Changeset 1bf4403 in git


Ignore:
Timestamp:
Mar 9, 2015, 7:56:50 PM (9 years ago)
Author:
Oleksandr Motsak <motsak@…>
Branches:
(u'spielwiese', '8e0ad00ce244dfd0756200662572aef8402f13d5')
Children:
2a413dd9aa8d8ec395a5eaebe7c9991725f19aa0
Parents:
de688442dfe449992dc14a000bca0691ecc7e106
git-author:
Oleksandr Motsak <motsak@mathematik.uni-kl.de>2015-03-09 19:56:50+01:00
git-committer:
Oleksandr Motsak <motsak@mathematik.uni-kl.de>2015-03-14 17:58:51+01:00
Message:
Added grpres/grgens and simple grtranspose

chg: also improved & clarified presentation/coker vs submodule gens conventions
Files:
4 edited

Legend:

Unmodified
Added
Removed
  • Singular/LIB/gradedModules.lib

    rde6884 r1bf4403  
    1010    The library contains several procedures for constructing and manipulating graded modules/matrices/resolutions.
    1111    Basics about graded objects can be found in [MO].
     12    Throughout this library graded objects are graded maps, that is,
     13    matrices with polynomials, together with grading weights for source and
     14    destination. Graded modules are implicitly given as coker of a graded map.
     15    Note that in special cases we may also consider submodules in S^r generated
     16    by columns of a graded polynomial matrix (or a graded map).
    1217NOTE:
    1318    set assumeLevel to positive integer value in order to auto-check all assumptions.
     19    We denote the current basering by S.
    1420REFERENCES:
    1521[MO]   Motsak, O.: Non-commutative Computer Algebra with applications: Graded commutative algebra and related
    1622       structures in Singular with applications, Ph.D. thesis, TU Kaiserslautern, 2010
    1723PROCEDURES:
    18     grobj(M,w[,d])  construct a graded object
     24    grobj(M,w[,d])  construct a graded object (map) given by matrix M
    1925    grtest(A)       check whether A is a valid graded object
    20     grdeg(M)        compute graded degrees of columns of M
    21     grview(M)       view the graded structure of M
    22     grshift(M,d)    shift graded module M by d
    23     grzero()        presentation of S(0)^1, where S = basering
    24     grtwist(r,d)    presentation of S(d)^r, where S = basering
    25     grsum(M,N)      direct sum of two graded modules M + N
    26     grpower(M,p)    direct p-th power of graded module M
     26    grdeg(M)        compute graded degrees of columns of the map M
     27    grview(M)       view the graded structure of map M
     28    grshift(M,d)    shift graded module coker(M) by d
     29    grzero()        presentation of S(0)^1
     30    grtwist(r,d)    presentation of S(-d)^r
     31    grtwists(v)     presentation of S(-v[1])+...+S(-v[size(v)])
     32    grsum(M,N)      direct sum of two graded modules coker(M) + coker(N)
     33    grpower(M,p)    direct p-th power of graded module coker(M)
     34    grtranspose(M)  un-ordered graded transpose of map M
     35    grgens(M)       try to compute submodule generators of coker(M)
     36    grpres(F)       presentation of submodule generated by columns of F
    2737    grorder(M)      reorder cols/rows of M for correct graded-block-structure
    28     grtranspose(M)  reordered graded transpose of M
    29     TestGRRes(n,I)  compute/order/transpose a graded resolution of I
     38    grtranspose1(M)  reordered graded transpose of map M
     39    TestGRRes(n,I)  compute/order/transpose a graded resolution of ideal I
    3040";
    3141
     
    303313}
    304314
    305 static proc gtranspose(M)
    306 "??????"
    307 {
    308   ASSUME(1, grtest(M) );
    309   return (  grobj(transpose(M), -grdeg(M), -attrib(M, "isHomog"))  );
    310 }
    311 
    312 
    313 proc grtranspose(def M)
    314 "USAGE:  grtranspose(M), graded object or list M
     315proc grtranspose1(def M)
     316"USAGE:  grtranspose1(M), graded object or list M
    315317RETURN:  same as input
    316318PURPOSE: graded transpose of graded object or chain complex M
    317319ASSUME:  M must be a graded object or a list of graded objects
    318 EXAMPLE: example grtranspose; shows an example
     320EXAMPLE: example grtranspose1; shows an example
    319321"
    320322{
     
    343345    {
    344346//      grview(M[i]);
    345       L[j] = grtranspose( grobj( M[i], attrib(M[i], "isHomog")) );
     347      L[j] = grtranspose1( grobj( M[i], attrib(M[i], "isHomog")) );
    346348//      grview(L[j]);
    347349
     
    398400  "Graded (original) structure of 'mres(Input,0)': "; grview(L);
    399401
    400   "Graded transpose of the previous resolution "; list LLL = grtranspose( L ); grview( LLL );
     402  "Graded transpose of the previous resolution "; list LLL = grtranspose1( L ); grview( LLL );
    401403
    402404  "Its non-minimal betti numbers: "; print(betti(LLL, 0), "betti");
     
    518520  // ordres(L, intvec(0)); // ?
    519521
    520   grview( LL ); " = TRANSPOSEed complex: %%%%%%%%%%%%%%";
    521   list LLL = grtranspose( LL ); //  resolution RR = LLL;
     522  grview( LL ); " = TRANSPOSE'd complex: %%%%%%%%%%%%%%";
     523  list LLL = grtranspose1( LL ); //  resolution RR = LLL;
    522524  print(betti(LLL, 0), "betti");
    523525
     
    11571159}
    11581160
     1161proc grtranspose(def M)
     1162"
     1163USAGE:   grtranspose(M), graded object M
     1164RETURN:  graded object
     1165PURPOSE: graded transpose of M
     1166NOTE:    no reordering is performend by this procedure   
     1167EXAMPLE: example grtranspose; shows an example
     1168"
     1169{
     1170  ASSUME(1, grtest(M) );
     1171  return (  grobj(transpose(M), -grdeg(M), -attrib(M, "isHomog"))  );
     1172}
     1173example
     1174{ "EXAMPLE:"; echo = 2;
     1175
     1176  ring r=32003,(x,y,z),dp;
     1177
     1178  module M = grtwists( intvec(-2, 0, 4, 4) ); grview(M);
     1179
     1180  module N = grsyz( grtranspose( M ) ); grview(N);
     1181
     1182  module L = grtranspose(N); grview( L );
     1183
     1184  module K = grsyz( L ); grview(K);
     1185 
     1186
     1187  module A = grobj( module([x+y, x, 0, 3], [0, x+y, y, 2], [y, y, z, 1]), intvec(0,0,0,1) ); A = grgroebner(A); grview(A);
     1188
     1189  "graded transpose: "; module B = grtranspose(A); grview( B ); print(B);
     1190
     1191  "... syzygy: "; module C = grsyz(B); grview(C);
     1192
     1193  "... transposed: "; module D = grtranspose(C); grview( D ); print (D);
     1194
     1195  "... and back to presentation: "; module E = grsyz( D ); grview(E); print(E);
     1196
     1197  module F = grgens( E ); grview(F); print(F);
     1198 
     1199  module G = grpres( F ); grview(G); print(G);
     1200}
     1201
     1202
     1203proc grgens(def M)
     1204"
     1205USAGE:   grgens(M), graded object M (map)
     1206RETURN:  graded object 
     1207PURPOSE: try compute graded generators of coker(M) and return them as columns
     1208         of a graded map.
     1209NOTE:    presentation of resulting generated submodule may be different to M!
     1210EXAMPLE: example grgens; shows an example
     1211"
     1212{
     1213  ASSUME(1, grtest(M) );
     1214
     1215  module N = grtranspose( grsyz( grtranspose(M) ) );
     1216 
     1217//  ASSUME(3, grisequal( grgroebner(M), grgroebner( grpres( N ) ) ) ); // FIXME: not always true!?
     1218 
     1219  return ( N );
     1220}
     1221example
     1222{ "EXAMPLE:"; echo = 2;
     1223
     1224  ring r=32003,(x,y,z),dp;
     1225
     1226  module M = grtwists( intvec(-2, 0, 4, 4) ); grview(M);
     1227
     1228  module N = grgens(M);
     1229 
     1230  grview( N ); print(N); // fine == M
     1231
     1232
     1233  module A = grobj( module([x+y, x, 0, 3], [0, x+y, y, 2], [y, y, z, 1]), intvec(0,0,0,1) );
     1234
     1235  A = grgroebner(A); grview(A);
     1236
     1237  module B = grgens(A);
     1238
     1239  grview( B ); print(B); // Ups :( != A
     1240 
     1241}
     1242
     1243
     1244proc grpres(def M)
     1245"
     1246USAGE:   grpres(M), graded object M (submodule gens)
     1247RETURN:  graded module (via coker)
     1248PURPOSE: compute graded presentation matrix of submodule generated by columns of M
     1249EXAMPLE: example grpres; shows an example
     1250"
     1251{
     1252  ASSUME(1, grtest(M) );
     1253
     1254  module N = grsyz(M);
     1255
     1256//  ASSUME(3, grisequal( M, grgens( N ) ) );
     1257 
     1258  return ( N );
     1259}
     1260example
     1261{ "EXAMPLE:"; echo = 2;
     1262
     1263  ring r=32003,(x,y,z),dp;
     1264
     1265  module M = grtwists( intvec(-2, 0, 4, 4) ); grview(M);
     1266
     1267  module N = grgens(M); grview( N ); print(N);
     1268
     1269  module L = grpres( N ); grview( L ); print(L);
     1270
     1271
     1272  module A = grobj( module([x+y, x, 0, 3], [0, x+y, y, 2], [y, y, z, 1]), intvec(0,0,0,1) );
     1273
     1274  A = grgroebner(A); grview(A);
     1275
     1276  module B = grgens(A); grview( B ); print(B);
     1277
     1278  module C = grpres( B ); grview( C ); print(C);
     1279}
     1280
     1281
    11591282proc grconcat(A, B)
    11601283"
     
    11681291{ "EXAMPLE:"; echo = 2;
    11691292}
     1293
     1294
  • Tst/New/grmodexamples.res.gz.uu

    rde6884 r1bf4403  
    11begin 644 grmodexamples.res.gz
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  • Tst/New/grmodexamples.stat

    rde6884 r1bf4403  
    1 1 >> tst_memory_0 :: 1425497158:4020, 64 bit:4.0.2:x86_64-Linux:dilbert:6314192
    2 1 >> tst_memory_1 :: 1425497158:4020, 64 bit:4.0.2:x86_64-Linux:dilbert:10620928
    3 1 >> tst_memory_2 :: 1425497158:4020, 64 bit:4.0.2:x86_64-Linux:dilbert:10639720
    4 1 >> tst_timer_1 :: 1425497158:4020, 64 bit:4.0.2:x86_64-Linux:dilbert:5566
     11 >> tst_memory_0 :: 1425927388:4020, 64 bit:4.0.2:x86_64-Linux:dilbert:6859600
     21 >> tst_memory_1 :: 1425927388:4020, 64 bit:4.0.2:x86_64-Linux:dilbert:8523776
     31 >> tst_memory_2 :: 1425927388:4020, 64 bit:4.0.2:x86_64-Linux:dilbert:10639728
     41 >> tst_timer_1 :: 1425927388:4020, 64 bit:4.0.2:x86_64-Linux:dilbert:6037
  • Tst/New/grmodexamples.tst

    rde6884 r1bf4403  
    1717
    1818  example grorder;
    19   example grtranspose;
     19  example grtranspose1;
    2020
    2121  example grobj;
     
    3333 
    3434  example grtwists;
     35
     36
     37  example grtranspose;
     38  example grgens;
     39  example grpres;
    3540 
    3641tst_status(1);$
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