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 | @* Christian Schilli, christian.schilli@rwth-aachen.de, |
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8 | @* Gerhard Pfister, pfister@mathematik.uni-kl.de |
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9 | |
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10 | OVERVIEW: In this library we present tools of homological algebra for |
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11 | finitely presented modules over GR-algebras. |
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12 | |
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13 | PROCEDURES: |
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14 | ncExt_R(k,M); computes presentation of Ext^k(M',R), M module, R basering, M'=coker(M) |
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15 | ncHom(M,N); computes presentation of Hom(M',N'), M,N modules, M'=coker(M), N'=coker(N) |
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16 | coHom(A,k); computes presentation of Hom(R^k,A), A matrix over basering R |
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17 | contraHom(A,k); computes presentation of Hom(A,R^k), A matrix over basering R |
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18 | dmodoublext(M, l); computes presentation of Ext_D^i(Ext_D^i(M,D),D), where D is a basering |
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19 | is_cenBimodule(M); checks whether a module presented by M is Artin-centralizing |
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20 | is_cenSubbimodule(M); checks whether a subbimodule M is Artin-centralizing |
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21 | "; |
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22 | |
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23 | LIB "dmod.lib"; |
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24 | LIB "gkdim.lib"; |
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25 | LIB "involut.lib"; |
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26 | LIB "nctools.lib"; |
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27 | LIB "ncalg.lib"; |
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28 | LIB "central.lib"; |
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29 | |
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30 | // ncExt(k,M,N); Ext^k(M',N'), M,N modules, M'=coker(M), N'=coker(N) |
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31 | // ncTensorMod(M,N); Tensor product of modules M'=coker(M), N'=coker(N) |
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32 | // ncTor(k,M,N); Tor_k(M',N'), M,N modules, M'=coker(M), N'=coker(N) |
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33 | // tensorMaps(M,N); tensor product of matrices |
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34 | |
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35 | |
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36 | /* LOG: |
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37 | 5.12.2012, VL: cleanup, is_cenSubbimodule and is_cenBimodule are added for assume checks; |
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38 | added doc for contraHom and coHom; assume check for ncHom etc. |
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39 | */ |
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40 | |
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41 | /* TODO: |
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42 | add noncomm examples to important precedures ncHom, |
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43 | */ |
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44 | |
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45 | proc contraHom(matrix M, int s) |
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46 | "USAGE: contraHom(A,k); A matrix, k int |
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47 | RETURN: matrix |
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48 | PURPOSE: compute the matrix of a homomorphism Hom(A,R^k), where R is the basering. Let A be a matrix defining a map F1-->F2 of free R-modules, then the matrix of Hom(F2,R^k)-->Hom(F1,R^k) is computed. |
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49 | NOTE: if A is matrix of a left (resp. right) R-module homomorphism, then Hom(A,R^k) is a right (resp. left) R-module R-module homomorphism |
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50 | EXAMPLE: example contraHom; shows an example. |
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51 | SEE ALSO: |
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52 | " |
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53 | { |
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54 | // also possible: compute with kontrahom from homolog_lib |
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55 | // and warn that the module changes its side |
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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*n][s*m]; |
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59 | for(b=1; b<=m; b++) |
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60 | { |
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61 | for(a=1; a<=s; 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)*n+c,(a-1)*m+b] = M[b,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 = contraHom(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 coHom(matrix M, int s) |
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82 | "USAGE: coHom(A,k); A matrix, k int |
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83 | PURPOSE: compute the matrix of a homomorphism Hom(R^k,A), where R is the basering. Let A be a matrix defining a map F1-->F2 of free R-modules, then the matrix of Hom(R^k,F1)-->Hom(R^k,F2) is computed. |
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84 | NOTE: Both A and Hom(A,R^k) are matrices for either left or right R-module homomorphisms |
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85 | EXAMPLE: example coHom; shows an example. |
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86 | " |
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87 | { |
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88 | int n,m=ncols(M),nrows(M); |
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89 | int a,b,c; |
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90 | matrix R[s*m][s*n]; |
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91 | for(b=1; b<=s; b++) |
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92 | { |
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93 | for(a=1; a<=m; a++) |
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94 | { |
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95 | for(c=1; c<=n; c++) |
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96 | { |
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97 | R[(a-1)*s+b,(c-1)*s+b] = M[a,c]; |
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98 | } |
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99 | } |
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100 | } |
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101 | return(R); |
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102 | } |
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103 | example |
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104 | { "EXAMPLE:"; echo = 2; |
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105 | ring A=0,(x,y,z),dp; |
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106 | matrix M[3][3]=1,2,3, |
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107 | 4,5,6, |
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108 | 7,8,9; |
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109 | module cM = coHom(M,2); |
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110 | print(cM); |
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111 | } |
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112 | |
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113 | |
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114 | proc ncHom(matrix M, matrix N) |
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115 | "USAGE: ncHom(M,N); M,N modules |
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116 | COMPUTE: A presentation of Hom(M',N'), M'=coker(M), N'=coker(N) |
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117 | ASSUME: M' is a left module, N' is a centralizing bimodule |
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118 | NOTE: ncHom(M,N) is a right module, hence a right presentation matrix |
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119 | is returned |
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120 | EXAMPLE: example ncHom; shows examples |
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121 | " |
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122 | { |
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123 | // assume: M is left module; nothing to check |
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124 | // assume: N is centralizing bimodule: to check |
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125 | if ( !is_cenBimodule(N) ) |
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126 | { |
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127 | ERROR("Second module in not centralizing."); |
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128 | } |
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129 | // returns a right presentation matrix (for a right module) |
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130 | matrix F = contraHom(M,nrows(N)); |
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131 | matrix B = coHom(N,ncols(M)); |
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132 | matrix C = coHom(N,nrows(M)); |
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133 | def Rbase = basering; |
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134 | def Rop = opposite(Rbase); |
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135 | setring Rop; |
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136 | matrix Bop = oppose(Rbase, B); |
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137 | matrix Cop = oppose(Rbase, C); |
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138 | matrix Fop = oppose(Rbase, F); |
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139 | matrix Dop = modulo(Fop, Bop); |
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140 | matrix Eop = modulo(Dop, Cop); |
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141 | setring Rbase; |
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142 | matrix E = oppose(Rop, Eop); |
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143 | kill Rop; |
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144 | return(E); |
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145 | } |
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146 | example |
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147 | { "EXAMPLE:"; echo = 2; |
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148 | ring A=0,(x,y,z),dp; |
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149 | matrix M[3][3]=1,2,3, |
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150 | 4,5,6, |
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151 | 7,8,9; |
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152 | matrix N[2][2]=x,y, |
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153 | z,0; |
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154 | module H = ncHom(M,N); |
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155 | print(H); |
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156 | } |
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157 | |
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158 | proc ncHom_alt(matrix M, matrix N) |
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159 | { |
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160 | // shorter but potentially slower |
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161 | matrix F = contraHom(M,nrows(N)); // \varphi^* |
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162 | matrix B = coHom(N,ncols(M)); // i |
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163 | matrix C = coHom(N,nrows(M)); // j |
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164 | matrix D = rightModulo(F,B); // D |
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165 | matrix E = rightModulo(D,C); // Hom(M,N) |
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166 | return(E); |
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167 | } |
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168 | example |
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169 | { "EXAMPLE:"; echo = 2; |
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170 | ring A=0,(x,y,z),dp; |
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171 | matrix M[3][3]=1,2,3, |
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172 | 4,5,6, |
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173 | 7,8,9; |
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174 | matrix N[2][2]=x,y, |
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175 | z,0; |
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176 | module H = ncHom_alt(M,N); |
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177 | print(H); |
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178 | } |
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179 | |
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180 | proc ncHom_R(matrix M) |
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181 | "USAGE: ncHom_R(M); M a module |
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182 | COMPUTE: A presentation of Hom_R(M',R), M'=coker(M) |
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183 | ASSUME: M' is a left module |
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184 | NOTE: ncHom_R(M) is a right module, hence a right presentation matrix is returned |
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185 | EXAMPLE: example ncHom_R; shows examples |
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186 | " |
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187 | { |
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188 | // assume: M is left module |
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189 | // returns a right presentation matrix |
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190 | // for a right module |
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191 | matrix F = transpose(M); |
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192 | def Rbase = basering; |
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193 | def Rop = opposite(Rbase); |
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194 | setring Rop; |
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195 | matrix Fop = oppose(Rbase, F); |
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196 | matrix Dop = modulo(Fop, std(0)); //ker Hom(A^n,A) -> Hom(A^m,A) |
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197 | matrix Eop = modulo(Dop, std(0)); // its presentation |
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198 | setring Rbase; |
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199 | matrix E = oppose(Rop, Eop); |
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200 | kill Rop; |
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201 | return(E); |
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202 | } |
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203 | example |
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204 | { "EXAMPLE:"; echo = 2; |
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205 | ring A=0,(x,t,dx,dt),dp; |
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206 | def W = Weyl(); setring W; |
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207 | matrix M[2][2] = |
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208 | dt, dx, |
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209 | t*dx,x*dt; |
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210 | module H = ncHom_R(M); |
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211 | print(H); |
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212 | matrix N[2][1] = x,dx; |
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213 | H = ncHom_R(N); |
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214 | print(H); |
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215 | } |
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216 | |
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217 | |
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218 | proc is_cenBimodule(module M) |
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219 | "USAGE: is_cenBimodule(M); M module |
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220 | COMPUTE: 1, if a module, presented by M can be centralizing in the sense of Artin and 0 otherwise |
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221 | NOTE: only one condition for centralizing factor module can be checked algorithmically |
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222 | EXAMPLE: example is_cenBimodule; shows examples |
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223 | " |
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224 | { |
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225 | // define in a ring R, for a module R: cen(M) ={ m in M: mr = rm for all r in R} |
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226 | // according to the definition, M is a centralizing bimodule <=> M is generated by cen(M) |
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227 | // if basering R is a G-algebra, then prop 6.4 of BGV indicates it's enough to provide |
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228 | // commutation of elements of M with the generators x_i of R |
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229 | // prop 6.4 verbatim generalizes to R = R'/I for a twosided I. |
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230 | // is M generates submodule, see the proc is_cenSubbimodule |
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231 | // let M be a presentation matrix for P=R*/R*M, then [e_i + M]x_j=x_j[e_i+M] |
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232 | // <=> Mx_j - x_jM in M must hold; thus forall j: Mx_j in M; thus M has to be |
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233 | // closed from the right, that is to be a two-sided submodule indeed |
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234 | // the rest of checks are complicated by now, so do the check only |
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235 | // *the algorithm *// |
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236 | if (isCommutative() ) { return(int(1));} |
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237 | int n = nvars(basering); |
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238 | int ans = 0; |
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239 | int i,j; |
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240 | vector P; |
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241 | module N; |
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242 | if ( attrib(M,"isSB") != 1) |
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243 | { |
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244 | N = std(M); |
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245 | } |
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246 | else |
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247 | { |
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248 | N = M; |
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249 | } |
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250 | // N is std(M) now |
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251 | for(i=1; i<=ncols(M); i++) |
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252 | { |
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253 | P = M[i]; |
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254 | if (P!=0) |
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255 | { |
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256 | for(j=1; j<=n; j++) |
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257 | { |
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258 | if ( NF(P*var(j) - var(j)*P, N) != 0) |
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259 | { |
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260 | return(ans); |
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261 | } |
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262 | } |
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263 | } |
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264 | } |
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265 | ans = 1; |
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266 | return(ans); |
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267 | } |
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268 | example |
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269 | { "EXAMPLE:"; echo = 2; |
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270 | def A = makeUsl2(); setring A; |
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271 | poly p = 4*e*f + h^2-2*h; // generator of the center |
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272 | matrix M[2][2] = p, p^2-7,0,p*(p+1); |
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273 | is_cenBimodule(M); // M is centralizing |
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274 | matrix N[2][2] = p, e*f,h,p*(p+1); |
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275 | is_cenBimodule(N); // N is not centralizing |
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276 | } |
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277 | |
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278 | proc is_cenSubbimodule(module M) |
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279 | "USAGE: is_cenSubbimodule(M); M module |
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280 | COMPUTE: 1, if a subbimodule, generated by the columns of M is |
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281 | centralizing in the sense of Artin and 0 otherwise |
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282 | EXAMPLE: example is_cenSubbimodule; shows examples |
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283 | " |
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284 | { |
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285 | // note: M in R^m is centralizing subbimodule iff it is generated by vectors, |
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286 | // each nonconstant component of which is central; 2 check: every entry of the |
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287 | // matrix M is central |
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288 | if (isCommutative()) { return(int(1));} |
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289 | return( inCenter(ideal(matrix(M))) ); |
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290 | } |
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291 | example |
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292 | { "EXAMPLE:"; echo = 2; |
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293 | def A = makeUsl2(); setring A; |
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294 | poly p = 4*e*f + h^2-2*h; // generator of the center |
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295 | matrix M[2][2] = p, p^2-7,0,p*(p+1); |
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296 | is_cenSubbimodule(M); // M is centralizing subbimodule |
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297 | matrix N[2][2] = p, e*f,h,p*(p+1); |
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298 | is_cenSubbimodule(N); // N is not centralizing subbimodule |
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299 | } |
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300 | |
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301 | |
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302 | proc ncExt(int i, matrix Ps, matrix Ph) |
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303 | "USAGE: Ext(i,M,N); i int, M,N matrices |
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304 | COMPUTE: A presentation of Ext^i(M',N'); for M'=coker(M) and N'=coker(N). |
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305 | ASSUME: M' is a left module, N' is a centralizing bimodule |
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306 | NOTE: ncExt(M,N) is a right module, hence a right presentation matrix |
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307 | is returned |
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308 | EXAMPLE: example ncExt; shows examples |
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309 | " |
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310 | { |
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311 | if ( !is_cenBimodule(Ph) ) |
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312 | { |
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313 | ERROR("Second module in not centralizing."); |
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314 | } |
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315 | |
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316 | if(i==0) { return(module(ncHom(Ps,Ph))); } |
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317 | list Phi = mres(Ps,i+1); |
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318 | module Im = coHom(Ph,ncols(Phi[i+1])); |
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319 | module f = contraHom(matrix(Phi[i+1]),nrows(Ph)); |
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320 | module Im1 = coHom(Ph,ncols(Phi[i])); |
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321 | module Im2 = contraHom(matrix(Phi[i]),nrows(Ph)); |
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322 | def Rbase = basering; |
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323 | def Rop = opposite(Rbase); |
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324 | setring Rop; |
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325 | module fop = oppose(Rbase,f); |
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326 | module Imop = oppose(Rbase,Im); |
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327 | module Im1op = oppose(Rbase,Im1); |
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328 | module Im2op = oppose(Rbase,Im2); |
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329 | module ker_op = modulo(fop,Imop); |
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330 | module ext_op = modulo(ker_op,Im1op+Im2op); |
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331 | // ext = prune(ext); |
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332 | // to be discussed and done prune_from_the_left |
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333 | setring Rbase; |
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334 | module ext = oppose(Rop,ext_op); |
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335 | kill Rop; |
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336 | return(ext); |
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337 | } |
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338 | example |
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339 | { "EXAMPLE:"; echo = 2; |
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340 | ring R = 0,(x,y),dp; |
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341 | ideal I = x2-y3; |
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342 | qring S = std(I); |
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343 | module M = [-x,y],[-y2,x]; |
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344 | module E1 = ncExt(1,M,M); |
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345 | E1; |
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346 | } |
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347 | |
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348 | proc ncExt_R(int i, matrix Ps) |
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349 | "USAGE: ncExt_R(i, M); i int, M module |
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350 | COMPUTE: a presentation of Ext^i(M',R); for M'=coker(M). |
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351 | RETURN: right module Ext, a presentation of Ext^i(M',R) |
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352 | EXAMPLE: example ncExt_R; shows an example |
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353 | "{ |
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354 | if (i==0) |
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355 | { |
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356 | return(ncHom_R(Ps)); // the rest is not needed |
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357 | } |
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358 | list Phi = nres(Ps,i+1); // left resolution |
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359 | module f = transpose(matrix(Phi[i+1])); // transp. because of Hom_R |
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360 | module Im2 = transpose(matrix(Phi[i])); |
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361 | def Rbase = basering; |
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362 | def Rop = opposite(Rbase); |
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363 | setring Rop; |
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364 | module fop = oppose(Rbase,f); |
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365 | module Im2op = oppose(Rbase,Im2); |
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366 | module ker_op = modulo(fop,std(0)); |
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367 | module ext_op = modulo(ker_op,Im2op); |
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368 | // ext = prune(ext); |
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369 | // to be discussed and done prune_from_the_left |
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370 | // necessary: compute SB! |
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371 | // "Computing SB of Ext"; |
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372 | // option(redSB); |
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373 | // option(redTail); |
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374 | // ext_op = std(ext_op); |
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375 | // int dimop = GKdim(ext_op); |
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376 | // printf("Ext has dimension %s",dimop); |
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377 | // if (dimop==0) |
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378 | // { |
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379 | // printf("of K-dimension %s",vdim(ext_op)); |
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380 | // } |
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381 | setring Rbase; |
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382 | module ext = oppose(Rop,ext_op); // a right module! |
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383 | kill Rop; |
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384 | return(ext); |
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385 | } |
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386 | example |
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387 | { "EXAMPLE:"; echo = 2; |
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388 | ring R = 0,(x,y),dp; |
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389 | poly F = x2-y2; |
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390 | def A = annfs(F); setring A; // A is the 2nd Weyl algebra |
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391 | matrix M[1][size(LD)] = LD; // ideal |
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392 | print(M); |
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393 | print(ncExt_R(1,M)); // hence the Ext^1 is zero |
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394 | module E = ncExt_R(2,M); // define the right module E |
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395 | print(E); // E is in the opposite algebra |
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396 | def Aop = opposite(A); setring Aop; |
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397 | module Eop = oppose(A,E); |
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398 | module T1 = ncExt_R(2,Eop); |
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399 | setring A; |
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400 | module T1 = oppose(Aop,T1); |
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401 | print(T1); // this is a left module Ext^2(Ext^2(M,A),A) |
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402 | print(M); // it is known that M holonomic implies Ext^2(Ext^2(M,A),A) iso to M |
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403 | } |
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404 | |
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405 | proc nctors(matrix M) |
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406 | { |
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407 | // ext^1_A(adj(M),A) |
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408 | def save = basering; |
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409 | matrix MM = M; // left |
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410 | def sop = opposite(save); |
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411 | setring sop; |
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412 | matrix MM = oppose(save,MM); // right |
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413 | MM = transpose(MM); // transposed |
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414 | list Phi = nres(MM,2); // i=1 |
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415 | module f = transpose(matrix(Phi[2])); // transp. because of Hom_R |
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416 | module Im2 = transpose(matrix(Phi[1])); |
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417 | setring save; |
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418 | module fop = oppose(sop,f); |
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419 | module Im2op = oppose(sop,Im2); |
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420 | module ker_op = modulo(fop,std(0)); |
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421 | module ext_op = modulo(ker_op,Im2op); |
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422 | // matrix E = ncExt_R(1,MM); |
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423 | // setring save; |
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424 | // matrix E = oppose(sop,E); |
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425 | return(ext_op); |
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426 | } |
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427 | |
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428 | proc altExt_R(int i, matrix Ps, map Invo) |
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429 | // TODO!!!!!!!! |
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430 | // matrix Ph |
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431 | // work thru Involutions; |
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432 | { |
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433 | if(i==0) |
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434 | { // return the formal adjoint |
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435 | matrix Ret = transpose(Ps); |
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436 | matrix Retop = involution(Ret, Invo); |
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437 | // "Computing prune of Hom"; |
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438 | // Retop = prune(Retop); |
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439 | // Retop = std(Retop); |
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440 | return(Retop); |
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441 | } |
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442 | list Phi = mres(Ps,i+1); |
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443 | // module Im = coHom(Ph,ncols(Phi[i+1])); |
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444 | module f = transpose(matrix(Phi[i+1])); |
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445 | f = involution(f, Invo); |
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446 | //= contraHom(matrix(Phi[i+1]),nrows(Ph)); |
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447 | // module Im1 = coHom(Ph,ncols(Phi[i])); |
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448 | module Im2 = transpose(matrix(Phi[i])); |
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449 | Im2 = involution(Im2, Invo); |
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450 | //contraHom(matrix(Phi[i]),nrows(Ph)); |
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451 | module ker_op = modulo(f,std(0)); |
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452 | module ext_op = modulo(ker_op,Im2); |
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453 | // ext = prune(ext); |
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454 | // to be discussed and done prune_from_the_left |
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455 | // optionally: compute SB! |
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456 | // "Computing prune of Ext"; |
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457 | ext_op = std(ext_op); |
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458 | int dimop = GKdim(ext_op); |
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459 | printf("Ext has dimension %s",dimop); |
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460 | if (dimop==0) |
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461 | { |
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462 | printf("of K-dimension %s",vdim(ext_op)); |
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463 | } |
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464 | module ext = involution(ext_op, Invo); // what about transpose? |
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465 | return(ext); |
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466 | } |
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467 | example |
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468 | { "EXAMPLE:"; echo = 2; |
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469 | ring R = 0,(x,y),dp; |
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470 | ideal I = x2-y3; |
---|
471 | qring S = std(I); |
---|
472 | module M = [-x,y],[-y2,x]; |
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473 | module E1 = ncExt(2,M,M); |
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474 | E1; |
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475 | } |
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476 | |
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477 | proc tensorMaps(matrix M, matrix N) |
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478 | { |
---|
479 | int r = ncols(M); |
---|
480 | int s = nrows(M); |
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481 | int p = ncols(N); |
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482 | int q = nrows(N); |
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483 | int a,b,c,d; |
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484 | matrix R[s*q][r*p]; |
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485 | for(b=1;b<=p;b++) |
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486 | { |
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487 | for(d=1;d<=q;d++) |
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488 | { |
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489 | for(a=1;a<=r;a++) |
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490 | { |
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491 | for(c=1;c<=s;c++) |
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492 | { |
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493 | R[(c-1)*q+d,(a-1)*p+b]=M[c,a]*N[d,b]; |
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494 | } |
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495 | } |
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496 | } |
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497 | } |
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498 | return(R); |
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499 | } |
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500 | |
---|
501 | proc ncTensorMod(matrix Phi, matrix Psi) |
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502 | { |
---|
503 | int s=nrows(Phi); |
---|
504 | int q=nrows(Psi); |
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505 | matrix A=tensorMaps(unitmat(s),Psi); //I_s tensor Psi |
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506 | matrix B=tensorMaps(Phi,unitmat(q)); //Phi tensor I_q |
---|
507 | matrix R=concat(A,B); //sum of A and B |
---|
508 | return(R); |
---|
509 | } |
---|
510 | |
---|
511 | |
---|
512 | proc ncTor(int i, matrix Ps, matrix Ph) |
---|
513 | { |
---|
514 | if(i==0) { return(module(ncTensorMod(Ps,Ph))); } |
---|
515 | // the tensor product |
---|
516 | list Phi = mres(Ph,i+1); // a resolution of Ph |
---|
517 | module Im = tensorMaps(unitmat(nrows(Phi[i])),Ps); |
---|
518 | module f = tensorMaps(matrix(Phi[i]),unitmat(nrows(Ps))); |
---|
519 | module Im1 = tensorMaps(unitmat(ncols(Phi[i])),Ps); |
---|
520 | module Im2 = tensorMaps(matrix(Phi[i+1]),unitmat(nrows(Ps))); |
---|
521 | module ker = modulo(f,Im); |
---|
522 | module tor = modulo(ker,Im1+Im2); |
---|
523 | // tor = prune(tor); |
---|
524 | return(tor); |
---|
525 | } |
---|
526 | |
---|
527 | |
---|
528 | static proc Hochschild() |
---|
529 | { |
---|
530 | ring A = 0,(x,y),dp; |
---|
531 | ideal I = x2-y3; |
---|
532 | qring B = std(I); |
---|
533 | module M = [-x,y],[-y2,x]; |
---|
534 | ring C = 0,(x,y,z,w),dp; // x->z, y->w |
---|
535 | ideal I = x2-y3,z3-w2; |
---|
536 | qring Be = std(I); //the enveloping algebra |
---|
537 | matrix AA[1][2] = x-z,y-w; //the presentation of the algebra B as Be-module |
---|
538 | module MM = imap(B,M); |
---|
539 | module E = ncExt(1,AA,MM); |
---|
540 | print(E); //the presentation of the H^1(A,M) |
---|
541 | |
---|
542 | ring A = 0,(x,y),dp; |
---|
543 | ideal I = x2-y3; |
---|
544 | qring B = std(I); |
---|
545 | ring C = 0,(x,y,z,w),dp; |
---|
546 | ideal I = x2-y3,z3-w2; |
---|
547 | qring Be = std(I); //the enveloping algebra |
---|
548 | matrix AA[1][2] = x-z,y-w; //the presentation of B as Be-module |
---|
549 | matrix AAA[1][2] = z,w; // equivalent? pres. of B |
---|
550 | print(ncExt(1,AA,AA)); //the presentation of the H^1(A,A) |
---|
551 | print(ncExt(1,AAA,AAA)); |
---|
552 | } |
---|
553 | |
---|
554 | static proc Lie() |
---|
555 | { |
---|
556 | // consider U(sl2)* U(sl2)^opp; |
---|
557 | LIB "ncalg.lib"; |
---|
558 | ring A = 0,(e,f,h,H,F,E),Dp; // any degree ordering |
---|
559 | int N = 6; // nvars(A); |
---|
560 | matrix @D[N][N]; |
---|
561 | @D[1,2] = -h; |
---|
562 | @D[1,3] = 2*e; |
---|
563 | @D[2,3] = -2*f; |
---|
564 | @D[4,5] = 2*F; |
---|
565 | @D[4,6] = -2*E; |
---|
566 | @D[5,6] = H; |
---|
567 | def AA = nc_algebra(1,@D); setring AA; |
---|
568 | ideal Q = E,F,H; |
---|
569 | poly Z = 4*e*f+h^2-2*h; // center |
---|
570 | poly Zo = 4*F*E+H^2+2*H; // center opposed |
---|
571 | ideal Qe = Z,Zo; |
---|
572 | //qring B = twostd(Qe); |
---|
573 | //ideal T = e-E,f-F,h-H; |
---|
574 | //ideal T2 = e-H,f-F,h-E; |
---|
575 | //Q = twostd(Q); // U is U(sl2) as left U(sl2)* U(sl2)^opp -- module |
---|
576 | matrix M[1][3] = E,F,H; |
---|
577 | module X0 = ncExt(0,M,M); |
---|
578 | print(X0); |
---|
579 | |
---|
580 | module X1 = ncExt(1,M,M); |
---|
581 | print(X1); |
---|
582 | module X2 = ncExt(2,M,M); // equal to Tor^Z_1(K,K) |
---|
583 | print(X2); |
---|
584 | |
---|
585 | // compute Tor^Z_1(K,K) |
---|
586 | ring r = 0,(z),dp; |
---|
587 | ideal i = z; |
---|
588 | matrix I[1][1]=z; |
---|
589 | Tor(1,I,I); |
---|
590 | } |
---|
591 | |
---|
592 | |
---|
593 | proc AllExts(module N, list #) |
---|
594 | // computes and shows everything |
---|
595 | // assumes we are in the opposite |
---|
596 | // and N is dual of some M |
---|
597 | // if # is given, map Invo and Ext_Invo are used |
---|
598 | { |
---|
599 | int UseInvo = 0; |
---|
600 | int sl = size(#); |
---|
601 | if (sl >0) |
---|
602 | { |
---|
603 | ideal I = ideal(#[1]); |
---|
604 | map Invo = basering, I; |
---|
605 | UseInvo = 1; |
---|
606 | "Using the involution"; |
---|
607 | } |
---|
608 | int nv = nvars(basering); |
---|
609 | int i,d; |
---|
610 | module E; |
---|
611 | list EE; |
---|
612 | print("--- module:"); print(matrix(N)); |
---|
613 | for (i=1; i<=nv; i++) |
---|
614 | { |
---|
615 | if (UseInvo) |
---|
616 | { |
---|
617 | E = altExt_R(i,N,Invo); |
---|
618 | } |
---|
619 | else |
---|
620 | { |
---|
621 | E = ncExt_R(i,N); |
---|
622 | } |
---|
623 | printf("--- Ext %s",i); |
---|
624 | print(matrix(E)); |
---|
625 | EE[i] = E; |
---|
626 | } |
---|
627 | return(E); |
---|
628 | } |
---|
629 | |
---|
630 | proc dmodualtest(module M, int n) |
---|
631 | { |
---|
632 | // computes the "dual" of the "dual" of a d-mod M |
---|
633 | // where n is the half-number of vars of Weyl algebra |
---|
634 | // assumed to be basering |
---|
635 | // returns the difference between M and Ext^n_D(Ext^n_D(M,D),D) |
---|
636 | def save = basering; |
---|
637 | setring save; |
---|
638 | module Md = ncExt_R(n,M); // right module |
---|
639 | // would be nice to use "prune"! |
---|
640 | // NO! prune performs left sided operations!!! |
---|
641 | // Md = prune(Md); |
---|
642 | // print(Md); |
---|
643 | def saveop = opposite(save); |
---|
644 | setring saveop; |
---|
645 | module Mdop = oppose(save,Md); // left module |
---|
646 | // here we're eligible to use prune |
---|
647 | Mdop = prune(Mdop); |
---|
648 | module Mopd = ncExt_R(n,Mdop); // right module |
---|
649 | setring save; |
---|
650 | module M2 = oppose(saveop,Mopd); // left module |
---|
651 | M2 = prune(M2); // eligible since M2 is a left mod |
---|
652 | M2 = groebner(M2); |
---|
653 | ideal tst = M2 - M; |
---|
654 | tst = groebner(tst); |
---|
655 | return(tst); |
---|
656 | } |
---|
657 | example |
---|
658 | { "EXAMPLE:"; echo = 2; |
---|
659 | ring R = 0,(x,y),dp; |
---|
660 | poly F = x3-y2; |
---|
661 | def A = annfs(F); |
---|
662 | setring A; |
---|
663 | dmodualtest(LD,2); |
---|
664 | } |
---|
665 | |
---|
666 | |
---|
667 | proc dmodoublext(module M, list #) |
---|
668 | "USAGE: dmodoublext(M [,i]); M module, i optional int |
---|
669 | COMPUTE: a presentation of Ext^i(Ext^i(M,D),D) for basering D |
---|
670 | RETURN: left module |
---|
671 | NOTE: by default, i is set to the integer part of the half of number of variables of D |
---|
672 | @* for holonomic modules over Weyl algebra, the double ext is known to be holonomic left module |
---|
673 | EXAMPLE: example dmodoublext; shows an example |
---|
674 | " |
---|
675 | { |
---|
676 | // assume: basering is a Weyl algebra? |
---|
677 | def save = basering; |
---|
678 | setring save; |
---|
679 | // if a list is nonempty and contains an integer N, n = N; otherwise n = nvars/2 |
---|
680 | int n; |
---|
681 | if (size(#) > 0) |
---|
682 | { |
---|
683 | // if (typeof(#) == "int") |
---|
684 | // { |
---|
685 | n = int(#[1]); |
---|
686 | // } |
---|
687 | // else |
---|
688 | // { |
---|
689 | // ERROR("the optional argument expected to have type int"); |
---|
690 | // } |
---|
691 | } |
---|
692 | else |
---|
693 | { |
---|
694 | n = nvars(save); n = n div 2; |
---|
695 | } |
---|
696 | // returns Ext^i_D(Ext^i_D(M,D),D), that is |
---|
697 | // computes the "dual" of the "dual" of a d-mod M (for n = nvars/2) |
---|
698 | module Md = ncExt_R(n,M); // right module |
---|
699 | // no prune yet! |
---|
700 | def saveop = opposite(save); |
---|
701 | setring saveop; |
---|
702 | module Mdop = oppose(save,Md); // left module |
---|
703 | // here we're eligible to use prune |
---|
704 | Mdop = prune(Mdop); |
---|
705 | module Mopd = ncExt_R(n,Mdop); // right module |
---|
706 | setring save; |
---|
707 | module M2 = oppose(saveop,Mopd); // left module |
---|
708 | kill saveop; |
---|
709 | M2 = prune(M2); // eligible since M2 is a left mod |
---|
710 | def M3; |
---|
711 | if (nrows(M2)==1) |
---|
712 | { |
---|
713 | M3 = ideal(M2); |
---|
714 | } |
---|
715 | else |
---|
716 | { |
---|
717 | M3 = M2; |
---|
718 | } |
---|
719 | M3 = groebner(M3); |
---|
720 | return(M3); |
---|
721 | } |
---|
722 | example |
---|
723 | { "EXAMPLE:"; echo = 2; |
---|
724 | ring R = 0,(x,y),dp; |
---|
725 | poly F = x3-y2; |
---|
726 | def A = annfs(F); |
---|
727 | setring A; |
---|
728 | dmodoublext(LD); |
---|
729 | LD; |
---|
730 | // fancier example: |
---|
731 | setring A; |
---|
732 | ideal I = Dx*(x2-y3),Dy*(x2-y3); |
---|
733 | I = groebner(I); |
---|
734 | print(dmodoublext(I,1)); |
---|
735 | print(dmodoublext(I,2)); |
---|
736 | } |
---|
737 | |
---|
738 | static proc part_Ext_R(matrix M) |
---|
739 | { |
---|
740 | // if i==0 |
---|
741 | matrix Ret = transpose(Ps); |
---|
742 | def Rbase = basering; |
---|
743 | def Rop = opposite(Rbase); |
---|
744 | setring Rop; |
---|
745 | module Retop = oppose(Rbase,Ret); |
---|
746 | module Hm = modulo(Retop,std(0)); // right kernel of transposed |
---|
747 | // "Computing prune of Hom"; |
---|
748 | // Retop = prune(Retop); |
---|
749 | // Retop = std(Retop); |
---|
750 | setring Rbase; |
---|
751 | Ret = oppose(Rop, Hm); |
---|
752 | kill Rop; |
---|
753 | return(Ret); |
---|
754 | // some checkz: |
---|
755 | // setring Rbase; |
---|
756 | // ker_op is the right Kernel of f^t: |
---|
757 | // module ker = oppose(Rop,ker_op); |
---|
758 | // print(f*ker); |
---|
759 | // module ext = oppose(Rop,ext_op); |
---|
760 | } |
---|