1 | // |
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2 | version="$Id$"; |
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3 | category="Noncommutative"; |
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4 | info=" |
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5 | LIBRARY: nchomolog.lib Procedures for Noncommutative Homological Algebra |
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6 | AUTHORS: Viktor Levandovskyy levandov@math.rwth-aachen.de, |
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7 | @* Gerhard Pfister, pfister@mathematik.uni-kl.de |
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8 | |
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9 | PROCEDURES: |
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10 | ncExt_R(k,M); Ext^k(M',R), M module, R basering, M'=coker(M) |
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11 | ncHom(M,N); Hom(M',N'), M,N modules, M'=coker(M), N'=coker(N) |
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12 | coHom(A,k); Hom(R^k,A), A matrix over basering R |
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13 | contraHom(A,k); Hom(A,R^k), A matrix over basering R |
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14 | dmodoublext(M, l); computes Ext_D^i(Ext_D^i(M,D),D), where D is a basering |
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15 | "; |
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16 | |
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17 | LIB "dmod.lib"; |
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18 | LIB "gkdim.lib"; |
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19 | LIB "involut.lib"; |
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20 | LIB "nctools.lib"; |
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21 | |
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22 | // ncExt(k,M,N); Ext^k(M',N'), M,N modules, M'=coker(M), N'=coker(N) |
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23 | // ncTensorMod(M,N); Tensor product of modules M'=coker(M), N'=coker(N) |
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24 | // ncTor(k,M,N); Tor_k(M',N'), M,N modules, M'=coker(M), N'=coker(N) |
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25 | // tensorMaps(M,N); tensor product of matrices |
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26 | |
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27 | proc contraHom(matrix M, int s) |
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28 | { |
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29 | int n,m=ncols(M),nrows(M); |
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30 | int a,b,c; |
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31 | matrix R[s*n][s*m]; |
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32 | for(b=1; b<=m; b++) |
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33 | { |
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34 | for(a=1; a<=s; a++) |
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35 | { |
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36 | for(c=1; c<=n; c++) |
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37 | { |
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38 | R[(a-1)*n+c,(a-1)*m+b] = M[b,c]; |
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39 | } |
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40 | } |
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41 | } |
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42 | return(R); |
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43 | } |
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44 | example |
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45 | { "EXAMPLE:"; echo = 2; |
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46 | ring A=0,(x,y,z),dp; |
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47 | matrix M[3][3]=1,2,3, |
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48 | 4,5,6, |
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49 | 7,8,9; |
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50 | module cM = contraHom(M,2); |
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51 | print(cM); |
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52 | } |
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53 | |
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54 | proc coHom(matrix M, int s) |
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55 | { |
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56 | int n,m=ncols(M),nrows(M); |
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57 | int a,b,c; |
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58 | matrix R[s*m][s*n]; |
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59 | for(b=1; b<=s; b++) |
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60 | { |
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61 | for(a=1; a<=m; a++) |
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62 | { |
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63 | for(c=1; c<=n; c++) |
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64 | { |
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65 | R[(a-1)*s+b,(c-1)*s+b] = M[a,c]; |
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66 | } |
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67 | } |
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68 | } |
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69 | return(R); |
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70 | } |
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71 | example |
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72 | { "EXAMPLE:"; echo = 2; |
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73 | ring A=0,(x,y,z),dp; |
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74 | matrix M[3][3]=1,2,3, |
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75 | 4,5,6, |
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76 | 7,8,9; |
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77 | module cM = coHom(M,2); |
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78 | print(cM); |
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79 | } |
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80 | |
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81 | proc ncHom(matrix M, matrix N) |
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82 | "USAGE: ncHom(M,N); M,N modules |
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83 | COMPUTE: A presentation of Hom(M',N'), M'=coker(M), N'=coker(N) |
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84 | ASSUME: M' is a left module, N' is a centralizing bimodule |
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85 | NOTE: ncHom(M,N) is a right module, hence a right presentation matrix |
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86 | is returned |
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87 | EXAMPLE: example ncHom; shows examples |
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88 | " |
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89 | { |
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90 | // assume: M is left module |
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91 | // assume: N is centralizing bimodule |
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92 | // returns a right presentation matrix |
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93 | // for a right module |
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94 | matrix F = contraHom(M,nrows(N)); |
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95 | matrix B = coHom(N,ncols(M)); |
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96 | matrix C = coHom(N,nrows(M)); |
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97 | def Rbase = basering; |
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98 | def Rop = opposite(Rbase); |
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99 | setring Rop; |
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100 | matrix Bop = oppose(Rbase, B); |
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101 | matrix Cop = oppose(Rbase, C); |
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102 | matrix Fop = oppose(Rbase, F); |
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103 | matrix Dop = modulo(Fop, Bop); |
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104 | matrix Eop = modulo(Dop, Cop); |
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105 | setring Rbase; |
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106 | matrix E = oppose(Rop, Eop); |
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107 | kill Rop; |
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108 | return(E); |
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109 | } |
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110 | example |
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111 | { "EXAMPLE:"; echo = 2; |
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112 | ring A=0,(x,y,z),dp; |
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113 | matrix M[3][3]=1,2,3, |
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114 | 4,5,6, |
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115 | 7,8,9; |
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116 | matrix N[2][2]=x,y, |
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117 | z,0; |
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118 | module H = ncHom(M,N); |
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119 | print(H); |
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120 | } |
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121 | |
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122 | proc ncHom_alt(matrix M, matrix N) |
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123 | { |
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124 | // shorter but potentially slower |
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125 | matrix F = contraHom(M,nrows(N)); // \varphi^* |
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126 | matrix B = coHom(N,ncols(M)); // i |
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127 | matrix C = coHom(N,nrows(M)); // j |
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128 | matrix D = rightModulo(F,B); // D |
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129 | matrix E = rightModulo(D,C); // Hom(M,N) |
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130 | return(E); |
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131 | } |
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132 | example |
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133 | { "EXAMPLE:"; echo = 2; |
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134 | ring A=0,(x,y,z),dp; |
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135 | matrix M[3][3]=1,2,3, |
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136 | 4,5,6, |
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137 | 7,8,9; |
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138 | matrix N[2][2]=x,y, |
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139 | z,0; |
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140 | module H = ncHom_alt(M,N); |
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141 | print(H); |
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142 | } |
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143 | |
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144 | proc ncHom_R(matrix M) |
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145 | "USAGE: ncHom_R(M); M a module |
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146 | COMPUTE: A presentation of Hom_R(M',R), M'=coker(M) |
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147 | ASSUME: M' is a left module |
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148 | NOTE: ncHom_R(M) is a right module, hence a right presentation matrix is returned |
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149 | EXAMPLE: example ncHom_R; shows examples |
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150 | " |
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151 | { |
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152 | // assume: M is left module |
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153 | // returns a right presentation matrix |
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154 | // for a right module |
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155 | matrix F = transpose(M); |
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156 | def Rbase = basering; |
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157 | def Rop = opposite(Rbase); |
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158 | setring Rop; |
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159 | matrix Fop = oppose(Rbase, F); |
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160 | matrix Dop = modulo(Fop, std(0)); //ker Hom(A^n,A) -> Hom(A^m,A) |
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161 | matrix Eop = modulo(Dop, std(0)); // its presentation |
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162 | setring Rbase; |
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163 | matrix E = oppose(Rop, Eop); |
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164 | kill Rop; |
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165 | return(E); |
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166 | } |
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167 | example |
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168 | { "EXAMPLE:"; echo = 2; |
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169 | ring A=0,(x,t,dx,dt),dp; |
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170 | def W = Weyl(); setring W; |
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171 | matrix M[2][2] = |
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172 | dt, dx, |
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173 | t*dx,x*dt; |
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174 | module H = ncHom_R(M); |
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175 | print(H); |
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176 | matrix N[2][1] = x,dx; |
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177 | H = ncHom_R(N); |
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178 | print(H); |
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179 | } |
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180 | |
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181 | |
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182 | proc ncExt(int i, matrix Ps, matrix Ph) |
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183 | "USAGE: Ext(i,M,N); i int, M,N modules |
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184 | COMPUTE: A presentation of Ext^i(M',N'); for M'=coker(M) and N'=coker(N). |
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185 | NOTE: ncExt(M,N) is a right module, hence a right presentation matrix |
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186 | is returned |
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187 | EXAMPLE: example ncExt; shows examples |
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188 | " |
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189 | { |
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190 | if(i==0) { return(module(ncHom(Ps,Ph))); } |
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191 | list Phi = mres(Ps,i+1); |
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192 | module Im = coHom(Ph,ncols(Phi[i+1])); |
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193 | module f = contraHom(matrix(Phi[i+1]),nrows(Ph)); |
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194 | module Im1 = coHom(Ph,ncols(Phi[i])); |
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195 | module Im2 = contraHom(matrix(Phi[i]),nrows(Ph)); |
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196 | def Rbase = basering; |
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197 | def Rop = opposite(Rbase); |
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198 | setring Rop; |
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199 | module fop = oppose(Rbase,f); |
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200 | module Imop = oppose(Rbase,Im); |
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201 | module Im1op = oppose(Rbase,Im1); |
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202 | module Im2op = oppose(Rbase,Im2); |
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203 | module ker_op = modulo(fop,Imop); |
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204 | module ext_op = modulo(ker_op,Im1op+Im2op); |
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205 | // ext = prune(ext); |
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206 | // to be discussed and done prune_from_the_left |
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207 | setring Rbase; |
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208 | module ext = oppose(Rop,ext_op); |
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209 | kill Rop; |
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210 | return(ext); |
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211 | } |
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212 | example |
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213 | { "EXAMPLE:"; echo = 2; |
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214 | ring R = 0,(x,y),dp; |
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215 | ideal I = x2-y3; |
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216 | qring S = std(I); |
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217 | module M = [-x,y],[-y2,x]; |
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218 | module E1 = ncExt(1,M,M); |
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219 | E1; |
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220 | } |
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221 | |
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222 | proc ncExt_R(int i, matrix Ps) |
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223 | "USAGE: ncExt_R(i, M); i int, M module |
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224 | COMPUTE: a presentation of Ext^i(M',R); for M'=coker(M). |
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225 | RETURN: right module Ext, a presentation of Ext^i(M',R) |
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226 | EXAMPLE: example ncExt_R; shows an example |
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227 | "{ |
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228 | if (i==0) |
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229 | { |
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230 | return(ncHom_R(Ps)); // the rest is not needed |
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231 | } |
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232 | list Phi = nres(Ps,i+1); // left resolution |
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233 | module f = transpose(matrix(Phi[i+1])); // transp. because of Hom_R |
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234 | module Im2 = transpose(matrix(Phi[i])); |
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235 | def Rbase = basering; |
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236 | def Rop = opposite(Rbase); |
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237 | setring Rop; |
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238 | module fop = oppose(Rbase,f); |
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239 | module Im2op = oppose(Rbase,Im2); |
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240 | module ker_op = modulo(fop,std(0)); |
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241 | module ext_op = modulo(ker_op,Im2op); |
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242 | // ext = prune(ext); |
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243 | // to be discussed and done prune_from_the_left |
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244 | // necessary: compute SB! |
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245 | // "Computing SB of Ext"; |
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246 | // option(redSB); |
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247 | // option(redTail); |
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248 | // ext_op = std(ext_op); |
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249 | // int dimop = GKdim(ext_op); |
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250 | // printf("Ext has dimension %s",dimop); |
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251 | // if (dimop==0) |
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252 | // { |
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253 | // printf("of K-dimension %s",vdim(ext_op)); |
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254 | // } |
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255 | setring Rbase; |
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256 | module ext = oppose(Rop,ext_op); // a right module! |
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257 | kill Rop; |
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258 | return(ext); |
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259 | } |
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260 | example |
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261 | { "EXAMPLE:"; echo = 2; |
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262 | ring R = 0,(x,y),dp; |
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263 | poly F = x2-y2; |
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264 | def A = annfs(F); |
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265 | setring A; |
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266 | matrix M[1][size(LD)] = LD; |
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267 | print(ncExt_R(1,M)); // hence the Ext^1 is zero |
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268 | module E = ncExt_R(2,M); // right module |
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269 | print(E); |
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270 | def Aop = opposite(A); |
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271 | setring Aop; |
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272 | module Eop = oppose(A,E); |
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273 | module T1 = ncExt_R(2,Eop); |
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274 | setring A; |
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275 | module T1 = oppose(Aop,T1); |
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276 | print(T1); // this is a left module Ext^2(Ext^2(M,A),A) |
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277 | } |
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278 | |
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279 | proc nctors(matrix M) |
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280 | { |
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281 | // ext^1_A(adj(M),A) |
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282 | def save = basering; |
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283 | matrix MM = M; // left |
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284 | def sop = opposite(save); |
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285 | setring sop; |
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286 | matrix MM = oppose(save,MM); // right |
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287 | MM = transpose(MM); // transposed |
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288 | list Phi = nres(MM,2); // i=1 |
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289 | module f = transpose(matrix(Phi[2])); // transp. because of Hom_R |
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290 | module Im2 = transpose(matrix(Phi[1])); |
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291 | setring save; |
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292 | module fop = oppose(sop,f); |
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293 | module Im2op = oppose(sop,Im2); |
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294 | module ker_op = modulo(fop,std(0)); |
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295 | module ext_op = modulo(ker_op,Im2op); |
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296 | // matrix E = ncExt_R(1,MM); |
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297 | // setring save; |
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298 | // matrix E = oppose(sop,E); |
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299 | return(ext_op); |
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300 | } |
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301 | |
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302 | proc altExt_R(int i, matrix Ps, map Invo) |
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303 | // TODO!!!!!!!! |
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304 | // matrix Ph |
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305 | // work thru Involutions; |
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306 | { |
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307 | if(i==0) |
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308 | { // return the formal adjoint |
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309 | matrix Ret = transpose(Ps); |
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310 | matrix Retop = involution(Ret, Invo); |
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311 | // "Computing prune of Hom"; |
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312 | // Retop = prune(Retop); |
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313 | // Retop = std(Retop); |
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314 | return(Retop); |
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315 | } |
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316 | list Phi = mres(Ps,i+1); |
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317 | // module Im = coHom(Ph,ncols(Phi[i+1])); |
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318 | module f = transpose(matrix(Phi[i+1])); |
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319 | f = involution(f, Invo); |
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320 | //= contraHom(matrix(Phi[i+1]),nrows(Ph)); |
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321 | // module Im1 = coHom(Ph,ncols(Phi[i])); |
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322 | module Im2 = transpose(matrix(Phi[i])); |
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323 | Im2 = involution(Im2, Invo); |
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324 | //contraHom(matrix(Phi[i]),nrows(Ph)); |
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325 | module ker_op = modulo(f,std(0)); |
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326 | module ext_op = modulo(ker_op,Im2); |
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327 | // ext = prune(ext); |
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328 | // to be discussed and done prune_from_the_left |
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329 | // optionally: compute SB! |
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330 | // "Computing prune of Ext"; |
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331 | ext_op = std(ext_op); |
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332 | int dimop = GKdim(ext_op); |
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333 | printf("Ext has dimension %s",dimop); |
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334 | if (dimop==0) |
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335 | { |
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336 | printf("of K-dimension %s",vdim(ext_op)); |
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337 | } |
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338 | module ext = involution(ext_op, Invo); // what about transpose? |
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339 | return(ext); |
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340 | } |
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341 | example |
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342 | { "EXAMPLE:"; echo = 2; |
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343 | ring R = 0,(x,y),dp; |
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344 | ideal I = x2-y3; |
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345 | qring S = std(I); |
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346 | module M = [-x,y],[-y2,x]; |
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347 | module E1 = ncExt(2,M,M); |
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348 | E1; |
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349 | } |
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350 | |
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351 | proc tensorMaps(matrix M, matrix N) |
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352 | { |
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353 | int r = ncols(M); |
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354 | int s = nrows(M); |
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355 | int p = ncols(N); |
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356 | int q = nrows(N); |
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357 | int a,b,c,d; |
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358 | matrix R[s*q][r*p]; |
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359 | for(b=1;b<=p;b++) |
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360 | { |
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361 | for(d=1;d<=q;d++) |
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362 | { |
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363 | for(a=1;a<=r;a++) |
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364 | { |
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365 | for(c=1;c<=s;c++) |
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366 | { |
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367 | R[(c-1)*q+d,(a-1)*p+b]=M[c,a]*N[d,b]; |
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368 | } |
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369 | } |
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370 | } |
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371 | } |
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372 | return(R); |
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373 | } |
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374 | |
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375 | proc ncTensorMod(matrix Phi, matrix Psi) |
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376 | { |
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377 | int s=nrows(Phi); |
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378 | int q=nrows(Psi); |
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379 | matrix A=tensorMaps(unitmat(s),Psi); //I_s tensor Psi |
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380 | matrix B=tensorMaps(Phi,unitmat(q)); //Phi tensor I_q |
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381 | matrix R=concat(A,B); //sum of A and B |
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382 | return(R); |
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383 | } |
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384 | |
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385 | |
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386 | proc ncTor(int i, matrix Ps, matrix Ph) |
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387 | { |
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388 | if(i==0) { return(module(ncTensorMod(Ps,Ph))); } |
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389 | // the tensor product |
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390 | list Phi = mres(Ph,i+1); // a resolution of Ph |
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391 | module Im = tensorMaps(unitmat(nrows(Phi[i])),Ps); |
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392 | module f = tensorMaps(matrix(Phi[i]),unitmat(nrows(Ps))); |
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393 | module Im1 = tensorMaps(unitmat(ncols(Phi[i])),Ps); |
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394 | module Im2 = tensorMaps(matrix(Phi[i+1]),unitmat(nrows(Ps))); |
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395 | module ker = modulo(f,Im); |
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396 | module tor = modulo(ker,Im1+Im2); |
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397 | // tor = prune(tor); |
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398 | return(tor); |
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399 | } |
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400 | |
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401 | |
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402 | static proc Hochschild() |
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403 | { |
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404 | ring A = 0,(x,y),dp; |
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405 | ideal I = x2-y3; |
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406 | qring B = std(I); |
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407 | module M = [-x,y],[-y2,x]; |
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408 | ring C = 0,(x,y,z,w),dp; // x->z, y->w |
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409 | ideal I = x2-y3,z3-w2; |
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410 | qring Be = std(I); //the enveloping algebra |
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411 | matrix AA[1][2] = x-z,y-w; //the presentation of the algebra B as Be-module |
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412 | module MM = imap(B,M); |
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413 | module E = ncExt(1,AA,MM); |
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414 | print(E); //the presentation of the H^1(A,M) |
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415 | |
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416 | ring A = 0,(x,y),dp; |
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417 | ideal I = x2-y3; |
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418 | qring B = std(I); |
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419 | ring C = 0,(x,y,z,w),dp; |
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420 | ideal I = x2-y3,z3-w2; |
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421 | qring Be = std(I); //the enveloping algebra |
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422 | matrix AA[1][2] = x-z,y-w; //the presentation of B as Be-module |
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423 | matrix AAA[1][2] = z,w; // equivalent? pres. of B |
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424 | print(ncExt(1,AA,AA)); //the presentation of the H^1(A,A) |
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425 | print(ncExt(1,AAA,AAA)); |
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426 | } |
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427 | |
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428 | static proc Lie() |
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429 | { |
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430 | // consider U(sl2)* U(sl2)^opp; |
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431 | LIB "ncalg.lib"; |
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432 | ring A = 0,(e,f,h,H,F,E),Dp; // any degree ordering |
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433 | int N = 6; // nvars(A); |
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434 | matrix @D[N][N]; |
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435 | @D[1,2] = -h; |
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436 | @D[1,3] = 2*e; |
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437 | @D[2,3] = -2*f; |
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438 | @D[4,5] = 2*F; |
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439 | @D[4,6] = -2*E; |
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440 | @D[5,6] = H; |
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441 | def AA = nc_algebra(1,@D); setring AA; |
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442 | ideal Q = E,F,H; |
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443 | poly Z = 4*e*f+h^2-2*h; // center |
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444 | poly Zo = 4*F*E+H^2+2*H; // center opposed |
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445 | ideal Qe = Z,Zo; |
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446 | //qring B = twostd(Qe); |
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447 | //ideal T = e-E,f-F,h-H; |
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448 | //ideal T2 = e-H,f-F,h-E; |
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449 | //Q = twostd(Q); // U is U(sl2) as left U(sl2)* U(sl2)^opp -- module |
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450 | matrix M[1][3] = E,F,H; |
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451 | module X0 = ncExt(0,M,M); |
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452 | print(X0); |
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453 | |
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454 | module X1 = ncExt(1,M,M); |
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455 | print(X1); |
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456 | module X2 = ncExt(2,M,M); // equal to Tor^Z_1(K,K) |
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457 | print(X2); |
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458 | |
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459 | // compute Tor^Z_1(K,K) |
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460 | ring r = 0,(z),dp; |
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461 | ideal i = z; |
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462 | matrix I[1][1]=z; |
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463 | Tor(1,I,I); |
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464 | } |
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465 | |
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466 | |
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467 | proc AllExts(module N, list #) |
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468 | // computes and shows everything |
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469 | // assumes we are in the opposite |
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470 | // and N is dual of some M |
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471 | // if # is given, map Invo and Ext_Invo are used |
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472 | { |
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473 | int UseInvo = 0; |
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474 | int sl = size(#); |
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475 | if (sl >0) |
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476 | { |
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477 | ideal I = ideal(#[1]); |
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478 | map Invo = basering, I; |
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479 | UseInvo = 1; |
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480 | "Using the involution"; |
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481 | } |
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482 | int nv = nvars(basering); |
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483 | int i,d; |
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484 | module E; |
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485 | list EE; |
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486 | print("--- module:"); print(matrix(N)); |
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487 | for (i=1; i<=nv; i++) |
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488 | { |
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489 | if (UseInvo) |
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490 | { |
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491 | E = altExt_R(i,N,Invo); |
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492 | } |
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493 | else |
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494 | { |
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495 | E = ncExt_R(i,N); |
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496 | } |
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497 | printf("--- Ext %s",i); |
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498 | print(matrix(E)); |
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499 | EE[i] = E; |
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500 | } |
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501 | return(E); |
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502 | } |
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503 | |
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504 | static proc dmod_exts(module M) |
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505 | { |
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506 | // return all Ext_R for a D-module M |
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507 | } |
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508 | |
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509 | proc dmodualtest(module M, int n) |
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510 | { |
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511 | // computes the "dual" of the "dual" of a d-mod M |
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512 | // where n is the half-number of vars of Weyl algebra |
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513 | // assumed to be basering |
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514 | // returns the difference between M and Ext^n_D(Ext^n_D(M,D),D) |
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515 | def save = basering; |
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516 | setring save; |
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517 | module Md = ncExt_R(n,M); // right module |
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518 | // would be nice to use "prune"! |
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519 | // NO! prune performs left sided operations!!! |
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520 | // Md = prune(Md); |
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521 | // print(Md); |
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522 | def saveop = opposite(save); |
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523 | setring saveop; |
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524 | module Mdop = oppose(save,Md); // left module |
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525 | // here we're eligible to use prune |
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526 | Mdop = prune(Mdop); |
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527 | module Mopd = ncExt_R(n,Mdop); // right module |
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528 | setring save; |
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529 | module M2 = oppose(saveop,Mopd); // left module |
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530 | M2 = prune(M2); // eligible since M2 is a left mod |
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531 | M2 = groebner(M2); |
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532 | ideal tst = M2 - M; |
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533 | tst = groebner(tst); |
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534 | return(tst); |
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535 | } |
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536 | example |
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537 | { "EXAMPLE:"; echo = 2; |
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538 | ring R = 0,(x,y),dp; |
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539 | poly F = x3-y2; |
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540 | def A = annfs(F); |
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541 | setring A; |
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542 | dmodualtest(LD,2); |
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543 | } |
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544 | |
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545 | |
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546 | proc dmodoublext(module M, list #) |
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547 | "USAGE: dmodoublext(M [,i]); M module, i optional int |
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548 | COMPUTE: a presentation of Ext^i(Ext^i(M,D),D); for basering D |
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549 | RETURN: left module |
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550 | NOTE: by default, i is set to the integer part of the half of number of variables of D |
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551 | @* for holonomic modules over Weyl algebra, the double ext is known to be holonomic |
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552 | EXAMPLE: example dmodoublext; shows an example |
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553 | " |
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554 | { |
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555 | // assume: basering is a Weyl algebra? |
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556 | def save = basering; |
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557 | setring save; |
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558 | // if a list is nonempty and contains an integer N, n = N; otherwise n = nvars/2 |
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559 | int n; |
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560 | if (size(#) > 0) |
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561 | { |
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562 | // if (typeof(#) == "int") |
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563 | // { |
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564 | n = int(#[1]); |
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565 | // } |
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566 | // else |
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567 | // { |
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568 | // ERROR("the optional argument expected to have type int"); |
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569 | // } |
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570 | } |
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571 | else |
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572 | { |
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573 | n = nvars(save); n = n div 2; |
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574 | } |
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575 | // returns Ext^i_D(Ext^i_D(M,D),D), that is |
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576 | // computes the "dual" of the "dual" of a d-mod M (for n = nvars/2) |
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577 | module Md = ncExt_R(n,M); // right module |
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578 | // no prune yet! |
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579 | def saveop = opposite(save); |
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580 | setring saveop; |
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581 | module Mdop = oppose(save,Md); // left module |
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582 | // here we're eligible to use prune |
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583 | Mdop = prune(Mdop); |
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584 | module Mopd = ncExt_R(n,Mdop); // right module |
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585 | setring save; |
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586 | module M2 = oppose(saveop,Mopd); // left module |
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587 | kill saveop; |
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588 | M2 = prune(M2); // eligible since M2 is a left mod |
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589 | def M3; |
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590 | if (nrows(M2)==1) |
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591 | { |
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592 | M3 = ideal(M2); |
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593 | } |
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594 | else |
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595 | { |
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596 | M3 = M2; |
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597 | } |
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598 | M3 = groebner(M3); |
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599 | return(M3); |
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600 | } |
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601 | example |
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602 | { "EXAMPLE:"; echo = 2; |
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603 | ring R = 0,(x,y),dp; |
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604 | poly F = x3-y2; |
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605 | def A = annfs(F); |
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606 | setring A; |
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607 | dmodoublext(LD); |
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608 | LD; |
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609 | // fancier example: |
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610 | setring A; |
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611 | ideal I = Dx*(x2-y3),Dy*(x2-y3); |
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612 | I = groebner(I); |
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613 | print(dmodoublext(I,1)); |
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614 | print(dmodoublext(I,2)); |
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615 | } |
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616 | |
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617 | static proc part_Ext_R(matrix M) |
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618 | { |
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619 | // if i==0 |
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620 | matrix Ret = transpose(Ps); |
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621 | def Rbase = basering; |
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622 | def Rop = opposite(Rbase); |
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623 | setring Rop; |
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624 | module Retop = oppose(Rbase,Ret); |
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625 | module Hm = modulo(Retop,std(0)); // right kernel of transposed |
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626 | // "Computing prune of Hom"; |
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627 | // Retop = prune(Retop); |
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628 | // Retop = std(Retop); |
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629 | setring Rbase; |
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630 | Ret = oppose(Rop, Hm); |
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631 | kill Rop; |
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632 | return(Ret); |
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633 | // some checkz: |
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634 | // setring Rbase; |
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635 | // ker_op is the right Kernel of f^t: |
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636 | // module ker = oppose(Rop,ker_op); |
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637 | // print(f*ker); |
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638 | // module ext = oppose(Rop,ext_op); |
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639 | } |
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