source: git/Singular/LIB/gradedModules.lib @ bf75121

spielwiese
Last change on this file since bf75121 was bf75121, checked in by Oleksandr Motsak <motsak@…>, 9 years ago
Changes in documentation and in grtwists
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1//////////////////////////////////////////////////////////////////////////
2version="version gradedModules.lib 4.0.1.1 Jan_2015 "; // $Id$
3category="Commutative Algebra";
4info="
5LIBRARY: gradedModules.lib     Operations with graded modules/matrices/resolutions
6AUTHORS:  Oleksandr Motsak <U@D>, where U=motsak, D=mathematik.uni-kl.de
7@*        Hanieh Keneshlou <hkeneshlou@yahoo.com>
8KEYWORDS: graded modules, graded homomorphisms, syzygies
9OVERVIEW:
10    The library contains several procedures for constructing and manipulating graded modules/matrices/resolutions.
11    Basics about graded objects can be found in [DL].
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).
17NOTE:
18    set assumeLevel to positive integer value in order to auto-check all assumptions.
19    We denote the current basering by S.
20REFERENCES:
21[DL] Decker, W., Lossen, Ch.: Computing in Algebraic Geometry, Springer, 2006
22PROCEDURES:
23    grobj(M,w[,d])  construct a graded object (map) given by matrix M
24    grtest(A)       check whether A is a valid graded object
25    grisequal(A,B)  check whether A is exactly eqal to B? TODO: isomorphic!
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
37    grorder(M)      reorder cols/rows of M for correct graded-block-structure
38    grtranspose1(M)  reordered graded transpose of map M
39    TestGRRes(n,I)  compute/order/transpose a graded resolution of ideal I
40    KeneshlouMatrixPresentation(v)  build some presentation with intvec v
41    grsyz(M)        syzygy of Im(M)
42    grres(M,l[,m])  resolution of Im(M) of length l... minimal?
43    grlift(A,B)     graded lift, gens!
44    grprod(A,B)     composition of graded maps (product of matrices?)
45    grgroebner(M)   Groebner Basis of Im(M) as a graded object
46    grconcat(M,N)   sum of maps into the same target module
47    grrndmat(s,d[,p,b])   generate random matrix compatible with src and dst gradings
48    grrndmap(S,D[,p,b])   generate random 0-deg homomorphism src(S) -> src(D)
49    grrndmap2(S,D[,p,b])   generate random 0-deg homomorphism dst(S) -> dst(D)
50    grlifting(A,B)     RND! chain lifting
51    grlifting2(A,B)    RND! chain lifting
52    mappingcone(M,N)   mapping cone?
53    grlifting3(A,B)    RND! chain lifting? probably wrong one
54    mappingcone3(A,B)  mapping cone3?
55    grrange(M)         get the row-weightings
56    grneg(A)           graded object given by -A
57    matrixpres(a)      matrix presentation of direct sum of Omega^{a[i]}(i)
58";
59
60LIB "matrix.lib"; // ?
61
62//////////////////////////////////////////////////////////////////////////////////////////////////////////
63// . view graded module/map
64// . reorder graded resolution
65// . transpose graded module/map?
66
67// draw helpers
68static proc repeat(int n, string c) { string r = ""; while( n > 0 ){ r = r + c; n--; } return(r); }
69static proc pad(int m, string s, string c){ string r = s; while( size(r) < m ){ r = c + r; } return(r); }
70static proc mstring( int m, string c){ if( m < 0 ) { return (c); }; return (string(m)); }
71
72static proc grsumstr(string R, intvec v)
73"direct sum_i=1^size R(-v[i]), for source and targets of graded objects"
74{
75  if (R == "")
76  {
77    R = nameof(basering);
78  }
79
80  ASSUME(0, defined(R) && (R != "") );
81
82  int n = size(v);
83
84  if (n == 0) { return (R); }
85
86  ASSUME(0, n > 0 );
87 
88  v = -v; // NOTE: due to Mathematical meanings of Singular data
89 
90
91  int lst = v[1];
92  int cnt = 1;
93
94  string p = R;
95  if( lst != 0 ) { p = p + "(" + string(lst) + ")"; }
96
97  int k, d;
98  for (k = 2; k <= n; k++ )
99  {
100    d = v[k];
101    if( d == lst ) { cnt = cnt + 1; }
102    else
103    {
104      if (cnt > 1){ p = p + "^" + string(cnt); }
105
106      cnt = 1; lst = d;
107
108      p = p + " + " + R;
109      if( lst != 0 ) { p = p + "(" + string(lst) + ")"; }
110    }
111  }
112  if (cnt > 1){ p = p + "^" + string(cnt); }
113
114  return (p);
115}
116example
117{ "EXAMPLE:"; echo = 2;
118
119  ring r=32003,(x,y,z),dp;
120
121  def E = grtwist(2, 0);
122  def v = grrange(E); // grdeg(E);
123  grsumstr("", v ); 
124}
125
126// view helper
127static proc draw ( intmat D, int d )
128{
129//  print(D); return ();
130  int nc = ncols(D); int nr = nrows(D);
131  int s, r, c; int max = 0;
132  // get maximum string-length among all {D[r,c]}
133  for (r = nr; r > 0; r-- ) { for (c = nc; c > 0; c-- ) { s = size( string(D[r, c]) ); if( max < s ) { max = s; } } }
134  max = max + 1;
135  string head = ""; string foot = ""; string middle = "";
136  for ( c = d+1; c < (nc-d); c++ )
137  {
138    head = head + pad(max, string(D[1 , c]), ".") + " ";
139    foot = foot + pad(max, string(D[nr, c]), " ") + " ";
140  }
141  // last head/foot enties:
142  head = head + pad(max, string(D[1 , c]), ".");
143  foot = foot + pad(max, string(D[nr, c]), " ");
144  // head/foot dash lines:
145  string dash  = "-"; string dash2  = "=";
146  dash = repeat( (nc - 2*d) - 1 , repeat(max, dash) + " " ) + repeat(max, dash) + " "; // dash  = repeat( (max + 1) * (nc - 2*d) , dash );
147  dash2 = repeat( (nc - 2*d) - 1 , repeat(max, dash2) + " " ) + repeat(max, dash2) + " "; // dash2 = repeat( (max + 1) * (nc - 2*d) , dash2);
148  for ( r = d+1; r <= (nr-d); r++ )
149  {
150    middle = middle + pad(max, string(D[r,1]), " ") + " :";
151    for ( c = d+1; c < (nc-d); c++ ) { middle = middle + pad(max, mstring(D[r,c], "-"), " ") + " "; }
152    middle = middle + pad(max, mstring(D[r,nc-d], "-"), " ") + " |" + pad(max, string(D[r,nc]), ".") + newline;
153  }
154  string corner_id = repeat(max, ".");
155  string corner = repeat(max, " ");
156  // print everything all at once:
157  print                                     (
158      corner + "  " + head  + " ." + corner_id + newline +
159      corner + "  " + dash  + "+"  + corner_id + newline +
160                      middle                          +
161      corner + "  " + dash2 + " "  + corner + newline +
162      corner + "  " + foot +  "  " + corner );
163}
164
165proc grview(N)
166"USAGE:  grview(M), graded object M
167RETURN:  nothing
168PURPOSE: print the degree/grading data about the GRADED matrix/module/ideal/mapping object M
169ASSUME:  M must be graded
170EXAMPLE: example grview; shows an example
171"
172{
173//  if( size(N) == 0 ) { return (); }
174  string msg = "Graded";
175  string lst;
176
177  string arrow = " <- ";
178  string R = nameof(basering);
179
180  if( typeof( N ) == "list" )
181  {
182    msg = msg + " resolution";
183    if( size(R) >= 2 )
184    {
185      msg = msg + "(let R:="+R+")";
186      R = "R";
187    }
188
189
190    int i = 1;  int n = size(N);
191    string dst;
192   
193    msg = msg + ": " + newline + grsumstr(R, grrange(N[i])) + " <-- d_" + string(i) + " -- " ;
194    for( ; i < n; i++ )
195    {
196      dst = grsumstr(R, grdeg(N[i])) +  " <-- d_" + string(i+1) + " --";
197      msg = msg + newline + dst;
198    };
199
200    msg = msg + newline + grsumstr(R, grdeg(N[i])) + ", given by maps: ";
201
202    print(msg);
203
204    for( i = 1; i <= size(N); i++ )
205    {
206      "d_" + string(i) + " :"; grview(N[i]);
207    };
208
209    return ();
210  }
211
212//  typeof( N ) ;  attrib( N );  grrange(N);
213
214  ASSUME(1, grtest(N) );
215
216  intvec G = grdeg(N);
217  matrix M = module(N);
218
219  int nc = ncols(M); int nr = nrows(M);
220  int r,c;
221  int d = 1; // number of extra cols/rows for extra info around the central degree(N) block in D
222  intmat D[nr+2*d][nc+2*d];
223
224  for( c = nc; c > 0; c-- )
225  {
226    D[1, c+d] = c; // top row indeces
227    D[nr+2*d, c+d] = G[c]; // deg(v) + gr[ leadexp(v)[m] ]; // bottom row with computed column induced degrees
228  }
229
230  intvec gr = grrange(N); // grading weights?
231
232  for( r = nr; r > 0; r-- )
233  {
234    D[r+d, 1] = gr[r]; // left-most column with grading data
235    for( c = nc; c > 0; c-- )
236    {
237      D[r+d, c+d] = deg(M[r, c]); // central block with degrees (-1 means zero entry)
238    }
239    D[r+d, nc+2*d] = r; // right-most block with indeces
240  }
241
242  msg = msg + " homomorphism";
243  if( size(R) >= 2 )
244  {
245    msg = msg + "(let R:="+R+")" ;
246    R = "R";
247  }
248   
249  msg = msg + ": ";
250
251  string dst = grsumstr(R, gr);
252  string src = grsumstr(R, G);
253
254  lst = msg;
255
256  if( (size(lst) + size(dst) + size(src) + 4) > 80 )
257  {
258    if( (size(lst) + size(dst)) > 80 ) { msg = msg + newline; lst = ""; }
259
260    msg = msg + dst + arrow;
261    lst = lst + dst + arrow;
262
263    if( (size(lst) + size(src)) > 80 ) { msg = msg + newline; lst = ""; }
264
265    msg = msg + src;
266    lst = lst + src;
267  } else
268  {
269    msg = msg + dst + arrow + src;
270    lst = lst + dst + arrow + src;
271  }
272
273  if( size(lst) > 70 ) { msg = msg + newline; } // lst = "";
274  msg = msg + ", given by ";
275//  lst = lst + ", given by ";
276
277  if( size(N) == 0 )
278  {
279    msg = msg + "zero ("+ string(nr);
280    if( nr == nc ) { msg = msg + "^2"; } else { msg = msg + " x " + string(nc); }
281    print( msg+") matrix." );
282  } else
283  {
284    if( nr == nc) // square matrix // detect diagonal?
285    {
286      for (c = nr; c > 0; c-- )
287      {
288        M[c,c] = 0;
289      }
290
291      if( size(module(M)) == 0 )
292      {
293        msg = msg + "a diagonal matrix";
294      } else
295      {
296        msg = msg + "a square matrix";
297      }
298
299    } else
300    {
301      msg = msg + "a matrix";
302    }
303   
304    print(msg + ", with degrees: " );
305    draw(D, d); // print it nicely!   
306  }
307}
308example
309{ "EXAMPLE:"; echo = 2;
310
311  ring r=32003,(x,y,z),dp;
312
313  module A = grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0,0,1) );
314  grview(A);
315
316  module B = grobj( module([0,x,y]), intvec(15,1,1) );
317  grview(B);
318
319  module D = grsum( grsum(grpower(A,2), grtwist(1,1)), grsum(grtwist(1,2), grpower(B,2)) );
320  grview(D);
321
322  ring R = 0,(w,x,y,z), dp; def I = grobj( ideal(y2-xz, xy-wz, x2z-wyz), intvec(0) );
323  list res1 = grres(I, 0); // non-minimal
324  grview(res1);
325  print(betti(res1,0), "betti");
326
327  list res2 = grres(grshift(I, -10), 0, 1); //  minimal!
328  grview(res2);
329  print(betti(res2,0), "betti");
330}
331
332static proc issorted( intvec g, int s )
333{
334  g = s * g; //  "g: ", g;
335  int i = size(g);
336
337  for(; i > 1; i--)
338  {
339    if( (g[i] - g[i-1]) < 0 )
340    {
341      return (0);
342    }
343  }
344
345  return (1);
346}
347
348static proc mysort( intvec gr, int s )
349"
350computes the permutation P of gr, such that (s*gr)[P] is ascendingly sorted
351NOTE: looks like a bubble sort (was taken from sort) and modified to ensure stability!
352TODO: replace with some kernel function if this turns out to be inefficient!!
353"
354{
355  gr = s * gr;
356  int m = size(gr);
357  intvec pivot;
358
359  int Bi;
360  // compute reordering permutation pivot such that gr[pivot] is (stably) sorted
361  for(Bi=m; Bi>0; Bi--) { pivot[Bi]=Bi; } // pivot = Id_m for starters
362
363  int Bn,Bb; int P, D;
364
365  int Bj = 0; int flag = 0;
366
367  // Bi == 0
368  while(Bj==0)
369  {
370    Bi++; Bj=1;
371    for(Bn=1; Bn <= (m-Bi); Bn++)
372    {
373      D = (gr[pivot[Bn]] - gr[pivot[Bn+1]]); // sort gr
374      P = (D > 0) or ( (D == 0) && ((pivot[Bn]-pivot[Bn+1]) > 0) ); // stability!?
375      if(P)
376      {
377        Bb=pivot[Bn];
378        pivot[Bn]=pivot[Bn+1];
379        pivot[Bn+1]=Bb;
380
381        Bj=0; flag = 1;
382      }
383    }
384  }
385
386/*
387  if( flag ) // output details in case of non-identical permutation?
388  {
389    // grades & ordering permutation : gr[pivot] should be sorted:
390    "s: ", s;
391    "gr: ", gr;
392    "pivot: ",    pivot;
393  }
394*/
395  ASSUME(1, issorted(intvec(gr[pivot]), 1));
396
397  return (pivot);
398}
399
400// Q@Wolfram: what should be done with zero gens!?
401proc grdeg(M)
402"USAGE:  grdeg(M), graded object M
403RETURN:  intvec of degrees
404PURPOSE: graded degrees of columns (generators) of M
405ASSUME:  M must be a graded object (matrix/module/ideal/mapping)
406NOTE:    if M has zero cols it shoud have attrib(M,'degHomog') set.
407EXAMPLE: example grdeg; shows an example
408"
409{
410  ASSUME(1, grtest(M) );
411
412  if ( typeof(attrib(M, "degHomog")) == "intvec" )
413  {
414    intvec t = attrib(M, "degHomog"); // graded degrees
415    ASSUME(0, ncols(M) == size(t) );
416
417    return (t);
418  }
419
420  ASSUME(0, ncols(M) == size(M) );
421
422  def w = grrange(M); // grading weights?
423
424  if( size(M) == 0 ){ return (w); } // TODO: Q@Wolfram!???
425
426  int m = ncols(M); // m > 0 in Singular!
427  int n = nvars(basering) + 1; // index of mod. column in the leadexp
428
429  module L = lead(M[1..m]); // leading module-terms for input column vectors
430  intvec d = deg(L[1..m]); // their degrees
431  intvec c = leadexp(L[1..m])[n]; // their module-components
432
433//   w = intvec(-6665), w; // 0?????
434  intvec gr = w[c]; //  + 1]; // weights?????
435
436  gr = gr + d; // finally we compute their graded degrees
437
438  return (gr);
439}
440example
441{ "EXAMPLE:"; echo = 2;
442
443  ring r=32003,(x,y,z),dp;
444
445  module A = grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0,0,1) );
446  grview(A);
447
448  module B = grobj( module([0,x,y]), intvec(15,1,1) );
449  grview(B);
450
451  module D = grsum(
452                   grsum(grpower(A,2), grtwist(1,1)),
453                   grsum(grtwist(1,2), grpower(B,2))
454                  );
455
456  grview(D);
457  grdeg(D);
458
459  def D10 = grshift(D, 10);
460
461  grview(D10);
462  grdeg(D10);
463}
464
465static proc reorder(def M, int s)
466"
467Reorder gens of M: compute graded degrees and the permutation to sort them
468"
469{
470  // input should be graded:
471  ASSUME(1, grtest(M) );
472
473  intvec w = grrange(M); // grading weights
474
475  intvec gr = grdeg( M );
476
477//  intvec d = deg(M[1..nocls(M)]); // no need to deal with un-weighted degrees??!
478
479  intvec pivot = mysort(gr, s);
480
481  // grades & ordering permutation for N.  gr[pivot] should be sorted!
482  ASSUME(1, issorted(gr[pivot], s));
483
484  module N = grobj(module(M[pivot]), w, intvec(gr[pivot]));  // reorder the starting ideal/module
485
486//  "reorder: "; grview(N);
487
488  return (N, intvec(gr[pivot]));
489}
490
491proc grtranspose1(def M)
492"USAGE:  grtranspose1(M), graded object or list M
493RETURN:  same as input
494PURPOSE: graded transpose of graded object or chain complex M
495ASSUME:  M must be a graded object or a list of graded objects
496EXAMPLE: example grtranspose1; shows an example
497"
498{
499  if( typeof( M ) == "list" )
500  {
501    if( size(M) == 0 ) { return (); }
502
503    int j = size(M);
504
505    int i = 1;
506
507    // TODO: extra grading argument???
508    while( i < j )
509    {
510     if( size(M[i]) == 0 ){ break; }
511     ASSUME(0, typeof(grrange(M[i])) == "intvec");
512     i++;
513    }
514
515    if( size(M[i]) == 0 ) { i--; }
516
517    j = i; i = 1;
518
519    list L;
520    while( j > 0 )
521    {
522//      grview(M[i]);
523      L[j] = grtranspose1( grobj( M[i], grrange(M[i])) );
524//      grview(L[j]);
525
526      if( (i > 1) && (j > 0) )
527      {
528//        grview(L[j+1]);
529        ASSUME(2, size( module( matrix(L[j])*matrix(L[j+1]) ) ) == 0 );
530      };
531//      grview(L[j]);
532      j--; i++;
533    };
534    return (L); // ?
535  }
536
537//////
538// "a";  grview(M);
539  ASSUME(1, grtest(M) );
540
541 intvec d; module N;
542
543 (N,d) = reorder(M, -1);
544
545 kill M; module M = grobj(transpose(N), -d, -grrange(N));
546
547// "b";  grview(M);
548
549 kill N,d; module N; intvec d;
550 // reverse order:
551 (N,d) = reorder(M, 1); kill M;
552
553// "e"; grview( N );
554
555 ASSUME(1, issorted( grrange(N), 1) );
556 ASSUME(1, issorted(grdeg(N), 1) );
557
558
559 return (N);
560}
561example
562{ "EXAMPLE:"; echo = 2;
563
564  "Surface Name: 'k3.d10.g9.quart2' in P^4";
565  int @p=31991; ring R = (@p),(x,y,z,u,v), dp;
566  ideal J = 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567
568  def I = grobj( groebner(J), intvec(0) ); // ASSUME: no zero entries in J!
569  ASSUME(0, grtest(I));
570  "Input degrees: "; grview(I);
571
572  def RR = grres(I, 0, 1); list L = RR;
573
574  " = Non-minimal betti numbers: "; print(betti(L, 0), "betti");
575
576  "Graded (original) structure of 'res(Input,0)': "; grview(L);
577
578  "Graded transpose of the previous resolution "; list LLL = grtranspose1( L ); grview( LLL );
579
580  "Its non-minimal betti numbers: "; print(betti(LLL, 0), "betti");
581
582}
583
584proc grorder(def M)
585"USAGE:  grorder(M), graded object or list M
586RETURN:  same as input
587PURPOSE: reorder/transform graded object or chain complex M into block form
588ASSUME:  M must be a graded object or a list of graded objects
589EXAMPLE: example grorder; shows an example
590"
591{
592  if( typeof(M) == "list" )
593  { // TODO: extra grading argument???
594    if( size(M) == 0 ) { return (); }
595
596    int j = size(M);  int i = 1;
597
598    while( i < j )
599    {
600      if( size(M[i]) == 0 ){ break; }
601      ASSUME(0, typeof(grrange(M[i])) == "intvec");
602      i++;
603    }
604
605    if( size(M[i]) == 0 ) { i--; }
606
607    list L; module Z = 0; L[i] = Z; j = i;
608
609    while( i > 0 )
610    {
611//      "i: ", i;      "A"; grview(M[i]);
612      L[i] = grorder( grobj( M[i], grrange(M[i])) );
613//      "B"; grview(L[i]);
614      if( i < j )
615      {
616        ASSUME(2, size( module( matrix(transpose(L[i+1]))*matrix(transpose(L[i])) ) ) == 0 );
617      };
618
619      i--;
620    };
621
622    return (L); // ?
623  }
624
625  ASSUME(1, grtest(M) );
626
627// "a";  grview(M);
628
629  intvec d; module N;
630
631  (N,d) = reorder(M, 1); kill M;
632
633  module M = grobj(transpose(N), -d, -grrange(N)); kill N,d;
634
635// "b";  grview(M);
636
637  module N; intvec d;
638  // reverse order:
639  (N,d) = reorder(M, -1); kill M;
640
641  module M = grobj(transpose(N), -d, -grrange(N));
642
643// "c";  grview(M);
644
645  ASSUME(1, issorted(grrange(M), 1) );
646  ASSUME(1, issorted(grdeg(M), 1) );
647
648  return (M);
649}
650example
651{ "EXAMPLE:"; echo = 2;
652
653  "Surface Name: 'rat.d10.g9.quart2' in P^4";
654  int @p=31991; ring R = (@p),(x,y,z,u,v), dp;
655  ideal J = 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656
657  def I = grobj( groebner(J), intvec(0) ); // ASSUME: no zero entries in J!
658  ASSUME(0, grtest(I));
659
660  "Input degrees: "; grview(I);
661
662  def RR = grres(I, 0, 1);
663  list L = RR;
664
665  " = Non-minimal betti numbers: ";  print(betti(L, 0), "betti");
666  "Graded reordered structure of 'res(Input,0)': ";  grview(grorder(L));
667}
668
669proc TestGRRes(Name, J)
670"USAGE:  TestGRRes(name, I), string name, ideal I
671RETURN:  nothing
672PURPOSE: compute/test/output/order/transpose a graded resolution of I
673EXAMPLE: example TestGRRes; shows an example
674"
675{
676  "==============================================";
677  "";
678  "=== Example: [", Name, "]";
679  " = Ring: ", string(basering);
680
681  def I = grobj( groebner(J), intvec(0) ); // ASSUME: no zero entries in J!
682  ASSUME(0, grtest(I));
683//  " = Input degrees: "; grview(I);
684
685  " ! Resolution via 'grres': ";
686  def R = grres(I, 0, 1); // sres, lres: no grading! // nres, mres - graded (with attrib(, "isHomog"))
687
688  " = Non-minimal betti numbers: ";
689  print(betti(R, 0), "betti");
690
691  list L = R ; // SRES_list(R); //  " = Degrees of maps: "; grview(L); // MUST BE GRADED!!!
692
693  " = Degrees of (ordered) maps: ";
694  def LL = grorder(L); // MUST BE GRADED
695
696  // ordres(L, intvec(0)); // ?
697
698  grview( LL ); " = TRANSPOSE'd complex: %%%%%%%%%%%%%%";
699  list LLL = grtranspose1( LL ); //  resolution RR = LLL;
700  print(betti(LLL, 0), "betti");
701
702  grview( LLL ); ""; //  "==============================================";
703
704  kill L, R;
705}
706example
707{ "EXAMPLE:"; echo = 2;
708//  if( defined(assumeLevel) ){ int assumeLevel0 = assumeLevel; } else { int assumeLevel; export(assumeLevel); }; assumeLevel = 5; // store the state of aL
709
710  // note: data from random generation 2
711  string Name = "castelnuovo"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = 5153xy2-98/23y3-101/51xyz+33/41y2z+99/79xz2+7136yz2-106/111z3+119/53xyu+34/57y2u-77/92xzu+84/73yzu-109/78z2u-27/56xu2+10023yu2+82/103zu2-34/25u3+3/2xyv-68/25y2v+12721xzv+4/63yzv-73/21z2v-7291xuv-91/53yuv-4/79zuv-34/91u2v-122/53xv2+123/70yv2-64/73zv2+44/65uv2+14/31v3,xy2-15202y3+10613xyz+13640y2z-107/103xz2+5292yz2+19/119z3-10042xyu+2770y2u+7957xzu+14008yzu+92/121z2u-92/51xu2+1178yu2+1/117zu2-12726u3+82/101xyv-92/17y2v-107/56xzv+14233yzv+79/28z2v+51/50xuv-31/5yuv+95/91zuv+19/108u2v+12151xv2-69/110yv2+37/89zv2-63/116uv2-88/23v3,-5153x2+37/23xy+8706y2-13160xz+68/115yz+5548z2-22/61xu-113/98yu+11818zu+2114u2-101/97xv+89/22yv-3355zv-113/5uv-5521v2;TestGRRes(Name, I); kill R, Name, @p;  "";
712
713  string Name = "ell.d8.g7"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = x2y2-47/69xy3+6059y4+78/85x2yz+55/124xy2z+13641y3z+8/17x2z2+7817xyz2-2746y2z2+85/124xz3+87yz3+13182z4+109/93x2yu-69/17xy2u+12089y3u+8769x2zu-53/36xyzu-14834y2zu+123/23xz2u+103/77yz2u-2344z3u-43/104x2u2-6198xyu2+47/115y2u2-39/19xzu2-29/24yzu2+51/89z2u2-65/37xu3-95/94yu3+11302zu3-53/57u4-2874x2yv+4347xy2v-25/77y3v+13819x2zv+29/34xyzv+474y2zv+33/107xz2v-3517yz2v+10617z3v+1834x2uv+54/113xyuv-8751y2uv+111/70xzuv-66/61yzuv+9195z2uv-14289xu2v-13/110yu2v+103/9zu2v+5113u3v+116/89x2v2+15142xyv2+13078y2v2-38/41xzv2-13/113yzv2-12824z2v2-57/11xuv2-114/17yuv2-125/31zuv2+11939u2v2+44/13xv3+56/69yv3+12/125zv3+643uv3+3530v4,-3454x2y-1285xy2-6182y3-8/69x2z+9/19xyz+64/49y2z+98/67xz2-13809yz2+21/44z3+77/47x2u+748xyu-41/77y2u+7318xzu+4217yzu+12562z2u-98/69xu2-14/85yu2+119/46zu2-61/121u3+5582x2v+108/77xyv-93/4y2v-65/49xzv-4135yzv+2477z2v+11114xuv+85/14yuv+51/125zuv-7572u2v-115/52xv2-7647yv2+4647zv2-5684uv2-1/55v3,3454x3-6645x2y-43/34xy2+14590y3+8/11x2z-117/112xyz+109/54y2z+6566xz2+23/57yz2-13078z3+95/61x2u+67/40xyu-4544y2u-95/72xzu-8/103yzu+100/77z2u+23/63xu2+69/61yu2-94/105zu2+8619u3+68/123x2v+8/117xyv+101/77y2v+124/125xzv+17/84yzv+23/67z2v+18/59xuv+3216yuv-77/59zuv-9/50u2v+96/109xv2-2491yv2+14089zv2+14067uv2-56/113v3;TestGRRes(Name, I); kill R, Name, @p;  "";
714
715  string Name = "ell.d7.g6"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = 4971xy3+3/101y4-12318xy2z-12835y3z+97/98xyz2+63y2z2-8056xz3+23/91yz3-9662z4-7398xy2u+69/71y3u-53/68xyzu-49/67y2zu-113/122xz2u-9/61yz2u+71/88z3u+11358xyu2-38/29y2u2-10232xzu2+14490yzu2+2274z2u2+3501xu3+10427yu3-109/38zu3-99/5u4-6605xy2v-1555y3v-648xyzv-2083y2zv-61/41xz2v+75/17yz2v-69/55z3v-6104xyuv-9582y2uv+69/2xzuv-12551yzuv+47/49z2uv-118/13xu2v+34/105yu2v+105/41zu2v+6533u3v+122/25xyv2+2/43y2v2+16/61xzv2+11524yzv2+113/99z2v2-71/26xuv2+7809yuv2-4865zuv2-2122u2v2+53/118xv3-13209yv3-11106zv3-49/79uv3+3006v4,xy3+15492y4-13742xy2z+112/117y3z+6/47xyz2+28/41y2z2+71/111xz3+49/57yz3-61/44z4-11759xy2u+4242y3u-109/18xyzu+2260y2zu-6873xz2u-41/112yz2u+12574z3u-10939xyu2+119/38y2u2-62/33xzu2-3699yzu2+2651z2u2-13194xu3-15185yu3-11/116zu3-61/83u4-10094xy2v+13/4y3v-74/73xyzv+43/20y2zv-11547xz2v+53/43yz2v-92/93z3v+32/41xyuv+118/33y2uv-121/39xzuv-15913yzuv+53/11z2uv+97/76xu2v+85/29yu2v-5183zu2v+8520u3v+121/28xyv2+64/51y2v2-15810xzv2+1/43yzv2-6160z2v2+13988xuv2+9/40yuv2+123/4zuv2+15024u2v2+73/95xv3+80/97yv3+57/25zv3-109/81uv3-121/87v4,-4971x2+14389xy+1607y2+59/119xz+12020yz+103/122z2+8894xu+7091yu+54/19zu-50/77u2+28/25xv-113/56yv+68/29zv-14620uv+79/107v2;TestGRRes(Name, I); kill R, Name, @p;  "";
716
717  string Name = "k3.d7.g5"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = -97/108x2y-31/118xy2-73/61y3-79/14x2z-15930xyz-2324y2z+1842xz2+656yz2-8852z3-89/38x2u-102/43xyu+14719y2u+70/67xzu+7335yzu+27/56z2u-10744xu2-55/83yu2+120/73zu2+120/61u3-126/125x2v+691xyv-15385y2v+117/16xzv-17/97yzv+80/121z2v-48/119xuv+21/34yuv-103/65zuv-49/32u2v-41/42xv2+11/75yv2-502zv2-7583uv2+26/69v3,97/108x3+77/114x2y+71/21xy2+13679y3-1645x2z-1/33xyz-79/7y2z-52/53xz2+11940yz2-5800z3+109/13x2u-115/64xyu-125/56y2u-2365xzu+2103yzu+56/87z2u-84/79xu2+107/106yu2-79/70zu2-419u3+5354x2v+92/53xyv-32/19y2v+11/74xzv+4193yzv+45/79z2v-113/72xuv+17/71yuv+11164zuv-17/33u2v+103/66xv2+55/79yv2+118/15zv2-2646uv2+57/106v3,x3-61/113x2y-64/21xy2-107/8y3-13/60x2z+43/35xyz+41/114y2z-13683xz2-5829yz2+71/38z3+90/17x2u-39/29xyu+42/5y2u-61/55xzu+111/77yzu-87/100z2u+10735xu2-83/91yu2-4884zu2-7965u3-65/12x2v+109/86xyv+10606y2v-14164xzv-6678yzv+83/18z2v-93/10xuv+120/49yuv-1592zuv-8710u2v-73/57xv2+10762yv2-2956zv2-89/63uv2-12/7v3;TestGRRes(Name, I); kill R, Name, @p;  "";
718
719  string Name = "rat.d8.g6"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = -19/125x2y2-87/119xy3-97/21y4+36/53x2yz+2069xy2z-59/50y3z-65/33x2z2-14322xyz2+79/60y2z2-9035xz3-14890yz3+87/47z4-23/48x2yu+45/44xy2u+1972y3u+79/118x2zu-5173xyzu+115/121y2zu+1239xz2u-115/17yz2u-15900z3u-78/95x2u2+67/101xyu2-12757y2u2+12752xzu2+68/21yzu2+103/90z2u2-12917xu3+97/92yu3-24/49zu3-13/79u4-51/61x2yv-3103xy2v+77/117y3v+73/115x2zv-79/33xyzv+123/110y2zv+11969xz2v-31/95yz2v-123/95z3v-105/124x2uv+12624xyuv+2/63y2uv+6579xzuv+13/62yzuv+4388z2uv-12747xu2v-26/105yu2v-78/61zu2v-125/53u3v-5/71xyv2+62/77y2v2+21/44xzv2-9806yzv2+3/91z2v2+361xuv2+568yuv2+2926zuv2+53/38u2v2-14523yv3+2082zv3+113/115uv3,108/73x2y2+4028xy3+38/43y4-1944x2yz+39/80xy2z+8/109y3z+52/27x2z2+103/45xyz2+5834y2z2+63/101xz3+107/80yz3+1178z4-1/6x2yu+78/25xy2u-21/43y3u+50/71x2zu-14693xyzu+15074y2zu+9/103xz2u-7396yz2u-14493z3u+93/25x2u2+61/4xyu2-11306y2u2-79/81xzu2+59/82yzu2-5/106z2u2+89/71xu3-34/11yu3+15/103zu3-115/52u4-54/65x2yv+67/16xy2v-7/68y3v-10/13x2zv+32/85xyzv+1/91y2zv+107/118xz2v+7594yz2v-98/103z3v+9919x2uv-965xyuv+53/34y2uv+119/11xzuv-3400yzuv-8329z2uv+75/98xu2v-24yu2v+55/87zu2v-82/71u3v-73/115x2v2+85/19xyv2-213y2v2-7704xzv2-15347yzv2+14960z2v2+15065xuv2-125/17yuv2+32/83zuv2-14/73u2v2-21/44xv3+79/2yv3-61/32zv3+46/119uv3-2082v4,9/20x2y2+113/71xy3-88/65y4+9983x2yz-6722xy2z+87/68y3z+1893x2z2+65/32xyz2+51/55y2z2-102/53xz3+58/5yz3-7187z4-96/7x2yu-14/87xy2u-3532y3u+95/54x2zu+19/65xyzu-6728y2zu+31/121xz2u+73/106yz2u-91/5z3u-12928x2u2+707xyu2-55/48y2u2-96/25xzu2+15869yzu2-20/107z2u2-10030xu3-13786yu3-122/9zu3+19/59u4-7/52x2yv+101/74xy2v+83/6y3v-91/55x2zv-5266xyzv+85/61y2zv+126/95xz2v+56/51yz2v+13073z3v-50/21x2uv-13553xyuv-116/53y2uv+68/71xzuv-111/98yzuv-11037z2uv+68/121xu2v-124/53yu2v+54/55zu2v+5862u3v+12318x2v2-119/29xyv2+101/17y2v2-51/40xzv2-82/33yzv2-30/41z2v2-29/52xuv2+7817yuv2+8121zuv2-28/99u2v2+1125xv3-73/55yv3-14141zv3+8742uv3-1203v4,x2y2+11357xy3+295y4+144x2yz-31/54xy2z+89/119y3z+1/46x2z2+29/26xyz2+1384y2z2+1461xz3+113/91yz3+9494z4-7/32x2yu+12850xy2u-3626y3u-33/106x2zu-7/60xyzu-5935y2zu-8597xz2u+5527yz2u+1708z3u+6182x2u2-15780xyu2+4669y2u2-38/69xzu2+8412yzu2+9265z2u2-5679xu3-67/18yu3-34/67zu3-7178u4+113/56x2yv-3669xy2v+17/113y3v-87/35x2zv-4871xyzv-111/11y2zv-1131xz2v-72/13yz2v+838z3v-115/4x2uv+3395xyuv-43/68y2uv-82/13xzuv+7042yzuv-88/119z2uv+100/19xu2v+24/11yu2v+89/3zu2v+7395u3v-119/109x2v2+1/104xyv2+18/25y2v2+700xzv2-59/9yzv2-92/87z2v2+2486xuv2-67/103yuv2+1469zuv2-101/91u2v2-79/33xv3+10838yv3+81/4zv3-11843uv3+7204v4,19/125x3-15698x2y-22/117xy2-95/107y3+2027x2z-7750xyz+85/104y2z-15326xz2+31/101yz2+67/81z3-7879x2u-112/115xyu+124/81y2u+99/61xzu-7458yzu+40/33z2u-1502xu2+6591yu2-7/73zu2-42/95u3+93/83x2v-15/112xyv-84/95y2v+35/36xzv+5/24yzv-12768z2v+13232xuv-76/103yuv-79/52zuv-7217u2v+75/92xv2-49/64yv2+17/14zv2-6109uv2+1695v3;TestGRRes(Name, I); kill R, Name, @p; "";
720
721  string Name = "k3.d14.g19"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = 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I); kill R, Name, @p;  "";
722
723  string Name = "k3.d11.g11.ss0"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = 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I); kill R, Name, @p; "";
724
725  string Name = "ell.d10.g9"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = 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I); kill R, Name, @p; "";
726
727  string Name = "k3.d10.g9.quart2"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = 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I); kill R, Name, @p; "";
728
729  string Name = "rat.d10.g9.quart2"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = 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I); kill R, Name, @p; "";
730
731//  if( defined(assumeLevel0) ){ assumeLevel = assumeLevel0; } else { kill assumeLevel; } // restore the state of aL
732}
733
734/////////////////////////////////////////////////////////
735
736// Q@Woflram?
737proc grzero()
738"USAGE:  grzero()
739RETURN:  graded object representing S(0)^1
740PURPOSE: compute presentation of S(0)^1
741EXAMPLE: example grzero; shows an example
742"
743{
744 return ( grobj(module([0]), intvec(0), intvec(0)) );
745}
746example
747{ "EXAMPLE:"; echo = 2;
748
749  ring r=32003,(x,y,z),dp;
750
751  def M = grpower( grshift( grzero(), 10), 5 );
752
753  print(M);
754
755  grview(M);
756}
757
758proc grpower(def A, int p)
759"USAGE:  grpower(A, p), graded object A, int p > 0
760RETURN:  graded direct power A^p
761PURPOSE: compute the graded direct power A^p
762NOTE:    the power p must be positive
763EXAMPLE: example grpower; shows an example
764"
765{
766  if(p==0){ ERROR("Sorry, we don't know what is A^0!?!?"); } // grzero!?
767
768  ASSUME(0, p > 0);
769  ASSUME(1, grtest(A) );
770
771  if(p==1){ return(A); }
772
773  def N = grsum(A,A);
774
775  if(p==2){ return(N); }
776
777  // TODO: replace recursion with a loop!
778  // see http://en.wikipedia.org/wiki/Exponentiation_by_squaring
779  if((p%2)==0)
780    { return ( grpower(N, p div 2) ); }
781  else
782    { return ( grsum( A, grpower(N, (p-1) div 2) )); }
783}
784example
785{ "EXAMPLE:"; echo = 2;
786
787  ring r=32003,(x,y,z),dp;
788
789  module A = grobj( module([x+y, x, 0], [0, x+y, y]), intvec(1,1,1) );
790  grview(A);
791
792  module B = grobj( module([x,y]), intvec(2,2) );
793  grview(B);
794
795  module D = grsum( grpower(A,2), grpower(B,2) );
796
797  print(D);
798  homog(D);
799  grview(D);
800}
801
802
803proc grsum(A,B)
804"USAGE:  grsum(A, B), graded objects A and B
805RETURN:  graded direct sum of input objects
806PURPOSE: compute the graded direct sum of A and B
807EXAMPLE: example grsum; shows an example
808"
809{
810  ASSUME(1, grtest(A) );
811  ASSUME(1, grtest(B) );
812
813  intvec a = grrange(A);
814  intvec b = grrange(B);
815  intvec c = a,b;
816  int r = nrows(A);
817
818  module T = align(module(B), r); //  T;  print(T);  nrows(T); // BUG!!!!
819  module S = module(A), T;
820
821  intvec da = grdeg(A);
822  intvec db = grdeg(B);
823  intvec dc = da, db;
824
825  return(grobj(S, c, dc));
826}
827example
828{ "EXAMPLE:"; echo = 2;
829
830//  if( defined(assumeLevel) ){ int assumeLevel0 = assumeLevel; } else { int assumeLevel; export(assumeLevel); }; assumeLevel = 5;
831
832  ring r=32003,(x,y,z),dp;
833
834  module A = grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0,0,1) );
835  grview(A);
836
837  module B = grobj( module([0,x,y]), intvec(15,1,1) );
838  grview(B);
839
840  module C = grsum(A,B);
841
842  print(C);
843  homog(C);
844  grview(C);
845
846  module D = grsum(
847     grsum(grpower(A,2), grtwist(1,1)),
848     grsum(grtwist(1,2), grpower(B,2))
849     );
850
851  print(D);
852  homog(D);
853  grview(D);
854
855  module F = grobj( module([x,y,0]), intvec(1,1,5) );
856  grview(F);
857
858  module T = grsum( F, grsum( grtwist(1, 10), B ) );
859  grview(T);
860
861//  if( defined(assumeLevel0) ){ assumeLevel = assumeLevel0; } else { kill assumeLevel; } // restore the state of aL
862}
863
864proc grshift( def M, int d)
865"USAGE:  grshift(A, d), graded objects A, int d
866RETURN:  shifted graded object
867PURPOSE: shift the grading on A by d: A(i) -> A(i+d)
868EXAMPLE: example grshift; shows an example
869"
870{
871  ASSUME(1, grtest(M) );
872
873  intvec a = grrange(M);
874  a = a - intvec(d:size(a));
875
876  intvec t = attrib(M, "degHomog");
877  t = t - intvec(d:size(t));
878
879  return (grobj(M, a, t));
880}
881example
882{ "EXAMPLE:"; echo = 2;
883
884  ring r=32003,(x,y,z),dp;
885
886  module A = grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0,0,1) );
887
888  grview(A);
889
890  module S = grshift( A,  6);
891
892  grview(S);
893}
894
895
896proc grisequal (def A, def B)
897"USAGE:  grisequal(A, B), graded objects A and B
898RETURN:  1 if A == B as graded objects, 0 otherwise
899PURPOSE: test the equality of two graded object
900NOTE: A and B should be literarly the same at the moment. TODO?
901EXAMPLE: example grisequal; shows an example
902"
903{
904  ASSUME(1, grtest(A) );
905  ASSUME(1, grtest(B) );
906
907  int ra = nrows(A);
908  int rb = nrows(B);
909
910  intvec wa = grrange(A);
911  intvec wb = grrange(B);
912
913  if( (ra != rb) || (ncols(A) != ncols(B)) ){ return (0); } // TODO: ???
914
915  intvec da = grdeg(A);
916  intvec db = grdeg(B);
917
918  return ( (da == db) && (wa == wb) &&
919           (size(module(matrix(A) - matrix(B))) == 0)    ); // TODO: ???
920}
921example
922{ "EXAMPLE:"; echo = 2;
923  TODO
924}
925
926proc grtwist(int a, int d)
927"USAGE:  grtwist(a,d), int a, d
928RETURN:  graded object representing S(d)^a
929PURPOSE: compute presentation of S(d)^a
930EXAMPLE: example grtwist; shows an example
931"
932{
933  ASSUME(0, a > 0);
934
935  module Z; Z[a] = [0];
936  intvec w = -intvec(d:a);
937  Z = grobj(Z, w, w); // will set the rank as well
938  ASSUME(2, grisequal(Z, grpower( grshift(grzero(), d), a ) )); // optional check
939  return(Z);
940}
941example
942{ "EXAMPLE:"; echo = 2;
943
944  ring r=32003,(x,y,z),dp;
945
946  grview(grpower( grshift(grzero(), 10), 5 ) );
947
948  grview( grtwist (5, 10) );
949}
950
951proc grobj(def A, intvec w, list #)
952"USAGE:  grobj(M, w[, d]), matrix/ideal/module M, intvec w, d
953RETURN:  graded object with matrix presentation M, row weighting w [and total graded degrees d of columns]
954PURPOSE: create a valid graded object with a given matrix presentation, weighting [and total graded degrees (in case of zero columns)]
955EXAMPLE: example grobj; shows an example
956"
957{
958  module M = module(A);
959  ASSUME(0, size(w) >= nrows(M) );
960
961  attrib( M, "rank", size(w) );
962  attrib( M, "isHomog", w );
963
964  if( size(#) > 0 )
965  {
966    ASSUME(0, typeof(#[1]) == "intvec" );
967    ASSUME(0, size(#[1]) == ncols(M) );
968    attrib(M, "degHomog", #[1]);
969  }
970  else
971  {
972    ASSUME(0, /* no zero cols please! */ size(M) == ncols(M) );
973    attrib(M, "degHomog", grdeg(M));
974  }
975
976  ASSUME(0, grtest(M) );
977  return (M);
978}
979example
980{ "EXAMPLE:"; echo = 2;
981
982  ring r=32003,(x,y,z),dp;
983
984  module A = grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0,0,1) );
985  grview(A);
986
987  module F = grobj( module([x,y,0]), intvec(1,1,5) );
988  grview(F);
989
990  int d = 666; // zero can have any degree...
991  module Z = grobj( module([x,0], [0,0,0], [0, y]), intvec(1,2,3), intvec(2, d, 3) );
992  grview(Z);
993
994  print(Z);
995  attrib(Z);
996  grrange(Z); // module weights
997  attrib(Z, "degHomog"); // total degrees
998
999}
1000
1001proc grtest(def N, list #)
1002"USAGE:  grtest(M[,b]), anyting M, optionally int b
1003RETURN:  1 if M is a valid graded object, 0 otherwise
1004PURPOSE: validate a graded object. Print an invalid object message if b is not given
1005NOTE: M should be an ideal or module or matrix, with weighting attribute
1006   'isHomog' and optionally total graded degrees attribute 'degHomog'.
1007   Attributes should be compatible with the presentation matrix.
1008EXAMPLE: example grtest; shows an example
1009"
1010{
1011  int b = (size(#) == 0);
1012  string t = typeof(N);
1013  if( (t != "ideal") && (t != "module") && (t != "matrix") )
1014  {
1015    if(b) { "   ? grtest: Input should be something like a matrix!"; };
1016    return (0);
1017  };
1018
1019  if ( typeof(grrange(N)) != "intvec" )
1020  {
1021    if(b) { type(N); attrib(N); "   ? grtest: Input must be graded!";  };
1022    return (0);
1023  };
1024
1025  intvec gr = grrange(N); // grading weights...
1026  if ( nrows(N) != size(gr) )
1027  {
1028    if(b) { "   ? grtest: Input has wrong number of rows!"; };   
1029    return (0);
1030  };
1031
1032//  if( attrib(N, "rank") != size(gr) ){ return (0); } // wrong rank :(
1033
1034  if ( typeof(attrib(N, "degHomog")) == "intvec" )
1035  {
1036    intvec T = attrib(N, "degHomog"); // graded degrees
1037   
1038    if ( ncols(N) != size(T) )
1039    {
1040      if(b) { "   ? grtest: Input has wrong number of cols!"; };   
1041      return (0);
1042    };
1043   
1044    int k = nvars(basering) + 1; // index of mod. column in the leadexp
1045
1046    module L = lead(module(N)); vector v;
1047
1048    // checking T for non-zero N[i]
1049    int i = size(T);
1050
1051    for (; i > 0; i-- )
1052    {
1053      v = L[i];
1054      if( v != 0 )
1055      {
1056        if( (deg(v) + gr[ leadexp(v)[k] ]) != T[i] )
1057        {
1058          if(b) { "   ? grtest: Input has wrong total grade of " + string(i) + "-th column!"; };
1059          return (0);
1060        };  // wrong T[i]
1061      }
1062    }
1063
1064    // TODO: check t on nonzero cols...
1065  } else
1066  {
1067    if( ncols(N) != size(N) )
1068    {
1069      if(b) { "   ? grtest: Input should have exclusively non-zero columns, please give total grades otherwise!"; };
1070      return (0);
1071    };
1072  }
1073
1074  if( !homog(N) )
1075  {
1076    if(b) { "   ? grtest: Input should be graded homogenous!"; };
1077    return (0);
1078  };
1079
1080//  if(b) { "Input seems to be a valid graded object (map)!"; };
1081  return (1);
1082}
1083example
1084{ "EXAMPLE:"; echo = 2;
1085
1086  ring r=32003,(x,y,z),dp;
1087
1088  // the following calls will fail due to tests in grtest:
1089
1090// grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0) ); // not enough row weights
1091// grobj( module([x,0], [0,0,0], [0, y]), intvec(1,2,3) ); // zero column needs (otherwise optional) total degrees
1092// grobj( module([x,0], [0,0,0], [0, y]), intvec(1,2,3), intvec(1, 1, 1) ); // incompatible total degrees (on non-zero columns)
1093
1094}
1095
1096
1097static proc align( def A, int d)
1098"analog of align kernel command for older Singular versions
1099 this is static since it should not be used by @code{align}-able (newer)
1100 Singular releases.
1101 Note that this proc does not care about any attributes (of A)
1102"
1103{
1104  module T; T[d] = 0;
1105  T = T, module(transpose(A));
1106  return( module(transpose(T)) );
1107}
1108
1109proc grgroebner(A)
1110"USAGE:  grgroebner(M), graded object M
1111RETURN:  graded object
1112PURPOSE: compute graded groebner basis of M
1113EXAMPLE: example grgroebner; shows an example
1114"
1115{
1116  ASSUME(1, grtest(A));
1117
1118  return ( grobj( groebner(A), grrange(A) ) );
1119}
1120example
1121{ "EXAMPLE:"; echo = 2;
1122
1123  ring r=32003,(x,y,z),dp;
1124
1125  module A = grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0,0,1) );
1126  grview(A);
1127 
1128  module B = grgroebner(A);
1129  grview(B);
1130}
1131
1132
1133proc grtwists(intvec v)
1134"USAGE:  grtwists(v), intvec v
1135RETURN:  graded object representing S(v[1]) + ... + S(v[size(v)])
1136PURPOSE: compute presentation of S(v[1]) + ... + S(v[size(v)])
1137EXAMPLE: example grtwists; shows an example
1138"
1139{
1140  int l = size(v);
1141  module Z; Z[l] = [0];
1142  Z = grobj(Z, -v, -v); // will set the rank as well
1143  return(Z);
1144}
1145example
1146{ "EXAMPLE:"; echo = 2;
1147
1148  ring r=32003,(x,y,z),dp;
1149 
1150  grview( grtwists ( intvec(-4, 1, 6 )) );
1151}
1152
1153
1154proc grsyz(A)
1155"USAGE:  grsyz(M), graded object M
1156RETURN:  graded object
1157PURPOSE: compute graded syzygy of M
1158EXAMPLE: example grsyz; shows an example
1159"
1160{
1161  ASSUME(1, grtest(A));
1162  intvec v = grdeg(A);
1163
1164//  grrange(A);
1165 
1166  module M = syz(A);
1167//  print(M);
1168 
1169  if( size(M) > 0 ) { return( grobj( M, v ) ); }
1170
1171  // zero syzygy?
1172  return( grtwists(-v) ); // ???
1173}
1174example
1175{ "EXAMPLE:"; echo = 2;
1176
1177  ring r=32003,(x,y,z),dp;
1178
1179  module A = grobj( module([x+y, x, 0, 3], [0, x+y, y, 2], [y, y, z, 1]), intvec(0,0,0,1) );
1180  grview(A);
1181 
1182  module B = grgroebner(A);
1183  grview(B);
1184
1185  module C = grsyz(B); 
1186  grview(C);
1187}
1188
1189
1190proc grprod(A, B)
1191"USAGE:  grprod(M, N), graded objects M and N
1192RETURN:  graded object
1193PURPOSE: compute graded product M * N (as composition of maps)
1194EXAMPLE: example grprod; shows an example
1195"
1196{
1197  ASSUME(1, grtest(A));
1198  ASSUME(1, grtest(B));
1199
1200  // TODO: ASSUME(0, grdeg() == grrange());!!!
1201
1202  return ( grobj( A*B, grrange(A), grdeg(B) ) );
1203}
1204example
1205{ "EXAMPLE:"; echo = 2;
1206
1207  ring r=32003,(x,y,z),dp;
1208
1209  module A = grobj( module([x+y, x, 0, 3], [0, x+y, y, 2], [y, y, z, 1]), intvec(0,0,0,1) );
1210  grview(A);
1211 
1212  A = grgroebner(A);
1213  grview(A);
1214 
1215  module B = grsyz(A);
1216  grview(B);
1217  print(B);
1218 
1219  module D = grprod( A, B );
1220  grview(D);
1221  print(D); // must be all zeroes due to syzygy property!
1222  ASSUME(0, size(D) == 0);
1223}
1224
1225
1226
1227
1228proc grres(def A, int l, list #)
1229"USAGE:  grres(M, l[, b]), graded object M, int l, int b
1230RETURN:  graded resolution = list of graded objects
1231PURPOSE: compute graded resolution of M (of length l) and minimise it if b was given                             
1232EXAMPLE: example grres; shows an example
1233"
1234{
1235  ASSUME(0, l >= 0);
1236  ASSUME(1, grtest(A));
1237
1238  intvec v = grrange(A);
1239 
1240  int b = (size(#) > 0);
1241  if(b) { list r = res(A, l, #[1]); } else { list r = res(A, l); }
1242
1243//  r;  v;
1244
1245  l = size(r);
1246 
1247  int i; module m;
1248 
1249  for ( i = 1; i <= l; i++ )
1250  {
1251    if( size(r[i]) == 0 ){ r[i] = grtwists(-v); i++; break;   }
1252
1253    r[i] = grobj(r[i], v); v = grdeg(r[i]);
1254  }
1255  i = i-1;
1256
1257  return( list(r[1..i]) );
1258}
1259example
1260{ "EXAMPLE:"; echo = 2;
1261
1262  ring r=32003,(x,y,z),dp;
1263
1264  module A = grobj( module([x+y, x, 0, 3], [0, x+y, y, 2], [y, y, z, 1]), intvec(0,0,0,1) );
1265  grview(A);
1266 
1267  module B = grgroebner(A);
1268  grview(B);
1269
1270  "graded resolution of B: "; def C = grres(B, 0); grview(C);
1271
1272  int i; int l = size(C);
1273
1274  "D^2 == 0: "; for (i = 1; i < l; i++ ) { i; grview( grprod(C[i], C[i+1]) ); }
1275}
1276
1277proc grtranspose(def M)
1278"
1279USAGE:   grtranspose(M), graded object M
1280RETURN:  graded object
1281PURPOSE: graded transpose of M
1282NOTE:    no reordering is performend by this procedure   
1283EXAMPLE: example grtranspose; shows an example
1284"
1285{
1286  ASSUME(1, grtest(M) );
1287  return (  grobj(transpose(M), -grdeg(M), -grrange(M))  );
1288}
1289example
1290{ "EXAMPLE:"; echo = 2;
1291
1292  ring r=32003,(x,y,z),dp;
1293
1294  module M = grtwists( intvec(-2, 0, 4, 4) ); grview(M);
1295
1296  module N = grsyz( grtranspose( M ) ); grview(N);
1297
1298  module L = grtranspose(N); grview( L );
1299
1300  module K = grsyz( L ); grview(K);
1301 
1302
1303  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);
1304
1305  "graded transpose: "; module B = grtranspose(A); grview( B ); print(B);
1306
1307  "... syzygy: "; module C = grsyz(B); grview(C);
1308
1309  "... transposed: "; module D = grtranspose(C); grview( D ); print (D);
1310
1311  "... and back to presentation: "; module E = grsyz( D ); grview(E); print(E);
1312
1313  module F = grgens( E ); grview(F); print(F);
1314 
1315  module G = grpres( F ); grview(G); print(G);
1316}
1317
1318
1319proc grgens(def M)
1320"
1321USAGE:   grgens(M), graded object M (map)
1322RETURN:  graded object 
1323PURPOSE: try compute graded generators of coker(M) and return them as columns
1324         of a graded map.
1325NOTE:    presentation of resulting generated submodule may be different to M!
1326EXAMPLE: example grgens; shows an example
1327"
1328{
1329  ASSUME(1, grtest(M) );
1330
1331  module N = grtranspose( grsyz( grtranspose(M) ) );
1332 
1333//  ASSUME(3, grisequal( grgroebner(M), grgroebner( grpres( N ) ) ) ); // FIXME: not always true!?
1334 
1335  return ( N );
1336}
1337example
1338{ "EXAMPLE:"; echo = 2;
1339
1340  ring r=32003,(x,y,z),dp;
1341
1342  module M = grtwists( intvec(-2, 0, 4, 4) ); grview(M);
1343
1344  module N = grgens(M);
1345 
1346  grview( N ); print(N); // fine == M
1347
1348
1349  module A = grobj( module([x+y, x, 0, 3], [0, x+y, y, 2], [y, y, z, 1]), intvec(0,0,0,1) );
1350
1351  A = grgroebner(A); grview(A);
1352
1353  module B = grgens(A);
1354
1355  grview( B ); print(B); // Ups :( != A
1356 
1357}
1358
1359
1360proc grpres(def M)
1361"
1362USAGE:   grpres(M), graded object M (submodule gens)
1363RETURN:  graded module (via coker)
1364PURPOSE: compute graded presentation matrix of submodule generated by columns of M
1365EXAMPLE: example grpres; shows an example
1366"
1367{
1368  ASSUME(1, grtest(M) );
1369
1370  module N = grsyz(M);
1371
1372//  ASSUME(3, grisequal( M, grgens( N ) ) );
1373 
1374  return ( N );
1375}
1376example
1377{ "EXAMPLE:"; echo = 2;
1378
1379  ring r=32003,(x,y,z),dp;
1380
1381  module M = grtwists( intvec(-2, 0, 4, 4) ); grview(M);
1382
1383  module N = grgens(M); grview( N ); print(N);
1384
1385  module L = grpres( N ); grview( L ); print(L);
1386
1387
1388  module A = grobj( module([x+y, x, 0, 3], [0, x+y, y, 2], [y, y, z, 1]), intvec(0,0,0,1) );
1389
1390  A = grgroebner(A); grview(A);
1391
1392  module B = grgens(A); grview( B ); print(B);
1393
1394  module C = grpres( B ); grview( C ); print(C);
1395}
1396
1397
1398
1399LIB "random.lib"; // for sparsepoly
1400
1401proc grrndmat(intvec w, intvec v, list #)
1402"USAGE:  grrndmat(src,dst[,p,b]), intvec src, dst[, int p, b]
1403RETURN:  matrix of polynomials
1404PURPOSE: generate random matrix compatible with src and dst gradings
1405NOTE:    optional arguments p, b are for 'sparsepoly' (by default: 75%, 30000).
1406TODO:    this is experimental at the moment!
1407EXAMPLE: example grrndmat; shows an example
1408"
1409{
1410  // defaults for sparsepoly
1411  int p = 75;
1412  int b = 30000;
1413
1414  if ( size(#) > 0 )
1415  {
1416    ASSUME( 0, (typeof(#[1]) == "int") || (typeof(#[1]) == "bigint") );
1417    p = #[1];
1418
1419    if ( size(#) > 1 )
1420    {
1421      ASSUME( 0, (typeof(#[2]) == "int") || (typeof(#[2]) == "bigint") );
1422      b = #[2];
1423    }
1424  }
1425
1426  int n = size(v); // destination: rows!
1427  int m = size(w); // source: cols
1428
1429  matrix M[n][m];
1430
1431  int r,c; intvec ww;
1432
1433  for( c = m; c > 0; c-- )
1434  {
1435    ww = v - intvec(w[c]:n);
1436    for( r = n; r > 0; r-- )
1437    {
1438      if( ww[r] >= 0)
1439      {
1440        M[r,c] = sparsepoly(ww[r], ww[r], p, b);
1441      }
1442
1443    }
1444  }
1445
1446  return(M);
1447}
1448example
1449{ "EXAMPLE:"; echo = 2;
1450
1451  ring r=32003,(x,y,z),dp;
1452
1453  print( grrndmat( intvec(0, 1), intvec(1, 2, 3) ) );
1454}
1455
1456//
1457
1458
1459proc KeneshlouMatrixPresentation(intvec a)
1460"USAGE:  intvec a.
1461RETURN: graded object
1462PURPOSE: matrix presentation for direct sum of omega^a[i](i) in form of a graded object
1463EXAMPLE: example KeneshlouMatrixPresentation; shows an example
1464"
1465{
1466  int n = size(a)-1;
1467  //  ring r = 32003,(x(0..n)),dp;
1468  ASSUME(0, nvars(basering)==(n+1));
1469  int i,j;
1470
1471  // find first nonzero exponent a_i
1472  for(i=1;i<=size(a);i++)
1473    {
1474      if(a[i]!=0) {break; };
1475    }
1476
1477  // all zeroes?
1478  if(i>size(a)) {return (grzero()); };
1479
1480  for(i=2;i<=n;i++)
1481    {
1482      if(a[i]!=0) {break; };
1483    }
1484
1485  module N;
1486
1487  if(i>n)
1488    { // no middle part
1489      if(a[1]>0)
1490        {
1491          N=grtwist(a[1],0);
1492
1493          if(a[n+1]>0)
1494            { N=grsum(N,grtwist(a[n+1],-1));}       
1495        } 
1496      else
1497        { N=grtwist(a[n+1],-1);}
1498     
1499      return (N); // grorder(N));
1500    }
1501  else // i <= n: middle part is present, a_i != 0
1502    { // a = a1  ... |  i:2, a_2 ..... i: n, a_n | .... i: n+1a_(n+1)
1503      j = i - 1;
1504      module I = maxideal(1); attrib(I,"isHomog", intvec(0)); list L = mres(I, 0); // TODO: use grres() instead!!!
1505      list kos = grorder(L);
1506      // make sure that graded maps  are represented by blocks corresponding to the betti diagram?
1507
1508      def S = grpower(grshift(grobj( kos[j+2], attrib(kos[j+2], "isHomog")), j), a[i]);
1509
1510      i++;
1511
1512      for(; i <= n; i++)
1513        {
1514          if(a[i]==0) { i++; continue; }
1515          j = i - 1;
1516          S = grsum( S, grpower(grshift( grobj( kos[j+2], attrib(kos[j+2], "isHomog")), j), a[i])  );
1517        }
1518
1519      // S is the middle (non-zero) part
1520
1521      if(a[1] > 0 )
1522        {
1523          N=grsum(grtwist(a[1],0), S);
1524        }
1525      else
1526        { N = S;}
1527
1528      if(a[n+1] > 0 )
1529        { N=grsum(N, grtwist(a[n+1],-1)); }
1530
1531
1532      return ((N)); //      return (grorder(N));
1533    }
1534}
1535example
1536{ "EXAMPLE:"; echo = 2;
1537  ring r = 32003,(x(0..4)),dp;
1538
1539  def N1 = KeneshlouMatrixPresentation(intvec(2,0,0,0,0));
1540  grview(N1);
1541
1542  def N2 = KeneshlouMatrixPresentation(intvec(0,0,0,0,3));
1543  grview(N2);
1544
1545  def N = KeneshlouMatrixPresentation(intvec(2,0,0,0,3));
1546  grview(N);
1547
1548
1549  def M1 = KeneshlouMatrixPresentation(intvec(0,1,0,0,0));
1550  grview(M1);
1551
1552  def M2 = KeneshlouMatrixPresentation(intvec(0,1,1,0,0));
1553  grview(M2);
1554
1555  def M3 = KeneshlouMatrixPresentation(intvec(0,0,0,1,0));
1556  grview(M3);
1557
1558  def M = KeneshlouMatrixPresentation(intvec(1,1,1,0,0));
1559  grview(M);
1560}
1561
1562proc grconcat(A,B)
1563"USAGE: grconcat(A, B), graded objects A and B, dst(A) == dst(B) =: dst
1564RETURN: graded object
1565PURPOSE: construct src(A) + src(B) -----> dst  given by (A|B)
1566EXAMPLE: example grconcat; shows an example
1567"
1568{
1569
1570  ASSUME(1, grtest(A));
1571  ASSUME(1, grtest(B));
1572  ASSUME(0, grrange(A)==grrange(B));
1573 
1574  intvec v = grrange(A);
1575  intvec w=grdeg(A),grdeg(B);
1576  return(grobj(concat(A,B),v,w));
1577}
1578example
1579{ "EXAMPLE:"; echo = 2;
1580  ring r;
1581 
1582  module R=grobj(module([x,y,z]),intvec(0:3));
1583  grview(R);
1584
1585  module S=grobj(module([x,0,y],[xy,zy+x2,0]),intvec(0:3));
1586  grview(S);
1587
1588  def Q=grconcat(R,S);
1589  grview(Q);
1590}
1591
1592proc grlift(A, B)
1593"USAGE: grlift(M, N), graded objects M and N
1594RETURN: transformation matrix (graded object???)
1595PURPOSE: compute graded matrix which the generators of submodule Im(N) in terms of Im(M).
1596EXAMPLE: example grlift; shows an example
1597"
1598{
1599  ASSUME(1, grtest(A));
1600  ASSUME(1, grtest(B));
1601  ASSUME(0, grrange(A) == grrange(B));
1602
1603//  matrix T;  module AA = liftstd(A, T); //  AA = module(A*T)
1604//  matrix U;
1605  matrix L =lift(A,B/*,U*/);  //  module(B*U) = module(matrix(A)*L)
1606 
1607  return(grobj(L, grdeg(A), grdeg(B)));
1608}
1609example
1610{ "EXAMPLE:"; echo = 2;
1611
1612  ring r=32003,(x,y,z),dp;
1613  module P=grobj(module([xy,0,xz]),intvec(0,1,0));
1614  grview(P);
1615
1616
1617  module D=grobj(module([y,0,z],[x2+y2,z,0]),intvec(0,1,0));
1618  grview(D);
1619
1620  def G=grlift(D,P);
1621  grview(G);
1622
1623  ASSUME(0, grisequal( grprod(D, G), P) ); 
1624}
1625
1626proc grrange(M)
1627{
1628//  ASSUME(1, grtest(M)); // Leads to recursive call due to grtest...
1629  return( attrib(M, "isHomog") );
1630}
1631
1632proc grlift0(M, N, alpha1)
1633"PURPOSE: generic random alpha0 : coker(M) -> coker(N) from random alpha1
1634NOTE: this proc can work only if some assumptions are fulfilled (due
1635to Wolfram)! e.g. at the end of a resolution for the source module...
1636"
1637{
1638
1639  ASSUME(1, grtest(M));
1640  ASSUME(1, grtest(N));
1641
1642  ASSUME(0, grdeg(M) == grdeg(alpha1) );
1643  ASSUME(0, grdeg(N) == grrange(alpha1) );
1644  return(
1645   grtranspose( grlift( grtranspose( M ),
1646       grtranspose( grprod( N,  alpha1 ) ) )
1647       ) ); // alpha0!
1648   
1649}
1650example
1651{ "EXAMPLE:"; echo = 2;
1652
1653  ring S = 0, (x(0..3)), dp;
1654  list kos = grres(grobj(maxideal(1), intvec(0)), 0);
1655  print( betti(kos), "betti");
1656  grview(kos);
1657
1658
1659  // source module:
1660//  module M = grshift(kos[4], 2); // phi, Syz_3(K(2))
1661  def M = KeneshlouMatrixPresentation(intvec(0,0,1,0));
1662//   grview( grres(M, 0) );
1663  grview(M);
1664
1665  // destination module:
1666//   module N = grshift(kos[3], 1); // psi, Syz_2(K(1))
1667  def N = KeneshlouMatrixPresentation(intvec(0,1,0,0));
1668//  grview( grres(N, 0) );
1669  grview(N);
1670
1671  // random graded of degree 0, homomorphism of free presentations:
1672  // alpha1: src(M) -> src (N)
1673  def alpha1 = grrndmap( M, N ); // alpha1
1674  grview(alpha1);
1675
1676  // random graded of degree 0, homomorphism of free presentations:
1677  // alpha0: dst(M) -> dst (N)
1678  def alpha0 = grlift0(M, N, alpha1);
1679  grview(alpha0);
1680
1681}
1682
1683
1684proc grlifting(M,N)
1685"USAGE:  graded objects M and N
1686RETURN: map of chain complexes (as a list)
1687PURPOSE: construct a map of chain complexes between free resolutions of Img(M) and Img(N).
1688EXAMPLE: example grlifting; shows an example
1689"
1690{  ASSUME(1, grtest(M));
1691   ASSUME(1, grtest(N));
1692
1693   list rM=grres(M,0,1);
1694   list rN=grres(N,0,1);
1695   int i,j,k;
1696
1697  for(i=1;i<=size(rM);i++)
1698  {
1699    if(size(rM[i])==0){break;}
1700  }
1701
1702  for(j=1;j<=size(rN);j++)
1703  {
1704    if(size(rN[j])==0){break;}
1705  }
1706  int t=min(i,j);
1707
1708  ASSUME(0, t >= 2);
1709 
1710  list P;
1711
1712  "t: ", t;
1713 
1714  P[1]= grrndmap( rM[1], rN[1] ); // alpha1
1715
1716  if(t==2){return(P[1]);} 
1717   
1718  for(k=2; k<=t; k++)
1719  {
1720    P[k] = grlift( grprod(P[k-1],rM[k]), rN[k] );
1721     grview(P[k]); 
1722   
1723  }
1724     
1725  return(P);
1726   
1727}
1728example
1729{ "EXAMPLE:"; echo = 2;
1730/*
1731  ring r=32003,(x,y,z),dp;
1732
1733  module P=grobj(module([xy,0,xz]),intvec(0,1,0));
1734  grview(P);
1735
1736  module D=grobj(module([y,0,z],[x2+y2,z,0]),intvec(0,1,0));
1737  grview(D);
1738
1739  def G=grlifting(D,P);
1740  grview(G);
1741
1742  kill r;
1743  ring r=32003,(x,y,z),dp;
1744
1745  module D=grobj(module([y,0,z],[x2+y2,z,0], [z3, xy, xy2]),intvec(0,1,0));
1746  D = grgroebner(D);
1747  grview( grres(D, 0));
1748
1749  def G=grlifting(D, D);
1750  grview(G);
1751*/
1752
1753  ring S = 0, (x(0..3)), dp;
1754  list kos = grres(grobj(maxideal(1), intvec(0)), 0);
1755  print( betti(kos), "betti");
1756  grview(kos);
1757
1758//  module M = grshift(kos[4], 2); // phi, Syz_3(K(2))
1759  def M = KeneshlouMatrixPresentation(intvec(0,0,1,0));
1760  grview( grres(M, 0) );
1761 
1762//   module N = grshift(kos[3], 1); // psi, Syz_2(K(1))
1763  def N = KeneshlouMatrixPresentation(intvec(0,1,0,0));
1764  grview( grres(N, 0) );
1765
1766  grlifting(M, N); // grview(G);
1767
1768
1769//  def G=grlifting( grgens(M), grgens(N) );  grview(G);
1770
1771
1772}
1773
1774proc mappingcone(M,N)
1775"USAGE:M,N graded objects
1776RETURN: chain complex (as a list)
1777PURPOSE: construct a free resolution of the cokernel of a random map between Img(M), and Img(N).
1778EXAMPLE: example mappingcone; shows an example
1779"
1780{
1781  ASSUME(1, grtest(M));
1782  ASSUME(1, grtest(N));
1783
1784  list P=grlifting(M,N);
1785  list rM=grres(M,1);
1786  list rN=grres(N,1);
1787
1788  int i;
1789  list T;
1790
1791  T[1]=grconcat(P[1],rN[2]);
1792
1793  for(i=2;i<=size(P);i++)
1794  {
1795    intvec v=grrange(rM[i]);
1796    intvec w=grdeg(rN[i+1]);
1797    int r=size(v);
1798    int s=size(w);
1799    module zero = (0:s);
1800
1801    module A=grconcat(P[i],rN[i+1]);
1802    module B=grobj(zero,v,w);
1803    module C=grconcat(-rM[i],B);
1804    module D=grconcat(grtranspose(C), grtranspose(A));
1805
1806    T[i]=grtranspose(D);
1807  }
1808   return(T);
1809}
1810example
1811{ "EXAMPLE:"; echo = 2;
1812
1813ring r=32003, (x(0..4)),dp;
1814def A=KeneshlouMatrixPresentation(intvec(0,0,0,0,3));
1815def M=grgens(A);
1816grview(M);
1817
1818def B=KeneshlouMatrixPresentation(intvec(0,1,0,0,0));
1819def N=grgens(B);
1820grview(N);
1821
1822def R=grlifting(M,N);
1823grview(R);
1824def T=mappingcone(M,N);
1825grview(T);
1826
1827def U=grtranspose(T[1]);
1828resolution G=mres(U,0);
1829print(betti(G),"betti");
1830ideal I=groebner(flatten(G[2]));
1831resolution GI=mres(I,0);
1832print(betti(GI),"betti");
1833}
1834
1835// correct
1836proc grrndmap(def S, def D, list #)
1837"USAGE: (S,D), graded objects S and D
1838RETURN: graded object
1839PURPOSE: construct a random 0-deg graded homomorphism src(S) -> src(D)
1840EXAMPLE: example grrndmap; shows an example
1841"
1842{
1843
1844ASSUME(1, grtest(S) );
1845ASSUME(1, grtest(D) );
1846
1847// "src: "; grview(S);"dst: "; grview(D);
1848
1849intvec v = -grdeg(S); // source
1850intvec w = -grdeg(D); // destination
1851
1852return (grobj(grrndmat(v, w, #), -w, -v ) );
1853}
1854example
1855{ "EXAMPLE:"; echo = 2;
1856
1857  ring r=32003,(x,y,z),dp;
1858
1859  module D=grobj(module([y,0,z],[x2+y2,z,0]),intvec(0,1,0));
1860  grview(D);
1861
1862  module S=grobj(module([x,0,y],[xy,zy+x2,0]),intvec(0,0,0));
1863  grview(S);
1864
1865  def H=grrndmap(D,S);
1866  grview(H);
1867
1868}
1869
1870proc grrndmap2(def D, def S, list #)
1871"USAGE: (D,S), graded objects S and D
1872RETURN: graded object
1873PURPOSE: construct a random 0-deg graded homomorphism between target of D and S.
1874EXAMPLE: example grrndmap2; shows an example
1875"
1876{
1877  ASSUME(1, grtest(D) );
1878  ASSUME(1, grtest(S) );
1879  intvec v = -grrange(D); // source
1880  intvec w = -grrange(S); // target
1881  return (grobj(grrndmat(v, w, #), -w, -v ) );
1882}
1883example
1884{ "EXAMPLE:"; echo = 2;
1885
1886  ring r=32003,(x,y,z),dp;
1887
1888  module D=grobj(module([y,0,z],[x2+y2,z,0]),intvec(0,1,0));
1889  grview(D);
1890
1891  module S=grobj(module([x,0,y],[xy,zy+x2,0]),intvec(0,0,0));
1892  grview(S);
1893
1894  def G=grrndmap2(D,S);
1895  grview(G);
1896}
1897
1898
1899//                A     f2     f3
1900// 0<---M<----F0<----F1<----F2<----F3<----
1901//              |p1   |p2
1902//
1903// 0<---N<----G0<----G1<----G2<----G3<----
1904//                B(g1)      g2     g3
1905//
1906proc grlifting2(A,B)
1907"USAGE: (A,B), graded objects A and B (matrices defining maps)
1908RETURN: map of chain complexes (as a list)
1909PURPOSE: construct a map of chain complexes between free resolution of
1910M=coker(A) and N=coker(B).
1911EXAMPLE: example grlifting2; shows an example
1912"
1913{  ASSUME(1, grtest(A));
1914   ASSUME(1, grtest(B));
1915
1916   list rM=grres(A,0);
1917   list rN=grres(B,0);
1918   int i,j,k;
1919   list P;
1920
1921  // find first zero matrix in rM
1922  for(i=1;i<=size(rM);i++)
1923  {
1924    if(size(rM[i])==0){break;}
1925  }
1926
1927  // find first zero matrix in rN
1928  for(j=1;j<=size(rN);j++)
1929  {
1930    if(size(rN[j])==0){break;}
1931  }
1932
1933  int t=min(i,j);
1934
1935  P[1]=grrndmap2(A,B);
1936
1937  // A(or B)=0
1938  if(t==1){return(P[1])};
1939 
1940  for(k=2;k<=t;k++)
1941  {
1942   def E=grprod(P[k-1],rM[k-1]);
1943   P[k]=grlift(rN[k-1],E); // ---------->
1944   /* let yi=(pi)o(fi); to take grlift(gi,yi)
1945      we should have img(yi) is contained in
1946      img(gi)=ker(gi-1),i.e (gi-1)oyi=0. we have
1947      (gi-1)oyi=(gi-1)o(pi)o(fi)=(pi-1)o(fi-1)ofi=0
1948  */
1949  }
1950  return(P);
1951}
1952example
1953{ "EXAMPLE:"; echo = 2;
1954
1955ring r;
1956module P=grobj(module([xy,0,xz]),intvec(0,1,0));
1957grview(P);
1958
1959module D=grobj(module([y,0,z],[x2+y2,z,0]),intvec(0,1,0));
1960grview(D);
1961
1962module PP = grpres(P);
1963grview(PP);
1964
1965module DD = grpres(D);
1966grview(DD);
1967
1968
1969def T=grlifting2(DD,PP); T;
1970
1971// def Z=grlifting2(P,D); Z; // WRONG!!! 
1972             
1973}
1974
1975/*
1976//              -f1 0       -f2 0
1977//   (p1 g1)     p2 g2       p3 g3
1978// G0<-----F0+G1<------F1+G2<-------F2+G3<-----
1979proc mappingcone2(A,B)
1980"USAGE: (A,B), graded objects A and B (matrices defining maps)
1981RETURN: chain complex (as a list)
1982PURPOSE: construct the free resolution of a cokernel of a random map between M=coker(A), and N=coker(B)
1983EXAMPLE: example mappingcone2;
1984"
1985{
1986  ASSUME(1, grtest(A));
1987  ASSUME(1, grtest(B));
1988
1989  list P=grlifting2(A,B);
1990  list rM=grres(A,1);
1991  list rN=grres(B,1);
1992
1993  int i;
1994  list T;
1995
1996  T[1]=grconcat(P[1],rN[1]);
1997
1998  for(i=2;i<=size(P);i++)
1999  {
2000    intvec v=grrange(rM[i-1]);
2001    intvec w=grdeg(rN[i]);
2002    int r=size(v);
2003    int s=size(w);
2004    matrix zero[r][s];
2005
2006    module A=grconcat(P[i],rN[i]);
2007    module B=grobj(zero,v,w);
2008
2009    module C=grconcat( grneg( rM[i-1] ) ,B);
2010    module D=grconcat(grtranspose(C), grtranspose(A));
2011
2012    T[i]=grtranspose(D);
2013  }
2014  return(T);
2015}
2016example
2017{ "EXAMPLE:"; echo = 2;
2018
2019ring r=32003,(x(0..4)),dp;
2020def I=maxideal(1);
2021module R=grobj(module(I), intvec(0));
2022resolution FR=mres(R,0);
2023print(betti(FR,0),"betti");
2024module K=grobj(module(FR[1]),intvec(-1),intvec(0:5));
2025grview(K);
2026
2027module S=grsyz(K);
2028grview(S);
2029S;
2030
2031module SS = grpres(S);
2032
2033module B=grobj(module([1,0,0],[0,1,0],[0,0,1]),intvec(1,1,1),intvec(1,1,1));
2034// module B=grobj(module([1],[0,1],[0,0,1],[0,0,0,1], [0,0,0,0,1]),intvec(0:5));
2035grview(B);
2036B;
2037module BB = grpres(B);
2038
2039def Z=grlifting2(SS,BB);Z;
2040def G=mappingcone2(SS,BB);G;
2041}
2042*/
2043
2044proc grlifting3(A,B)
2045"TODO: grlifting4 was newer and had more documentation than this proc, but was removed... Please verify and update!
2046"
2047{
2048  ASSUME(1, grtest(A));
2049  ASSUME(1, grtest(B));
2050
2051
2052  list rM = grres(A,0,1);
2053
2054  print( betti(rM), "betti");
2055  list rN = grres(B,0,1);
2056  print( betti(rN), "betti");
2057 
2058  int i,j,k;
2059
2060  for(i=1;i<=size(rM);i++)
2061  {
2062    if(size(rM[i])==0){break;}
2063  }
2064
2065  for(j=1;j<=size(rN);j++)
2066  {
2067    if(size(rN[j])==0){break;}
2068  }
2069  int t=min(i,j);
2070 
2071  list P;
2072
2073  "t: ", t;
2074//  grview(rM[t]);  grview(rN[t]);
2075 
2076  P[t]= grrndmap2(rM[t],rN[t]);
2077  grview(P[t]);
2078
2079  if(t==1){return(P)};
2080
2081  for(k=t-1; k>=1; k--)
2082  {
2083     "k: ", k;
2084//  grview(rM[k]);  grview(rN[k]);
2085
2086// def C = grtranspose(rM[k]); def T= grprod(rN[k],P[k+1]);
2087// def tT = grtranspose(T);
2088
2089    P[k]= grlift0( rM[k], rN[k], P[k+1] ); // grtranspose(grlift(C,tT));
2090
2091     grview(P[k]);
2092
2093   }
2094   return(P);
2095}
2096example
2097{"EXAMPLE:"; echo = 2;
2098
2099ring r=32003, x(0..4),dp;
2100
2101def A=grtwist(3,1);
2102grview(A);
2103
2104def T=KeneshlouMatrixPresentation(intvec(0,1,0,0,0));
2105grview(T);
2106
2107def F=grlifting3(T,A);
2108grview(F);
2109
2110def R=KeneshlouMatrixPresentation(intvec(0,0,0,2,0));
2111def S=KeneshlouMatrixPresentation(intvec(1,2,0,0,0));
2112
2113def H=grlifting3(R, S);
2114// grview(H);
2115
2116// 2nd module does not lie in the first:
2117// def H=grlifting3(S, R);
2118
2119
2120//def I=KeneshlouMatrixPresentation(intvec(2,3,0,6,2));
2121//def J=KeneshlouMatrixPresentation(intvec(4,0,1,2,1));
2122//def N=grlifting3(I,J); grview(N);
2123}
2124
2125proc grneg(A)
2126"USAGE: grneg(A), graded object A
2127RETURN: graded object
2128PURPOSE: graded map defined by -A
2129EXAMPLE: example grneg; shows an example
2130"
2131{
2132  ASSUME(1, grtest(A));
2133  return( grobj(-A, grrange(A), grdeg(A)) );
2134}
2135example
2136{ "EXAMPLE:"; echo = 2;
2137
2138   ring r=0,(x,y,z),dp;
2139   def A=grobj([x2,yz,xyz],intvec(1,1,0));
2140   grview(A);
2141
2142   def F=grneg(A);
2143   grview(F);
2144}
2145
2146//             -f1 0        -f2 0
2147//  (p1 g1)     p2 g2        p3 g3
2148//G0<-----F0+G1<------F1+G2<-------F2+G3<-----
2149proc mappingcone3(A,B)
2150"USAGE: (A,B), graded objects A and B (matrices defining maps)
2151RETURN: chain complex (as a list)
2152PURPOSE: construct a free resolution of the cokernel of a random map between M=coker(A), and N=coker(B)
2153EXAMPLE: example mappingcone3; shows an example
2154"
2155{
2156  ASSUME(1, grtest(A));
2157  ASSUME(1, grtest(B));
2158
2159  list P=grlifting3(A,B);
2160  list rM=grres(A,0,1);
2161  list rN=grres(B,0,1);
2162
2163  int i;
2164  list T;
2165
2166  T[1]=grconcat(P[1],rN[1]);
2167
2168  for(i=2;i<=size(P);i++)
2169  {
2170    intvec v= grrange(rM[i-1]);
2171    intvec w=grdeg(rN[i]);
2172    int r=size(v);
2173    int s=size(w);
2174    matrix zero[r][s];
2175
2176//    ASSUME( 0, grtest(P[i]) );
2177//    ASSUME( 0, grtest(rN[i]) );
2178
2179    module A=grconcat(P[i],rN[i]);
2180    module B=grobj(zero,v,w);
2181
2182    module C=grconcat( grneg( rM[i-1] ) ,B);
2183    module D=grconcat(grtranspose(C), grtranspose(A));
2184
2185    T[i]=grtranspose(D);
2186   
2187    kill A, B, C, D, v, w, r, s, zero;
2188  }
2189   return(T);
2190}
2191example
2192{ "EXAMPLE:"; echo = 2;
2193
2194ring r=32003,x(0..4),dp;
2195
2196def A=KeneshlouMatrixPresentation(intvec(0,0,0,0,3));
2197grview(A);
2198
2199def T= KeneshlouMatrixPresentation(intvec(0,1,0,0,0));
2200grview(T);
2201
2202def F=grlifting3(A,T); grview(F);
2203
2204// BUG in the proc 
2205def G=mappingcone3(A,T); grview(G);
2206
2207/*
2208module W=grtranspose(G[1]);
2209resolution U=mres(W,0);
2210print(betti(U,0),"betti"); // ?
2211ideal P=groebner(flatten(U[2]));
2212resolution L=mres(P,0);
2213print(betti(L),"betti");   
2214*/
2215
2216
2217def R=KeneshlouMatrixPresentation(intvec(0,0,0,2,0));
2218grview(R);
2219
2220def S=KeneshlouMatrixPresentation(intvec(1,2,0,0,0));
2221grview(S);
2222
2223def H=grlifting3(R,S); grview(H);
2224
2225// BUG in the proc
2226def G=mappingcone3(R,S);
2227
2228
2229def I=KeneshlouMatrixPresentation(intvec(2,3,0,6,2));
2230def J=KeneshlouMatrixPresentation(intvec(4,0,1,2,1));
2231// def N=grlifting3(I,J);
2232// 2nd module does not lie in the first:
2233// def NN=mappingcone3(I,J); // ????????
2234
2235}
2236
2237
2238
2239
2240// TODO: Please decide between KeneshlouMatrixPresentation and matrixpres, and replace one with the other!
2241proc matrixpres(intvec a)
2242"USAGE:  intvec a
2243RETURN:  graded object
2244PURPOSE: matrix presentation for direct sum of omega^a[i](i) in form of a graded object
2245EXAMPLE: example matrixpres; shows an example
2246"
2247{
2248  int n = size(a)-1;
2249  //  ring r = 32003,(x(0..n)),dp;
2250  ASSUME(0, nvars(basering)==(n+1));
2251  int i,j;
2252
2253  // find first nonzero exponent a_i
2254  for(i=1;i<=size(a);i++)
2255    {
2256      if(a[i]!=0) {break; };
2257    }
2258
2259  // all zeroes?
2260  if(i>size(a)) {return (grzero()); };
2261   for(i=2;i<=n;i++)
2262    {
2263      if(a[i]!=0) {break; };
2264    }
2265
2266  module N;
2267
2268  if(i>n)
2269    { // no middle part
2270      if(a[1]>0)
2271        {
2272          N=grtwist(a[1],-1);
2273
2274          if(a[n+1]>0)
2275            { N=grsum(N,grtwist(a[n+1],0));}         
2276        } 
2277      else
2278        { N=grtwist(a[n+1],0);}
2279     
2280      return (N); // grorder(N));
2281    }
2282
2283else // i <= n: middle part is present, a_i != 0
2284    { // a = a1  ... |  i:2, a_2 ..... i: n, a_n | .... i: n+1a_(n+1)
2285      module I = maxideal(1);
2286      attrib(I,"isHomog", intvec(0));
2287      list L = mres(I, 0);
2288      list kos = grorder(L);
2289      // make sure that graded maps  are represented by blocks corresponding to the betti diagram?
2290      int j=size(a)-i;
2291      def S = grpower(grshift(grobj( kos[j+2], attrib(kos[j+2], "isHomog")),j ), a[i]);
2292
2293      i++;
2294
2295      for(; i <= n; i++)
2296        {
2297          if(a[i]==0) { i++; continue; }
2298          int j=size(a)-i;
2299          S = grsum( S, grpower(grshift( grobj( kos[j+2], attrib(kos[j+2], "isHomog")), j), a[i])  );
2300        }
2301
2302      // S is the middle (non-zero) part
2303
2304      if(a[1] > 0 )
2305        {
2306          N=grsum(grtwist(a[1],-1), S);
2307        }
2308      else
2309        { N = S;}
2310
2311      if(a[n+1] > 0 )
2312        { N=grsum(N, grtwist(a[n+1],0)); }
2313
2314
2315      return ((N)); //      return (grorder(N));
2316    }
2317}
2318example
2319{ "EXAMPLE:"; echo = 2;
2320
2321ring r = 32003,(x(0..4)),dp;
2322
2323def R=matrixpres(intvec(1,4,0,0,0));
2324grview(R);
2325def S=matrixpres(intvec(0,0,3,0,0));
2326grview(S);
2327
2328def N1 = matrixpres(intvec(2,0,0,0,0));
2329grview(N1);
2330
2331def N2 = matrixpres(intvec(0,0,0,0,3));
2332grview(N2);
2333
2334def N = matrixpres(intvec(2,0,0,0,3));
2335grview(N);
2336
2337
2338def M1 = matrixpres(intvec(0,1,0,0,0));
2339grview(M1);
2340
2341def M2 = matrixpres(intvec(0,1,1,0,0));
2342grview(M2);
2343
2344def M3 = matrixpres(intvec(0,0,0,1,0));
2345grview(M3);
2346
2347def M = matrixpres(intvec(1,1,1,0,0));
2348grview(M);
2349}
2350
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