $M$ $B$+$i;O$^$k40A47ONs$N$3$H$r;X$7$^$9!#(B
$$
\cdots \longrightarrow P_n \stackrel{\delta_n}{\longrightarrow}
P_{n-1} \stackrel{\delta_{n-1}}{\longrightarrow} \cdots 
\stackrel{\delta_3}{\longrightarrow}
P_2 \stackrel{\delta_2}{\longrightarrow}
P_1 \stackrel{\delta_1}{\longrightarrow}
P_0 \stackrel{\varepsilon}{\longrightarrow}
M \stackrel{}{\longrightarrow}
0
$$
$BFC$K!"(B
Im $\delta_n \subset $ Rad($P_{n-1}$) $(i=1,2,3,\cdots)$
$B$,@.$jN)$D$H$-!"(Bminimal projective resolution $B$H8F$S$^$9!#(B

$B$3$N7W;;$r(BMAGMA$B$r;H$C$F!"Dj5ADL$j$K7W;;$7$F9T$/$3$H$O2DG=$G$9$,!"(B
$B7W;;$r?J$a$F9T$/$H8=$l$k<M1F2C72$N<!85$,GzH/E*$KBg$-$/$J$j(B
$B$9$0$K%a%b%j!<ITB-$+$i7W;;$,=PMh$J$/$J$C$F$7$^$$$^$9!#(B

$B$=$3$G!"724D(B $A$ $B$H(B Morita$BF1CM$H$J$k(B Quotient algebra  $k\Gamma/I$
$B$r;H$&$3$H$G!"(B
$B4{Ls(B $A$--$B2C72$N<!85$r$9$Y$F(B1$B<!85$KMn$9$3$H$G!"(B$A$--$B2C72$N<!85$r>.$5$/$7$F(B
$B7W;;$r$9$kJ}K!$,9M$(=P$5$l$^$7$?!#(B
( C.D. Feustel, E.L.Green \cite{F}, \cite{A})

$B:#2s$O$3$NJ}K!$r>R2p$7!"$5$i$K$$$/$D$+$N7W;;Nc$r<($7$F9T$-$?$$$H;W$$$^$9!#(B

\section{Path Algebra}

$B0J8e%0%i%U(B$\Gamma$$B$H$O!"(B
$B$$$/$D$+$NE@(B(points)$B$N=89g(B$I$$B$HE@$N4V$r7k$VLp(B(arrows)$B$N=89g(B$E$$B$G9=@.$5$l$?$b$N(B
$B$rI=$9$3$H$K$7$^$9!#(B
$B%0%i%U(B$\Gamma$$B$,(B$n$$B8D$NE@$G9=@.$5$l$F$$$k$H$-E@$r(B$p(i)$
$(i\in \{1,\ldots,n\})$$B$GI=$7!"(B
$BE@(B$p(i)$$B$+$i!"E@(B$p(j)$$B$X$N(B$k$$BHVL\$NLp$r(B$e_k(i,j)$$B$GI=$9$3$H$K$7$^$9!#(B
$B$^$?!"$$$/$D$+$NLp$r7R$2$?$b$N(B
$e_{k_1}(i_1,i_2)e_{k_2}(i_2,i_3)\cdots e_{k_l}(i_l,i_{l+1})$$B$r(B
$BD9$5(B$l$$B$N7PO)(B(path)$B$H8F$V$3$H$K$7$^$9!#(B

$BM-8BBN(B$k$$B$H%0%i%U(B$\Gamma$$B$,M?$($i$l$?$H$-!"(B
$B%0%i%U$NE@5Z$S7PO)$r(B$k$$B>e$N4pDl$H$9$k%Y%/%H%k6u4V$r9M$($^$9!#(B
$B$3$N%Y%/%H%k6u4V$K<!$N$h$&$K@Q$rDj5A$9$k$3$H$GF@$i$l$k(B$k$--algebra$B$r(B
Path algebra$B$r8F$V$3$H$K$7$^$9!#(B
($BCm0U!'E@(B$p(i)$$B$O!"(B$p(i)$$B$+$i(B$p(i)$$B$X$ND9$5(B0$B$N7PO)$H9M$($k$3$H$K$7$^$9!#(B)

$B7PO)(B
$P_1=e_{s_1}(i_1,i_2)e_{s_2}(i_2,i_3)\cdots e_{s_l}(i_l,i_{l+1})$$B$H(B
$P_2=e_{t_1}(j_1,j_2)e_{t_2}(j_2,j_3)\cdots e_{t_m}(j_m,j_{m+1})$$B$K$D$$$F(B

$$
P1*P2=\left\{ \begin{array}{cl}
e_{s_1}(i_1,i_2)\cdots e_{s_l}(i_l,i_{l+1})
e_{t_1}(j_1,j_2)\cdots e_{t_m}(j_m,j_{m+1}) & i_{l+1}=j_1 \\
0 & i_{l+1} \ne j_1 
\end{array}\right.
$$
$B$HDj5A$7$^$9!#Nc$($P!"(B
\begin{center}
        \epsfile{file=quiver02.eps,scale=1}
\end{center}
$B$N$h$&$J%0%i%U(B$\Gamma$ $B$+$iF@$i$l$k(B Path algebra $k\Gamma$$B$r9M$($k$H!"(B
$B$=$N4pDl$N4V$N@Q$O!"<!$N$h$&$K$J$j$^$9!#(B
$$
\begin{array}{|c|c|c|c|c|c|c|} \hline
	     & p(1) & p(2) & p(3) & a & b   & a*b \\  \hline
	p(1) & p(1) &   0  &   0  & a & 0   & a*b \\  \hline
	p(2) &    0 & p(2) &   0  & 0 & b   &   0 \\  \hline
	p(3) &    0 &   0  & p(3) & 0 & 0   &   0 \\  \hline
	a    &    0 & a    &   0  & 0 & a*b &   0 \\  \hline
	b    &    0 &   0  & b    & 0 & 0   &   0 \\  \hline
	a*b  &    0 &   0  & a*b  & 0 & 0   &   0 \\  \hline
		\end{array}
$$

$B$3$N(BPath algebra $k\Gamma$$B$NC10L85$O!"(B$p(1)+p(2)+p(3)$$B$H$J$j$^$9!#(B
$B0lHL$K(B Path algebra $k\Gamma$$B$H(B$k$--algebra $A$$B$N4V$G(B
$B<!$N$h$&$JDjM}$,@.$jN)$A$^$9!#(B

\begin{thm}
	$BG$0U$N(B$k$--algebra $A$$B$KBP$7!"(B
	$B$"$k%0%i%U(B$\Gamma$$B$r;H$C$F(B
	Path algebra $k\Gamma$$B$N%$%G%"%k(B $I$ $B$K$h$k(B Quotient algebra
	$k\Gamma / I$ $B$,(B $A$ $B$H(B  Morita $BF1CM$H$J$k$h$&$K=PMh$k!#(B
	$BFC$K!"N><T$N(Bprojective resolutions$B$OF1$8!#(B
\end{thm}

$k$--algebra $A$$B$+$i(B$k\Gamma / I$$B$r9=@.$9$kJ}K!$r35@b$9$k$H<!$N$h$&$K$J$j$^$9!#(B

\begin{itemize}
\item[(1)] $B$9$Y$F$N4{Ls(B $A$--$B2C72(B $S$ $B$r5a$a$k!#(B ($B%0%i%U$NE@(B)
\item[(2)] $B4{Ls(B $A$--$B2C72(B $S$ $B$KBP1~$9$k<M1FD>4{Ls2C72(B $P_S$$B$r5a$a$k!#(B
\item[(3)] $BG$0U$N4{Ls(B $A$--$B2C72(B $S_i$$B$H(B$S_j$$B$KBP$7!"(B
		$\Hom_A(P_{S_i}, P_{S_j})$$B$r7W;;$9$k!#(B($B%0%i%U$NLp(B)
\item[(4)] (3)$B$G5a$a$?=`F17?<LA|$r9g@.$7$F(B0$B$H$J$k$b$N$r5a$a$k!#(B
		($B%$%G%"%k(B$I$$B$N@8@.85$H$J$k7PO)(B)
\end{itemize}

$B$3$3$G!"(B(1), (2), (3) $B$O!"(BMeatAxe$B$d(BMAGMA$B$rMxMQ$7$F7W;;$r9T$$$^$9!#(B
$B$^$?!"(B(4)$B$O!"(B$k\Gamma / I$$B$r7W;;$9$k$?$a(BGr\"obner$B4pDl$bI,MW$K$J$j$^$9!#(B

$BNc$($P!"BN(B$k=GF(3)$$B$H$7$F!"(B3$B<!$NBP>N72(B $G$$B$N724D(B $kG$$B$r9M$($k$H!"(B
$B$3$l$KBP1~$9$k%0%i%U(B$\Gamma$$B$O!"(B
\begin{center}
	        \epsfile{file=quiver03.eps}
\end{center}
$B$HI=$5$l!"(B$I=<aba, bab>$$B$H$7$?$H$-!"(B
Quotient algebra $k\Gamma / I$
$B$,(B$kG$$B$H(BMorita$BF1CM$H$J$j$^$9!#(B
$B$^$?!"(B$k\Gamma / I$ $B$r1&(B$k\Gamma /I$--$B2C72$H8+$k$H!"<!$N$h$&$K(B2$B$D$N(B
uniserial$B$JD>4{Ls2C72$N$H$7$FI=$5$l$^$9!#(B
$$
k\Gamma /I \cong 
		\left(
			\begin{array}{c} 1 \\ a \\ ab \end{array}
		\right)_k 
	\oplus
		\left(
			\begin{array}{c} 2 \\ b \\ ba \end{array}
		\right)_k
$$

\section{ Minimal Projective Resolution $B$N7W;;(B}
\subsection{ $B5-9f(B }
$B$=$l$G$O!"(BQuotient algebra $k\Gamma / I$$B$+$i$I$N$h$&$K(B
minimal projective resolution$B$r7W;;$7$F9T$/$+@bL@$7$^$9!#(B
$B$=$N$?$a$K0J2<$N$h$&$J5-9f$rMQ0U$7$^$9!#(B
$$
\begin{array}{rcl}
\Lambda		& = &	k\Gamma / I \\
\Gamma_0 	& : & \Gamma\hbox{ $B$NE@$N=89g(B} \\
\Gamma_1 	& : & \Gamma\hbox{ $B$NLp$N=89g(B} \\
R		& = & k\Gamma / I \hbox{( $Lambda$$B$N(BRadical)} \\
{\cal B}	& : & \Gamma\hbox{ $B$N7PO)A4BN$N=89g(B} \\
{\cal G}(I)	& : & I \hbox{ $B$N(Breduced Gr\"obner$B4pDl(B ($B=g=x$O!"(Blength-lex)} \\
f 		& : & k\Gamma \rightarrow \Lambda
			\hbox{ ( $k$-algebra homomorphism ) } \\
N(I) 		& : & k\Gamma \hbox{ $B$NItJ,=89g$G(B$f(N(I))$$B$O!"(B 
			$\Lambda$$B$N4pDl$K$J$k$b$N(B} \\
\Gamma_2	& = & \{ Lt(p) | p \in {\cal G}(I) \}
		\hbox{ $B$?$@$7!"(B$Lt(p)$$B$O!"(B$p$$B$N(BLeading term} \\
\langle \ldots \rangle_k & : & \ldots \hbox{$B$G@8@.$5$l$k(B$k$--$B%Y%/%H%k6u4V(B}
\end{array}
$$
$B$5$i$K(B$\gamma\in \Gamma_3$$B$r0J2<$N>r7o$rK~$?$9=89g$H$7$FDj5A$7$^$9!#(B
\begin{itemize}
\item[(a)] $\gamma=\gamma_1\gamma_2$, $\gamma_1\in\Gamma_2$, 
$\gamma_2\in N(I)-\Gamma_0$
\item[(b)] $\gamma_1 = \beta_1 \beta_2$, $\beta_1\in\Gamma_1$, 
$\beta_2 \gamma_2\in{\cal B}-N(I)$
\item[(c)] $\gamma'$$B$,(B(a),(b)$B$rK~$?$9$H$-!"(B$\gamma''\in{\cal B}-\Gamma_0$
$B$G!"(B$\gamma=\gamma'\gamma''$$B$H$J$k(B$\gamma''$$B$O!"B8:_$7$J$$!#(B
\end{itemize}
$B$3$l$i$N5-9f$r>e$G<($7$?Nc$KEv$F$O$a$k$H!"(B
$$
\begin{array}{ccc}
\Gamma_0 & = & \{p(1),p(2)\} \\
\Gamma_1 & = & \{a,b\} \\
{\cal B} & = &\left\{\begin{array}{c}
		p(1) , p(2) , a , b , \quad \\
		\quad  ab , ba , \ldots \\
	     \end{array}\right\} \\
\Gamma_2 & = & \{aba, bab \} = {\cal G}(I) \\
N(I)     & = & \{p(1), p(2), a, b, ab, ba \} \\
\Gamma_3 & = & \{ abab , baba \}
\end{array}
$$

$B<!$K(B$\Lambda$$B$N<M1FD>4{Ls2C72$NI=$7J}$r9M$($^$9!#(B
$B0lHL$K(B$k$--algebra $A$$B$H$=$N86;O6REy85(B$e$$B$,M?$($i$l$?$H$-!"(B
$B=89g(B $eA:=\{ ea | a \in A\}$ $B$O!"1&<M1FD>4{Ls(B $A$--$B2C72$K$J$j$^$9!#(B
Path algebra $B$G$O!"86;O6REy85$O!"%0%i%U$NE@(B$p(i)$$B$GI=$5$l$F$$$^$9$N$G!"(B

$$
\begin{array}{ccl}
 v\in \Gamma_0 & \Rightarrow & v\hbox{ : $\Lambda$$B$N86;O6REy85(B } \\
       & \Rightarrow & v.\Lambda \hbox{$B$O!"<M1FD>4{Ls2C72(B} \\
       & \Rightarrow & v.\Lambda \hbox{ $B$KBP1~$9$k4{Ls2C72$O!"(B} 
	S_v:=v.\Lambda/v.R
\end{array}
$$

$B$H$J$j$^$9!#(B
$\Gamma_0=\{p(1), p(2)\}$ $B$H$J$k>e$NNc$G$O!"(B
$$
\begin{array}{cccc}
p(1).\Lambda & = & \langle 1, a, ab \rangle_k & \Leftrightarrow 
	\qquad S_1 \cong \langle \bar{1} \rangle_k\\
p(1).R  & = & \langle a, ab \rangle_k \\
p(2).\Lambda & = & \langle 2, b, ba \rangle_k & \Leftrightarrow 
	\qquad S_2 \cong \langle \bar{2} \rangle_k\\
p(2).R  & = & \langle b, ba \rangle_k \\
\end{array}
$$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{$B<M1F2C724V$N=`F17?<LA|(B}
$B<!$K<M1F(B $\Lambda$--$B2C72$N4V$N=`F17?<LA|$NI=$7J}$r8+$F$_$^$7$g$&!#(B
$v_1$,...,$v_n$, $w_1$,...,$w_m\in \Gamma_0$$B$K$D$$$F!"(B
$$
P=\bigoplus_i v_i\Lambda\hbox{ , }Q=\bigoplus_j w_j\Lambda
$$
$B$H$9$k$H!"(B
$B=`F17?<LA|(B$\varphi$ : $P\rightarrow Q$$B$O!"(B$\Lambda$$B$N85$r@.J,$H$9$k(B
$n\times m$$B9TNs(B $F=\{ f_{i,j} \}$
$B$G<!$N$h$&$KI=$5$l$^$9!#(B
$$f(v_i)=w_1f_{i,1}+w_2f_{i,2}+...+w_mf_{i,m}$$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$BNc$($P!"(B
$$
\begin{array}{cccc}
\varphi
   : & p(1).\Lambda\oplus p(2).\Lambda & \longrightarrow 
			& p(2).\Lambda\oplus p(2).\Lambda \\
    \\
    & (p(1),0) & \longrightarrow & (b,0) \\
    & (0,p(2)) & \longrightarrow & (0,ba)
\end{array}
$$
\qquad $B$H$9$k$H!"=`F17?<LA|(B$\varphi$$B$H$=$NI=8=9TNs$O!"<!$N$h$&$K$J$j$^$9!#(B
$$
\varphi \Leftrightarrow \left(
			\begin{array}{cc} b & 0 \\ 0 & ba \end{array}
			\right)
$$
$B$^$?!"=`F17?<LA|(B$\varphi$$B$N3K$HA|$O!"(B
$\Ker ( \varphi )=\langle (0,b), (0,ba), (ab,0) \rangle_k$ $B$H(B
$\Img ( \varphi )=\langle (b,0), (ba,0), (0,ba) \rangle_k$ $B$H$J$j$^$9!#(B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{$\Lambda$--$B2C72(B $M$ $B$NI=8=(B}

$\Lambda$--$B2C72(B $M$$B$O!"(B2$B$D$N<M1F2C72(B$P$$B$H(B$Q$$B$=$7$F(B
$B=`F17?<LA|(B$\varphi_M$ : $P\rightarrow Q$$B$r;H$C$F(B
$M\cong Q/\varphi_M(P)$$B$HDj5A$G$-$^$9!#(B
$B$D$^$j!"(B$\Lambda$--$B2C72(B $M$$B$O>o$K<M1F2C72$N4V$N=`F17?<LA|$rDL$7$F!"(B
$\Lambda$$B$r@.J,$H$9$k9TNs$N7A$GI=8=$5$l$k$3$H$K$J$j$^$9!#(B


$BNc$($P!">e$GDj5A$7$?(B$\varphi$ $B$NI=$9(B$\Lambda$--$B2C72(B $M$$B$O!"(B
$$
M\cong \frac{
		p(2).\Lambda\oplus p(2).\Lambda
	}
	{
		\varphi(p(1).\Lambda\oplus p(2).\Lambda)
	}
 \cong S_2\oplus \left(\begin{array}{c} S_2 \\ S_1 \end{array}\right)
$$
$B$H$J$j$^$9!#(B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{$B4{Ls2C72(B$S_v$$B$N(BProjective Resolution $B$N9=@.(B} 
\label{simple_proj_resol}
$B$=$l$G$O!"(B
$B4{Ls(B$\Lambda$--$B2C72(B$S_v$$B$N(Bprojective resolution $B$N9=@.$7$F$_$^$7$g$&!#(B

$$
\bigoplus_{\gamma\in\Gamma_3,o(\gamma)=v} e(\gamma)\Lambda
\stackrel{\Delta_3}{\longrightarrow}
\bigoplus_{t\in\Gamma_2,o(t)=v} e(t)\Lambda
\stackrel{\Delta_2}{\longrightarrow}
\bigoplus_{p\in\Gamma_1,o(p)=v} e(p)\Lambda
\stackrel{\Delta_1}{\longrightarrow}
v\Lambda
\stackrel{\varepsilon}{\longrightarrow}
S_v
\stackrel{}{\longrightarrow}0
$$

$B$?$@$7!"(B$o(p)$: $B7PO)(B $p$ $B$N;OE@!"(B$e(p)$: $B7PO)(B $p$ $B$N=*E@$rI=$7$^$9!#(B
$B$^$?!"=`F17?<LA|(B $\Delta_1$, $\Delta_2$, $\Delta_3$ $B$O<!$N$h$&$KDj5A$5$l$^$9!#(B
$$
\begin{array}{cl} 
p\in \Gamma_1 & \Delta_1(e(p))=vp \\ 
\\
t\in \Gamma_2 & \exists f_t\in {\cal G}(I) ; 
f_t=t+\sum_j \alpha_jp_j \quad \hbox{( $\alpha_j\in k$, $p_j\in N(I)$)} \\
              & \Delta_2(e(t))=e(a')b'+\sum_je(a_j)\alpha_jq_j \\
              & \hbox{($B$?$@$7(B $t=a'b'$, $p_j=a_jq_j$ \quad 
		($a',a_j\in\Gamma_1$, $b',q_j\in {\cal B}$))} \\
\\
\gamma\in \Gamma_3 & \exists t, t' \in\Gamma_2; \quad \gamma=tb=b't' \quad
		    \exists f_t, f_{t'} \in {\cal G}(I) \\
              & b'f_{t'}-f_tb=\sum_j\alpha_jc_jf_jd_j \quad
            ( \alpha_j\in k,\quad  c_j,d_j\in{\cal B}, f_j\in {\cal G}(I)) \\
               & \Delta_3(e(\gamma))
		=e(t)b+\sum_{j',c_{j'}=v}e(Lt(f_{j'}))\alpha_{j'}d_{j'} 
\end{array}
$$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{$\Lambda$--$B2C72(B$M$$B$N(BProjective Resolution$B$N9=@.(B}

$B0lHL$N(B$\Lambda$--$B2C72(B $M$ $B$N>l9g$N(B projective resolution $B$N7W;;$K$O!"(B
$B<!$N$h$&$J9)IW$,I,MW$K$J$j$^$9!#(B
$\Lambda$--$B2C72(B$M$$B$r:8(B$k$-- $B1&(B$\Lambda$--$B2C72$H8+$F!"(B
$\Lambda$$B$r4^$`(B$k$--algebra
$\Lambda^*=\left(\begin{array}{cc} k & M \\ 0 & \Lambda \end{array} \right)$
$B$r9=@.$7$^$9!#(B($BOB$H@Q$O!"(B$2\times 2$--$B9TNs$N$b$N$r:NMQ$9$k(B)
$B$3$N(B$\Lambda^*$$B$KBP1~$9$k(BQuotient algebra$B$r(B$k\Gamma^*/I^*$$B$H$9$k$H!"(B
$\Gamma^*$$B$O!"(B$\Gamma$$B$K!"(B1$BE@(B$v^*$$B$r2C$($?%0%i%U$G!"E@(B$v^*$$B$KBP1~$9$k(B
$B4{Ls2C72(B$S^*$$B$+$i!"(B\ref{simple_proj_resol}$B$NJ}K!$G(Bprojective resolution
$$
{P_3}^*
\stackrel{\Delta_3^*}{\longrightarrow}
{P_2}^*
\stackrel{\Delta_2^*}{\longrightarrow}
{P_1}^*
\stackrel{\Delta_1^*}{\longrightarrow}
{P_0}^*
\stackrel{\varepsilon^*}{\longrightarrow}
S^*
\stackrel{}{\longrightarrow}0
$$
$B$r:n$k$3$H$,$G$-$^$9!#(B
$B$3$N$H$-!"$3$N(Bprojective resolution $B$r(B$\Lambda$$B$K@)8B$9$k$H!"(B
$M$$B$N(Bprojective resolution
$
{P_3}
\stackrel{\delta_3}{\longrightarrow}
{P_2}
\stackrel{\delta_2}{\longrightarrow}
{P_1}
\stackrel{\delta_1}{\longrightarrow}
M
\stackrel{}{\rightarrow}0
$
$B$,=PMh>e$,$j$^$9!#(B
$B8e$O!"(B$P_2/\delta_3(P_3)$$B$r2~$a$F(B$M$$B$HCV$-D>$7$F!"(B
$B7W;;$r7+$jJV$;$PI,MW$JD9$5$N(Bprojective resolution $B$r9=@.$9$k$3$H$,=PMh$^$9!#(B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Minimal Projective Resolution $B$N9=@.(B}

$\Lambda$--$B2C72(B$M$$B$N(B projective resolution $B$,M?$($i$l$?$H$-!"(B
$$
\begin{array}{ccccc}
\rightarrow & P_n & \stackrel{\delta_n}{\longrightarrow} & P_{n-1} &\rightarrow \\
 & \Vert & & \Vert \\
 & \oplus P_i & & \oplus Q_i 
\end{array}
$$
$B$KBP$7$F!"(B
Im $\delta_n \subset $ Rad($P_{n-1}$)$B$H$J$k$3$H$O!"(B
$\delta_n$$B$NG$0U$N@.J,(B$f_{i,j}$$B$,6RNm85$G$"$k$3$H$HF1CM$H$J$j$^$9!#(B
$B2>$K$"$k(B$f_{i,j}$$B$,6RNm85$G$J$$$H$9$k$HE,Ev$JE@(B$ v\in \Gamma_0$$B$K$D$$$F(B
$B4pDl$rJQ49$7$F=`F17?<LA|(B$\delta_n$$B$rI=$99TNs$r(B
$$
\begin{array}{rc}
	& \quad (j) 					\\
(i)	&	\left(\begin{array}{ccc} 
$B!!!!(B			& 0	 	&	\\
			& \vdots	&	\\
		$B!v(B	& v		& $B!v(B	\\
			& \vdots	&	\\
			& 0		& 
		\end{array}\right) \\
\end{array}
$$
$B$H$$$&7A$KJQ7A$G$-$^$9!#(B
$B$3$N$H$-!"(B$P_n$$B$+$i<M1FD>4{Ls2C72(B$P_i$$B$r(B$P_{n-1}$$B$+$i<M1FD>4{Ls2C72(B$Q_j$$B$r(B
$B$=$l$>$l=|$$$?<M1F2C72$r(B
$B2~$a$F(B$P_n$$B$H(B$P_{n-1}$$B$H$7$F>e$N9TNs$+$i(B$i$$B9T(B$j$$BNs$r=|$$$?$b$N$G(B
$B=`F17?<LA|(B$\delta_n$$B$rI=$9$3$H$K$7$^$9!#(B
$B$3$NA`:n$r!"7+$jJV$7$F@.J,$N$9$Y$F$N85$,6RNm85$K$9$l$P!"(B
minimal projective resolution $B$,9=@.$G$-$?$3$H$K$J$j$^$9!#(B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{ $B6qBNE*$J(B Minimal Projective Resolution $B$N7W;;(B}
$B0J2<$K<($7$?$b$N$O!"72(B$M_{22}$$B>e$NI8?t(B3$B$NJ,2rBN$r(B$k$$B$H$7$F!"(B
$B3F4{Ls2C72$N(B $B$N(B minimal projective resolution $B$r5a$a$?7k2L$G$9!#(B
($I$$B$O<+L@$J2C72$=$NB>$N?t;z$O4{Ls2C72$r$=$N<!85$r;H$C$FI=$7$F$$$^$9!#(B)

$$ \qquad \qquad
 \stackrel{\delta_3}{\longrightarrow}
P_{I}\oplus P_{55} \stackrel{\delta_2}{\longrightarrow}
P_{I}\oplus P_{231} \stackrel{\delta_1}{\longrightarrow}
P_{49}\oplus P_{49^*} \stackrel{\varepsilon}{\longrightarrow}
I \stackrel{}{\longrightarrow}
0
$$
$$
P_{55}\oplus P_{49}\oplus P_{49^*} \stackrel{\delta_5}{\longrightarrow}
P_{231}\oplus P_{49}\oplus P_{49^*} \stackrel{\delta_4}{\longrightarrow}
P_{55}\oplus P_{49}\oplus P_{49^*}
\qquad \qquad
$$

$$ \qquad \qquad
 \stackrel{\delta_3}{\longrightarrow}
P_{I}\oplus P_{55} \stackrel{\delta_2}{\longrightarrow}
P_{55}\oplus P_{231} \stackrel{\delta_1}{\longrightarrow}
P_{49}\oplus P_{49^*} \stackrel{\varepsilon}{\longrightarrow}
55 \stackrel{}{\longrightarrow}
0
$$
$$
P_{I}\oplus P_{49}\oplus P_{49^*} \stackrel{\delta_5}{\longrightarrow}
P_{231}\oplus P_{49}\oplus P_{49^*} \stackrel{\delta_4}{\longrightarrow}
P_{I}\oplus P_{49}\oplus P_{49^*}
\qquad \qquad
$$

$$ \qquad \qquad
 \stackrel{\delta_3}{\longrightarrow}
P_{49}\oplus P_{49^*} \oplus P_{231} \stackrel{\delta_2}{\longrightarrow}
P_{I}\oplus P_{55}\oplus P_{231} \stackrel{\delta_1}{\longrightarrow}
P_{49}\oplus P_{49^*} \stackrel{\varepsilon}{\longrightarrow}
231 \stackrel{}{\longrightarrow}
0
$$
$$
231 \stackrel{\delta_7}{\longrightarrow}
231 \stackrel{\delta_6}{\longrightarrow}
P_{49}\oplus P_{49^*} \stackrel{\delta_5}{\longrightarrow}
P_{I}\oplus P_{55}\oplus P_{231} \stackrel{\delta_4}{\longrightarrow}
P_{49}\oplus P_{49^*} \oplus P_{231} 
\qquad \qquad
$$

$B:G8e$K!":#2s7W;;=PMh$?72$H$=$N4{Ls2C72$N(B minimal projective resolution 
$B$ND9$5!"$*$h$S(B $B7W;;$N4p$K$J$C$?(B Quiver $B$rIUO?$K:\$;$^$9!#(B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%  R E F E R E N C E S                                                  %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{thebibliography}{99}
%
\bibitem{A}
David J. Anick, Edward L. Green,
On the homology of quotients of path algebras,
Communications in Algebra, 15(1\&2), 309-341 (1987)
%
\bibitem{F}
Charles D. Feustel, Edward L. Green,
Constructing projective resolutions,
Communications in Algebra, 21(6), 1869-1887 (1993)
%
\end{thebibliography}
\newpage
{\LARGE
\begin{table}
%	\caption{$B7W;;7k2L(B}
	\begin{center}
        \begin{tabular}[t]{|c|c|c|} \hline
	$B72(B	& $B2C72(B 	& {\Large $B7W;;$G$-$?D9$5!!(B} \\ \hline
	$A_6$ 	& $I$ , 4		& 28 		\\
		& $3a$, $3b$ 		& 50$B0J>e(B 	\\ \hline
	$A_7$ 	& $I$			& 20 		\\
		& $10a$, $10b$,13	& 22 		\\ \hline
	$A_8$ 	& $I$, 28		& 11 		\\
		& 7	 		& 12 		\\
		& 13			& 10		\\ 
		& 35			& 8 		\\ \hline
	$S_6$ 	& $I$ , 1		& 36		\\
		& $4a$, $4b$ 		& 34	 	\\
		& 6	 		& 50$B0J>e(B 	\\ \hline
     $M_{11}$ 	& $I$, 5, $5^*$, 10, $10^*$, $10_a$, 24	& 50$B0J>e(B 	\\ \hline
     $M_{21}$ 	& $I$ 			& 44 		\\
		& $15a$, $15b$, $15c$, 19	& 50$B0J>e(B 	\\ \hline
     $M_{22}$ 	& $I$ , 55		& 36		\\
		& $49$, $49*$ 		& 34	 	\\
		& 231	 		& 50$B0J>e(B 	\\ \hline
     $M_{23}$ 	& $I$, 22, 256		& 33		\\
		& $104$			& 32	 	\\
		& $104^*$		& 30	 	\\
		& $770$			& 15	 	\\
		& $770^*$		& 16	 	\\ \hline
     $HS$ 	& $I$, 22, 154		& 40		\\
		& 748			& 42	 	\\
		& 1253			& 18	 	\\
		& 321, 1176 		& 50$B0J>e(B 	\\ \hline
	\end{tabular}
	\end{center}
\end{table}
}

\newpage
{\Huge

\begin{table}
\caption{$BM-8B72$KBP1~$9$k%0%i%U#1(B}
\begin{center}
        \begin{tabular}[t]{cc}
	        $B72(B			& $B%0%i%U(B \\  \hline \hline
	\raisebox{1.5cm}{$A_6$}		& \epsfile{file=a6.eps,scale=1.6} \\
 \hline
	\raisebox{1.5cm}{$A_7$}		& \epsfile{file=a7.eps,scale=1.6} \\
 \hline
	\raisebox{2.5cm}{$A_8$}		& \epsfile{file=a8.eps,scale=1.6} \\
 \hline
	\raisebox{2.5cm}{$S_6$}		& \epsfile{file=s6.eps,scale=1.6} 
        \end{tabular}
\end{center}
\end{table}
}
\newpage
{\Huge
%        \raisebox{7cm}{}
\begin{table}
\caption{$BM-8B72$KBP1~$9$k%0%i%U#2(B}
\begin{center}
        \begin{tabular}[t]{cc}
	        $B72(B			& $B%0%i%U(B \\ \hline \hline
	\raisebox{1.5cm}{$M_{11}$}	& \epsfile{file=m11.eps,scale=1.6} \\
 \hline
	\raisebox{1.5cm}{$M_{21}$}	& \epsfile{file=m21.eps,scale=1.6} \\
 \hline
	\raisebox{1.5cm}{$M_{22}$}	& \epsfile{file=m22.eps,scale=1.6} 
        \end{tabular}
\end{center}
\end{table}
}
\newpage
{\Huge
%        \raisebox{7cm}{}
\begin{table}
\caption{$BM-8B72$KBP1~$9$k%0%i%U#3(B}
\begin{center}
        \begin{tabular}[t]{cc}
	        $B72(B			& $B%0%i%U(B \\  \hline \hline
	\raisebox{3cm}{$M_{23}$}	& \epsfile{file=m23.eps,scale=2} \\
 \hline
	\raisebox{3cm}{$HS$}		& \epsfile{file=hs.eps,scale=2} 
        \end{tabular}
\end{center}
\end{table}
}

\end{document}