source: git/Singular/LIB/gradedModules.lib @ 04b3ef

//////////////////////////////////////////////////////////////////////////
version="version gradedModules.lib 4.0.1.1 Jan_2015 "; // $Id$
category="Commutative Algebra";
info="
LIBRARY: gradedModules.lib     Operations with graded modules/matrices/resolutions
AUTHORS:  Oleksandr Motsak <U@D>, where U=motsak, D=mathematik.uni-kl.de
@*        Hanieh Keneshlou <hkeneshlou@yahoo.com>
KEYWORDS: graded modules, graded homomorphisms, syzygies
OVERVIEW:
    The library contains several procedures for constructing and manipulating graded modules/matrices/resolutions.
    Basics about graded objects can be found in [DL].
    Throughout this library graded objects are graded maps, that is,
    matrices with polynomials, together with grading weights for source and
    destination. Graded modules are implicitly given as coker of a graded map.
    Note that in special cases we may also consider submodules in S^r generated
    by columns of a graded polynomial matrix (or a graded map).
NOTE:
    set assumeLevel to positive integer value in order to auto-check all assumptions.
    We denote the current basering by S.
REFERENCES:
[DL] Decker, W., Lossen, Ch.: Computing in Algebraic Geometry, Springer, 2006
PROCEDURES:
    grobj(M,w[,d])  construct a graded object (map) given by matrix M
    grtest(A)       check whether A is a valid graded object
    grisequal(A,B)  check whether A is exactly eqal to B? TODO: isomorphic!
    grdeg(M)        compute graded degrees of columns of the map M
    grview(M)       view the graded structure of map M
    grshift(M,d)    shift graded module coker(M) by d
    grzero()        presentation of S(0)^1
    grtwist(r,d)    presentation of S(-d)^r
    grtwists(v)     presentation of S(-v[1])+...+S(-v[size(v)])
    grsum(M,N)      direct sum of two graded modules coker(M) + coker(N)
    grpower(M,p)    direct p-th power of graded module coker(M)
    grtranspose(M)  un-ordered graded transpose of map M        
    grgens(M)       try to compute submodule generators of coker(M)
    grpres(F)       presentation of submodule generated by columns of F
    grorder(M)      reorder cols/rows of M for correct graded-block-structure
    grtranspose1(M)  reordered graded transpose of map M
    TestGRRes(n,I)  compute/order/transpose a graded resolution of ideal I
    KeneshlouMatrixPresentation(v)  build some presentation with intvec v
    grsyz(M)        syzygy of Im(M)
    grres(M,l[,m])  resolution of Im(M) of length l... minimal?
    grlift(A,B)     graded lift, gens!
    grprod(A,B)     composition of graded maps (product of matrices?)
    grgroebner(M)   Groebner Basis of Im(M) as a graded object
    grconcat(M,N)   sum of maps into the same target module
    grrndmat(s,d[,p,b])   generate random matrix compatible with src and dst gradings
    grrndmap(S,D[,p,b])   generate random 0-deg homomorphism src(S) -> src(D)
    grrndmap2(S,D[,p,b])   generate random 0-deg homomorphism dst(S) -> dst(D)
    grlifting(A,B)     RND! chain lifting
    grlifting2(A,B)    RND! chain lifting
    mappingcone(M,N)   mapping cone?
    mappingcone2(M,N)  mapping cone2?
    grlifting3(A,B)    RND! chain lifting? probably wrong one
    mappingcone3(A,B)  mapping cone3? 
    grrange(M)         get the row-weightings
    grneg(A)           graded object given by -A
    matrixpres(a)      matrix presentation of direct sum of Omega^{a[i]}(i)
";

LIB "matrix.lib"; // ?

//////////////////////////////////////////////////////////////////////////////////////////////////////////
// . view graded module/map
// . reorder graded resolution
// . transpose graded module/map?

// draw helpers
static proc repeat(int n, string c) { string r = ""; while( n > 0 ){ r = r + c; n--; } return(r); }
static proc pad(int m, string s, string c){ string r = s; while( size(r) < m ){ r = c + r; } return(r); }
static proc mstring( int m, string c){ if( m < 0 ) { return (c); }; return (string(m)); }

static proc grsumstr(string R, intvec v)
"direct sum_i=1^size R(-v[i]), for source and targets of graded objects"
{
  if (R == "")
  {
    R = nameof(basering);
  }

  ASSUME(0, defined(R) && (R != "") );

  int n = size(v);

  if (n == 0) { return (R); }

  ASSUME(0, n > 0 );
  
  v = -v; // NOTE: due to Mathematical meanings of Singular data
  

  int lst = v[1];
  int cnt = 1; 

  string p = R;
  if( lst != 0 ) { p = p + "(" + string(lst) + ")"; }

  int k, d;
  for (k = 2; k <= n; k++ ) 
  {
    d = v[k];
    if( d == lst ) { cnt = cnt + 1; }
    else 
    {
      if (cnt > 1){ p = p + "^" + string(cnt); }

      cnt = 1; lst = d;

      p = p + " + " + R;
      if( lst != 0 ) { p = p + "(" + string(lst) + ")"; }
    }
  }
  if (cnt > 1){ p = p + "^" + string(cnt); }

  return (p);
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  def E = grtwist(2, 0);
  def v = grrange(E); // grdeg(E);
  grsumstr("", v );  
}

// view helper
static proc draw ( intmat D, int d )
{
//  print(D); return ();
  int nc = ncols(D); int nr = nrows(D);
  int s, r, c; int max = 0;
  // get maximum string-length among all {D[r,c]}
  for (r = nr; r > 0; r-- ) { for (c = nc; c > 0; c-- ) { s = size( string(D[r, c]) ); if( max < s ) { max = s; } } }
  max = max + 1;
  string head = ""; string foot = ""; string middle = "";
  for ( c = d+1; c < (nc-d); c++ )
  {
    head = head + pad(max, string(D[1 , c]), ".") + " ";
    foot = foot + pad(max, string(D[nr, c]), " ") + " ";
  }
  // last head/foot enties:
  head = head + pad(max, string(D[1 , c]), ".");
  foot = foot + pad(max, string(D[nr, c]), " ");
  // head/foot dash lines:
  string dash  = "-"; string dash2  = "=";
  dash = repeat( (nc - 2*d) - 1 , repeat(max, dash) + " " ) + repeat(max, dash) + " "; // dash  = repeat( (max + 1) * (nc - 2*d) , dash );
  dash2 = repeat( (nc - 2*d) - 1 , repeat(max, dash2) + " " ) + repeat(max, dash2) + " "; // dash2 = repeat( (max + 1) * (nc - 2*d) , dash2);
  for ( r = d+1; r <= (nr-d); r++ )
  {
    middle = middle + pad(max, string(D[r,1]), " ") + " :";
    for ( c = d+1; c < (nc-d); c++ ) { middle = middle + pad(max, mstring(D[r,c], "-"), " ") + " "; }
    middle = middle + pad(max, mstring(D[r,nc-d], "-"), " ") + " |" + pad(max, string(D[r,nc]), ".") + newline;
  }
  string corner_id = repeat(max, ".");
  string corner = repeat(max, " ");
  // print everything all at once:
  print                                     (
      corner + "  " + head  + " ." + corner_id + newline +
      corner + "  " + dash  + "+"  + corner_id + newline +
                      middle                          +
      corner + "  " + dash2 + " "  + corner + newline +
      corner + "  " + foot +  "  " + corner );
}

proc grview(N)
"USAGE:  grview(M), graded object M
RETURN:  nothing
PURPOSE: print the degree/grading data about the GRADED matrix/module/ideal/mapping object M
ASSUME:  M must be graded
EXAMPLE: example grview; shows an example
"
{
//  if( size(N) == 0 ) { return (); }
  string msg = "Graded";
  string lst;

  string arrow = " <- ";
  string R = nameof(basering);

  if( typeof( N ) == "list" )
  {
    msg = msg + " resolution"; 
    if( size(R) >= 2 )
    {
      msg = msg + "(let R:="+R+")";
      R = "R";
    }


    int i = 1;  int n = size(N);
    string dst;
    
    msg = msg + ": " + newline + grsumstr(R, grrange(N[i])) + " <-- d_" + string(i) + " -- " ;
    for( ; i < n; i++ )
    { 
      dst = grsumstr(R, grdeg(N[i])) +  " <-- d_" + string(i+1) + " --";
      msg = msg + newline + dst; 
    };

    msg = msg + newline + grsumstr(R, grdeg(N[i])) + ", given by maps: "; 

    print(msg);

    for( i = 1; i <= size(N); i++ )
    { 
      "d_" + string(i) + " :"; grview(N[i]); 
    };

    return ();
  }

//  typeof( N ) ;  attrib( N );  grrange(N);

  ASSUME(1, grtest(N) ); 

  intvec G = grdeg(N);
  matrix M = module(N);

  int nc = ncols(M); int nr = nrows(M);
  int r,c;
  int d = 1; // number of extra cols/rows for extra info around the central degree(N) block in D
  intmat D[nr+2*d][nc+2*d];

  for( c = nc; c > 0; c-- )
  {
    D[1, c+d] = c; // top row indeces
    D[nr+2*d, c+d] = G[c]; // deg(v) + gr[ leadexp(v)[m] ]; // bottom row with computed column induced degrees
  }

  intvec gr = grrange(N); // grading weights?

  for( r = nr; r > 0; r-- )
  {
    D[r+d, 1] = gr[r]; // left-most column with grading data
    for( c = nc; c > 0; c-- )
    {
      D[r+d, c+d] = deg(M[r, c]); // central block with degrees (-1 means zero entry)
    }
    D[r+d, nc+2*d] = r; // right-most block with indeces
  }

  msg = msg + " homomorphism";
  if( size(R) >= 2 )
  {
    msg = msg + "(let R:="+R+")" ;
    R = "R";
  }
    
  msg = msg + ": ";

  string dst = grsumstr(R, gr);
  string src = grsumstr(R, G);

  lst = msg;

  if( (size(lst) + size(dst) + size(src) + 4) > 80 ) 
  { 
    if( (size(lst) + size(dst)) > 80 ) { msg = msg + newline; lst = ""; }

    msg = msg + dst + arrow;
    lst = lst + dst + arrow;

    if( (size(lst) + size(src)) > 80 ) { msg = msg + newline; lst = ""; }

    msg = msg + src;
    lst = lst + src;
  } else
  {
    msg = msg + dst + arrow + src;
    lst = lst + dst + arrow + src;
  }

  if( size(lst) > 70 ) { msg = msg + newline; } // lst = "";
  msg = msg + ", given by ";
//  lst = lst + ", given by ";

  if( size(N) == 0 )
  {
    msg = msg + "zero ("+ string(nr);
    if( nr == nc ) { msg = msg + "^2"; } else { msg = msg + " x " + string(nc); }
    print( msg+") matrix." );
  } else
  {
    if( nr == nc) // square matrix // detect diagonal?
    {
      for (c = nr; c > 0; c-- )
      {
        M[c,c] = 0;
      }

      if( size(module(M)) == 0 )
      {
        msg = msg + "a diagonal matrix";
      } else
      {
        msg = msg + "a square matrix";
      }

    } else 
    { 
      msg = msg + "a matrix";
    }
    
    print(msg + ", with degrees: " );
    draw(D, d); // print it nicely!    
  }
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module A = grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0,0,1) );
  grview(A);

  module B = grobj( module([0,x,y]), intvec(15,1,1) );
  grview(B);

  module D = grsum( grsum(grpower(A,2), grtwist(1,1)), grsum(grtwist(1,2), grpower(B,2)) );
  grview(D);

  ring R = 0,(w,x,y,z), dp; def I = grobj( ideal(y2-xz, xy-wz, x2z-wyz), intvec(0) );
  list res1 = grres(I, 0); // non-minimal
  grview(res1);
  print(betti(res1,0), "betti");

  list res2 = grres(grshift(I, -10), 0, 1); //  minimal!
  grview(res2);
  print(betti(res2,0), "betti");
}

static proc issorted( intvec g, int s )
{
  g = s * g; //  "g: ", g;
  int i = size(g);

  for(; i > 1; i--)
  {
    if( (g[i] - g[i-1]) < 0 )
    {
      return (0);
    }
  }

  return (1);
}

static proc mysort( intvec gr, int s )
"
computes the permutation P of gr, such that (s*gr)[P] is ascendingly sorted
NOTE: looks like a bubble sort (was taken from sort) and modified to ensure stability!
TODO: replace with some kernel function if this turns out to be inefficient!!
"
{
  gr = s * gr;
  int m = size(gr);
  intvec pivot;

  int Bi;
  // compute reordering permutation pivot such that gr[pivot] is (stably) sorted
  for(Bi=m; Bi>0; Bi--) { pivot[Bi]=Bi; } // pivot = Id_m for starters

  int Bn,Bb; int P, D;

  int Bj = 0; int flag = 0;

  // Bi == 0
  while(Bj==0)
  {
    Bi++; Bj=1;
    for(Bn=1; Bn <= (m-Bi); Bn++)
    {
      D = (gr[pivot[Bn]] - gr[pivot[Bn+1]]); // sort gr
      P = (D > 0) or ( (D == 0) && ((pivot[Bn]-pivot[Bn+1]) > 0) ); // stability!?
      if(P)
      {
        Bb=pivot[Bn];
        pivot[Bn]=pivot[Bn+1];
        pivot[Bn+1]=Bb;

        Bj=0; flag = 1;
      }
    }
  }

/*
  if( flag ) // output details in case of non-identical permutation?
  {
    // grades & ordering permutation : gr[pivot] should be sorted:
    "s: ", s;
    "gr: ", gr;
    "pivot: ",    pivot;
  }
*/
  ASSUME(1, issorted(intvec(gr[pivot]), 1));

  return (pivot);
}

// Q@Wolfram: what should be done with zero gens!?
proc grdeg(M)
"USAGE:  grdeg(M), graded object M
RETURN:  intvec of degrees
PURPOSE: graded degrees of columns (generators) of M
ASSUME:  M must be a graded object (matrix/module/ideal/mapping)
NOTE:    if M has zero cols it shoud have attrib(M,'degHomog') set.
EXAMPLE: example grdeg; shows an example
"
{
  ASSUME(1, grtest(M) );

  if ( typeof(attrib(M, "degHomog")) == "intvec" )
  {
    intvec t = attrib(M, "degHomog"); // graded degrees
    ASSUME(0, ncols(M) == size(t) );

    return (t);
  }

  ASSUME(0, ncols(M) == size(M) );

  def w = grrange(M); // grading weights?

  if( size(M) == 0 ){ return (w); } // TODO: Q@Wolfram!???

  int m = ncols(M); // m > 0 in Singular!
  int n = nvars(basering) + 1; // index of mod. column in the leadexp

  module L = lead(M[1..m]); // leading module-terms for input column vectors
  intvec d = deg(L[1..m]); // their degrees
  intvec c = leadexp(L[1..m])[n]; // their module-components

//   w = intvec(-6665), w; // 0?????
  intvec gr = w[c]; //  + 1]; // weights?????

  gr = gr + d; // finally we compute their graded degrees

  return (gr);
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module A = grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0,0,1) );
  grview(A);

  module B = grobj( module([0,x,y]), intvec(15,1,1) );
  grview(B);

  module D = grsum(
                   grsum(grpower(A,2), grtwist(1,1)),
                   grsum(grtwist(1,2), grpower(B,2))
                  );

  grview(D);
  grdeg(D);

  def D10 = grshift(D, 10);

  grview(D10);
  grdeg(D10);
}

static proc reorder(def M, int s)
"
Reorder gens of M: compute graded degrees and the permutation to sort them
"
{
  // input should be graded:
  ASSUME(1, grtest(M) );

  intvec w = grrange(M); // grading weights

  intvec gr = grdeg( M );

//  intvec d = deg(M[1..nocls(M)]); // no need to deal with un-weighted degrees??!

  intvec pivot = mysort(gr, s);

  // grades & ordering permutation for N.  gr[pivot] should be sorted!
  ASSUME(1, issorted(gr[pivot], s));

  module N = grobj(module(M[pivot]), w, intvec(gr[pivot]));  // reorder the starting ideal/module

//  "reorder: "; grview(N);

  return (N, intvec(gr[pivot]));
}

proc grtranspose1(def M)
"USAGE:  grtranspose1(M), graded object or list M
RETURN:  same as input
PURPOSE: graded transpose of graded object or chain complex M
ASSUME:  M must be a graded object or a list of graded objects
EXAMPLE: example grtranspose1; shows an example
"
{
  if( typeof( M ) == "list" )
  {
    if( size(M) == 0 ) { return (); }

    int j = size(M);

    int i = 1;

    // TODO: extra grading argument???
    while( i < j )
    {
     if( size(M[i]) == 0 ){ break; }
     ASSUME(0, typeof(grrange(M[i])) == "intvec");
     i++;
    }

    if( size(M[i]) == 0 ) { i--; }

    j = i; i = 1;

    list L;
    while( j > 0 )
    {
//      grview(M[i]);
      L[j] = grtranspose1( grobj( M[i], grrange(M[i])) );
//      grview(L[j]);

      if( (i > 1) && (j > 0) )
      {
//        grview(L[j+1]);
        ASSUME(2, size( module( matrix(L[j])*matrix(L[j+1]) ) ) == 0 );
      };
//      grview(L[j]);
      j--; i++;
    };
    return (L); // ?
  }

//////
// "a";  grview(M);
  ASSUME(1, grtest(M) );

 intvec d; module N;

 (N,d) = reorder(M, -1);

 kill M; module M = grobj(transpose(N), -d, -grrange(N));

// "b";  grview(M);

 kill N,d; module N; intvec d;
 // reverse order:
 (N,d) = reorder(M, 1); kill M;

// "e"; grview( N );

 ASSUME(1, issorted( grrange(N), 1) );
 ASSUME(1, issorted(grdeg(N), 1) );


 return (N);
}
example
{ "EXAMPLE:"; echo = 2;

  "Surface Name: 'k3.d10.g9.quart2' in P^4";
  int @p=31991; ring R = (@p),(x,y,z,u,v), dp;
  ideal J = 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  def I = grobj( groebner(J), intvec(0) ); // ASSUME: no zero entries in J!
  ASSUME(0, grtest(I));
  "Input degrees: "; grview(I);

  def RR = grres(I, 0, 1); list L = RR; 

  " = Non-minimal betti numbers: "; print(betti(L, 0), "betti");

  "Graded (original) structure of 'res(Input,0)': "; grview(L);

  "Graded transpose of the previous resolution "; list LLL = grtranspose1( L ); grview( LLL );

  "Its non-minimal betti numbers: "; print(betti(LLL, 0), "betti");

}

proc grorder(def M)
"USAGE:  grorder(M), graded object or list M
RETURN:  same as input
PURPOSE: reorder/transform graded object or chain complex M into block form
ASSUME:  M must be a graded object or a list of graded objects
EXAMPLE: example grorder; shows an example
"
{
  if( typeof(M) == "list" )
  { // TODO: extra grading argument???
    if( size(M) == 0 ) { return (); }

    int j = size(M);  int i = 1;

    while( i < j )
    {
      if( size(M[i]) == 0 ){ break; }
      ASSUME(0, typeof(grrange(M[i])) == "intvec");
      i++;
    }

    if( size(M[i]) == 0 ) { i--; }

    list L; module Z = 0; L[i] = Z; j = i;

    while( i > 0 )
    {
//      "i: ", i;      "A"; grview(M[i]);
      L[i] = grorder( grobj( M[i], grrange(M[i])) );
//      "B"; grview(L[i]);
      if( i < j )
      {
        ASSUME(2, size( module( matrix(transpose(L[i+1]))*matrix(transpose(L[i])) ) ) == 0 );
      };

      i--;
    };

    return (L); // ?
  }

  ASSUME(1, grtest(M) );

// "a";  grview(M);

  intvec d; module N;

  (N,d) = reorder(M, 1); kill M;

  module M = grobj(transpose(N), -d, -grrange(N)); kill N,d;

// "b";  grview(M);

  module N; intvec d;
  // reverse order:
  (N,d) = reorder(M, -1); kill M;

  module M = grobj(transpose(N), -d, -grrange(N));

// "c";  grview(M);

  ASSUME(1, issorted(grrange(M), 1) );
  ASSUME(1, issorted(grdeg(M), 1) );

  return (M);
}
example
{ "EXAMPLE:"; echo = 2;

  "Surface Name: 'rat.d10.g9.quart2' in P^4";
  int @p=31991; ring R = (@p),(x,y,z,u,v), dp;
  ideal J = 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  def I = grobj( groebner(J), intvec(0) ); // ASSUME: no zero entries in J!
  ASSUME(0, grtest(I));

  "Input degrees: "; grview(I);

  def RR = grres(I, 0, 1);
  list L = RR;

  " = Non-minimal betti numbers: ";  print(betti(L, 0), "betti");
  "Graded reordered structure of 'res(Input,0)': ";  grview(grorder(L));
}

proc TestGRRes(Name, J)
"USAGE:  TestGRRes(name, I), string name, ideal I
RETURN:  nothing
PURPOSE: compute/test/output/order/transpose a graded resolution of I
EXAMPLE: example TestGRRes; shows an example
"
{
  "==============================================";
  "";
  "=== Example: [", Name, "]";
  " = Ring: ", string(basering);

  def I = grobj( groebner(J), intvec(0) ); // ASSUME: no zero entries in J!
  ASSUME(0, grtest(I));
//  " = Input degrees: "; grview(I);

  " ! Resolution via 'grres': ";
  def R = grres(I, 0, 1); // sres, lres: no grading! // nres, mres - graded (with attrib(, "isHomog"))

  " = Non-minimal betti numbers: ";
  print(betti(R, 0), "betti");

  list L = R ; // SRES_list(R); //  " = Degrees of maps: "; grview(L); // MUST BE GRADED!!!

  " = Degrees of (ordered) maps: ";
  def LL = grorder(L); // MUST BE GRADED

  // ordres(L, intvec(0)); // ?

  grview( LL ); " = TRANSPOSE'd complex: %%%%%%%%%%%%%%";
  list LLL = grtranspose1( LL ); //  resolution RR = LLL;
  print(betti(LLL, 0), "betti");

  grview( LLL ); ""; //  "==============================================";

  kill L, R;
}
example
{ "EXAMPLE:"; echo = 2;
//  if( defined(assumeLevel) ){ int assumeLevel0 = assumeLevel; } else { int assumeLevel; export(assumeLevel); }; assumeLevel = 5; // store the state of aL

  // note: data from random generation 2
  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;  "";

  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;  "";

  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;  "";

  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;  "";

  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; "";

  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;  "";

  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; "";

  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; "";

  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; "";

  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; "";

//  if( defined(assumeLevel0) ){ assumeLevel = assumeLevel0; } else { kill assumeLevel; } // restore the state of aL
}

/////////////////////////////////////////////////////////

// Q@Woflram?
proc grzero()
"USAGE:  grzero()
RETURN:  graded object representing S(0)^1
PURPOSE: compute presentation of S(0)^1
EXAMPLE: example grzero; shows an example
"
{
 return ( grobj(module([0]), intvec(0), intvec(0)) );
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  def M = grpower( grshift( grzero(), 10), 5 );

  print(M);

  grview(M);
}

proc grpower(def A, int p)
"USAGE:  grpower(A, p), graded object A, int p > 0
RETURN:  graded direct power A^p
PURPOSE: compute the graded direct power A^p
NOTE:    the power p must be positive
EXAMPLE: example grpower; shows an example
"
{
  if(p==0){ ERROR("Sorry, we don't know what is A^0!?!?"); } // grzero!?

  ASSUME(0, p > 0);
  ASSUME(1, grtest(A) );

  if(p==1){ return(A); }

  def N = grsum(A,A);

  if(p==2){ return(N); }

  // TODO: replace recursion with a loop!
  // see http://en.wikipedia.org/wiki/Exponentiation_by_squaring
  if((p%2)==0)
    { return ( grpower(N, p div 2) ); }
  else
    { return ( grsum( A, grpower(N, (p-1) div 2) )); }
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module A = grobj( module([x+y, x, 0], [0, x+y, y]), intvec(1,1,1) );
  grview(A);

  module B = grobj( module([x,y]), intvec(2,2) );
  grview(B);

  module D = grsum( grpower(A,2), grpower(B,2) );

  print(D);
  homog(D);
  grview(D);
}


proc grsum(A,B)
"USAGE:  grsum(A, B), graded objects A and B
RETURN:  graded direct sum of input objects
PURPOSE: compute the graded direct sum of A and B
EXAMPLE: example grsum; shows an example
"
{
  ASSUME(1, grtest(A) );
  ASSUME(1, grtest(B) );

  intvec a = grrange(A);
  intvec b = grrange(B);
  intvec c = a,b;
  int r = nrows(A);

  module T = align(module(B), r); //  T;  print(T);  nrows(T); // BUG!!!!
  module S = module(A), T;

  intvec da = grdeg(A);
  intvec db = grdeg(B);
  intvec dc = da, db;

  return(grobj(S, c, dc));
}
example
{ "EXAMPLE:"; echo = 2;

//  if( defined(assumeLevel) ){ int assumeLevel0 = assumeLevel; } else { int assumeLevel; export(assumeLevel); }; assumeLevel = 5;

  ring r=32003,(x,y,z),dp;

  module A = grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0,0,1) );
  grview(A);

  module B = grobj( module([0,x,y]), intvec(15,1,1) );
  grview(B);

  module C = grsum(A,B);

  print(C);
  homog(C);
  grview(C);

  module D = grsum(
     grsum(grpower(A,2), grtwist(1,1)),
     grsum(grtwist(1,2), grpower(B,2))
     );

  print(D);
  homog(D);
  grview(D);

  module F = grobj( module([x,y,0]), intvec(1,1,5) );
  grview(F);

  module T = grsum( F, grsum( grtwist(1, 10), B ) );
  grview(T);

//  if( defined(assumeLevel0) ){ assumeLevel = assumeLevel0; } else { kill assumeLevel; } // restore the state of aL
}

proc grshift( def M, int d)
"USAGE:  grshift(A, d), graded objects A, int d
RETURN:  shifted graded object
PURPOSE: shift the grading on A by d: A_i -> A_{i+/-d}?
EXAMPLE: example grshift; shows an example
"
{
  ASSUME(1, grtest(M) );

  intvec a = grrange(M);
  a = a - intvec(d:size(a));

  intvec t = attrib(M, "degHomog");
  t = t - intvec(d:size(t));

  return (grobj(M, a, t));
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module A = grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0,0,1) );

  grview(A);

  module S = grshift( A,  6);

  grview(S);
}


proc grisequal (def A, def B)
"USAGE:  grisequal(A, B), graded objects A and B
RETURN:  1 if A == B as graded objects, 0 otherwise
PURPOSE: test the equality of two graded object
NOTE: A and B should be literarly the same at the moment. TODO?
EXAMPLE: example grisequal; shows an example
"
{
  ASSUME(1, grtest(A) );
  ASSUME(1, grtest(B) );

  int ra = nrows(A);
  int rb = nrows(B);

  intvec wa = grrange(A);
  intvec wb = grrange(B);

  if( (ra != rb) || (ncols(A) != ncols(B)) ){ return (0); } // TODO: ???

  intvec da = grdeg(A);
  intvec db = grdeg(B);

  return ( (da == db) && (wa == wb) &&
           (size(module(matrix(A) - matrix(B))) == 0)    ); // TODO: ???
}
example
{ "EXAMPLE:"; echo = 2;
  TODO
}

proc grtwist(int a, int d)
"USAGE:  grtwist(a,d), int a, d
RETURN:  graded object representing S(d)^a
PURPOSE: compute presentation of S(d)^a
EXAMPLE: example grtwist; shows an example
"
{
  ASSUME(0, a > 0);

  module Z; Z[a] = [0];
  intvec w = -intvec(d:a);
  Z = grobj(Z, w, w); // will set the rank as well
  ASSUME(2, grisequal(Z, grpower( grshift(grzero(), d), a ) )); // optional check
  return(Z);
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  grview(grpower( grshift(grzero(), 10), 5 ) );

  grview( grtwist (5, 10) );
}

proc grobj(def A, intvec w, list #)
"USAGE:  grobj(M, w[, d]), matrix/ideal/module M, intvec w, d
RETURN:  graded object with matrix presentation M, row weighting w [and total graded degrees d of columns]
PURPOSE: create a valid graded object with a given matrix presentation, weighting [and total graded degrees (in case of zero columns)]
EXAMPLE: example grobj; shows an example
"
{
  module M = module(A);
  ASSUME(0, size(w) >= nrows(M) );

  attrib( M, "rank", size(w) );
  attrib( M, "isHomog", w );

  if( size(#) > 0 )
  {
    ASSUME(0, typeof(#[1]) == "intvec" );
    ASSUME(0, size(#[1]) == ncols(M) );
    attrib(M, "degHomog", #[1]);
  }
  else
  {
    ASSUME(0, /* no zero cols please! */ size(M) == ncols(M) );
    attrib(M, "degHomog", grdeg(M));
  }

  ASSUME(0, grtest(M) );
  return (M);
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module A = grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0,0,1) );
  grview(A);

  module F = grobj( module([x,y,0]), intvec(1,1,5) );
  grview(F);

  int d = 666; // zero can have any degree...
  module Z = grobj( module([x,0], [0,0,0], [0, y]), intvec(1,2,3), intvec(2, d, 3) );
  grview(Z);

  print(Z);
  attrib(Z);
  grrange(Z); // module weights
  attrib(Z, "degHomog"); // total degrees

}

proc grtest(def N, list #)
"USAGE:  grtest(M[,b]), anyting M, optionally int b
RETURN:  1 if M is a valid graded object, 0 otherwise
PURPOSE: validate a graded object. Print an invalid object message if b is not given 
NOTE: M should be an ideal or module or matrix, with weighting attribute
   'isHomog' and optionally total graded degrees attribute 'degHomog'.
   Attributes should be compatible with the presentation matrix.
EXAMPLE: example grtest; shows an example
"
{
  int b = (size(#) == 0);
  string t = typeof(N);
  if( (t != "ideal") && (t != "module") && (t != "matrix") )
  {
    if(b) { "   ? grtest: Input should be something like a matrix!"; };
    return (0);
  };

  if ( typeof(grrange(N)) != "intvec" )
  {
    if(b) { type(N); attrib(N); "   ? grtest: Input must be graded!";  };
    return (0);
  };

  intvec gr = grrange(N); // grading weights...
  if ( nrows(N) != size(gr) )
  {
    if(b) { "   ? grtest: Input has wrong number of rows!"; };    
    return (0);
  };

//  if( attrib(N, "rank") != size(gr) ){ return (0); } // wrong rank :(

  if ( typeof(attrib(N, "degHomog")) == "intvec" )
  {
    intvec T = attrib(N, "degHomog"); // graded degrees
    
    if ( ncols(N) != size(T) )
    {
      if(b) { "   ? grtest: Input has wrong number of cols!"; };    
      return (0);
    };
    
    int k = nvars(basering) + 1; // index of mod. column in the leadexp

    module L = lead(module(N)); vector v;

    // checking T for non-zero N[i]
    int i = size(T);

    for (; i > 0; i-- )
    {
      v = L[i];
      if( v != 0 )
      {
        if( (deg(v) + gr[ leadexp(v)[k] ]) != T[i] )
        {
          if(b) { "   ? grtest: Input has wrong total grade of " + string(i) + "-th column!"; };
          return (0);
        };  // wrong T[i]
      }
    }

    // TODO: check t on nonzero cols...
  } else
  {
    if( ncols(N) != size(N) )
    {
      if(b) { "   ? grtest: Input should have exclusively non-zero columns, please give total grades otherwise!"; };
      return (0);
    };
  }

  if( !homog(N) )
  {
    if(b) { "   ? grtest: Input should be graded homogenous!"; };
    return (0);
  };

//  if(b) { "Input seems to be a valid graded object (map)!"; };
  return (1);
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  // the following calls will fail due to tests in grtest:

// grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0) ); // not enough row weights
// grobj( module([x,0], [0,0,0], [0, y]), intvec(1,2,3) ); // zero column needs (otherwise optional) total degrees
// grobj( module([x,0], [0,0,0], [0, y]), intvec(1,2,3), intvec(1, 1, 1) ); // incompatible total degrees (on non-zero columns)

}


static proc align( def A, int d)
"analog of align kernel command for older Singular versions
 this is static since it should not be used by @code{align}-able (newer)
 Singular releases.
 Note that this proc does not care about any attributes (of A)
"
{
  module T; T[d] = 0;
  T = T, module(transpose(A));
  return( module(transpose(T)) );
}

proc grgroebner(A)
"USAGE:  grgroebner(M), graded object M
RETURN:  graded object
PURPOSE: compute graded groebner basis of M
EXAMPLE: example grgroebner; shows an example
"
{
  ASSUME(1, grtest(A));

  return ( grobj( groebner(A), grrange(A) ) );
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module A = grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0,0,1) );
  grview(A);
  
  module B = grgroebner(A);
  grview(B);
}


proc grtwists(intvec v)
"USAGE:  grtwists(v), intvec v
RETURN:  graded object representing S(v[1]) + ... + S(v[size(v)])
PURPOSE: compute presentation of S(v[1]) + ... + S(v[size(v)])
EXAMPLE: example grtwists; shows an example
"
{
  int l = size(v);
  module Z; Z[l] = [0];
  Z = grobj(Z, v, v); // will set the rank as well
  return(Z);
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;
  
  grview( grtwists ( intvec(-4, 1, 6 )) );
}


proc grsyz(A)
"USAGE:  grsyz(M), graded object M
RETURN:  graded object
PURPOSE: compute graded syzygy of M
EXAMPLE: example grsyz; shows an example
"
{
  ASSUME(1, grtest(A));
  intvec v = grdeg(A);

//  grrange(A);
  
  module M = syz(A); 
//  print(M);
  
  if( size(M) > 0 ) { return( grobj( M, v ) ); }

  // zero syzygy?
  return( grtwists(v) ); // ???
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module A = grobj( module([x+y, x, 0, 3], [0, x+y, y, 2], [y, y, z, 1]), intvec(0,0,0,1) ); 
  grview(A);
  
  module B = grgroebner(A);
  grview(B);

  module C = grsyz(B);  
  grview(C);
}


proc grprod(A, B)
"USAGE:  grprod(M, N), graded objects M and N
RETURN:  graded object
PURPOSE: compute graded product M * N (as composition of maps)
EXAMPLE: example grprod; shows an example
"
{
  ASSUME(1, grtest(A));
  ASSUME(1, grtest(B));

  // TODO: ASSUME(0, grdeg() == grrange());!!! 

  return ( grobj( A*B, grrange(A), grdeg(B) ) );
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module A = grobj( module([x+y, x, 0, 3], [0, x+y, y, 2], [y, y, z, 1]), intvec(0,0,0,1) ); 
  grview(A);
  
  A = grgroebner(A);
  grview(A);
  
  module B = grsyz(A);
  grview(B);
  print(B);
  
  module D = grprod( A, B );
  grview(D);
  print(D); // must be all zeroes due to syzygy property!
  ASSUME(0, size(D) == 0);
}




proc grres(def A, int l, list #)
"USAGE:  grres(M, l[, b]), graded object M, int l, int b
RETURN:  graded resolution = list of graded objects
PURPOSE: compute graded resolution of M (of length l) and minimise it if b was given                             
EXAMPLE: example grres; shows an example
"
{
  ASSUME(0, l >= 0);
  ASSUME(1, grtest(A));

  intvec v = grrange(A);
  
  int b = (size(#) > 0);
  if(b) { list r = res(A, l, #[1]); } else { list r = res(A, l); }

//  r;  v;

  l = size(r);
  
  int i; module m;
  
  for ( i = 1; i <= l; i++ )
  {
    if( size(r[i]) == 0 ){ r[i] = grtwists(v); i++; break;   }

    r[i] = grobj(r[i], v); v = grdeg(r[i]);
  }
  i = i-1;

  return( list(r[1..i]) );
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module A = grobj( module([x+y, x, 0, 3], [0, x+y, y, 2], [y, y, z, 1]), intvec(0,0,0,1) ); 
  grview(A);
  
  module B = grgroebner(A);
  grview(B);

  "graded resolution of B: "; def C = grres(B, 0); grview(C);

  int i; int l = size(C);

  "D^2 == 0: "; for (i = 1; i < l; i++ ) { i; grview( grprod(C[i], C[i+1]) ); }
}

proc grtranspose(def M)
"
USAGE:   grtranspose(M), graded object M
RETURN:  graded object
PURPOSE: graded transpose of M
NOTE:    no reordering is performend by this procedure   
EXAMPLE: example grtranspose; shows an example
"
{
  ASSUME(1, grtest(M) );
  return (  grobj(transpose(M), -grdeg(M), -grrange(M))  );
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module M = grtwists( intvec(-2, 0, 4, 4) ); grview(M);

  module N = grsyz( grtranspose( M ) ); grview(N);

  module L = grtranspose(N); grview( L );

  module K = grsyz( L ); grview(K);
  

  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);

  "graded transpose: "; module B = grtranspose(A); grview( B ); print(B); 

  "... syzygy: "; module C = grsyz(B); grview(C);

  "... transposed: "; module D = grtranspose(C); grview( D ); print (D);

  "... and back to presentation: "; module E = grsyz( D ); grview(E); print(E);

  module F = grgens( E ); grview(F); print(F);
 
  module G = grpres( F ); grview(G); print(G);
}


proc grgens(def M)
"
USAGE:   grgens(M), graded object M (map)
RETURN:  graded object  
PURPOSE: try compute graded generators of coker(M) and return them as columns
         of a graded map.
NOTE:    presentation of resulting generated submodule may be different to M!
EXAMPLE: example grgens; shows an example
"
{
  ASSUME(1, grtest(M) );

  module N = grtranspose( grsyz( grtranspose(M) ) );
  
//  ASSUME(3, grisequal( grgroebner(M), grgroebner( grpres( N ) ) ) ); // FIXME: not always true!?
  
  return ( N );
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module M = grtwists( intvec(-2, 0, 4, 4) ); grview(M);

  module N = grgens(M);
  
  grview( N ); print(N); // fine == M


  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);

  module B = grgens(A);

  grview( B ); print(B); // Ups :( != A
  
}


proc grpres(def M)
"
USAGE:   grpres(M), graded object M (submodule gens)
RETURN:  graded module (via coker)
PURPOSE: compute graded presentation matrix of submodule generated by columns of M
EXAMPLE: example grpres; shows an example
"
{
  ASSUME(1, grtest(M) );

  module N = grsyz(M);

//  ASSUME(3, grisequal( M, grgens( N ) ) );
  
  return ( N );
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module M = grtwists( intvec(-2, 0, 4, 4) ); grview(M);

  module N = grgens(M); grview( N ); print(N);

  module L = grpres( N ); grview( L ); print(L);


  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);

  module B = grgens(A); grview( B ); print(B);

  module C = grpres( B ); grview( C ); print(C);
}



LIB "random.lib"; // for sparsepoly

proc grrndmat(intvec w, intvec v, list #)
"USAGE:  grrndmat(src,dst[,p,b]), intvec src, dst[, int p, b]
RETURN:  matrix of polynomials
PURPOSE: generate random matrix compatible with src and dst gradings
NOTE:    optional arguments p, b are for 'sparsepoly' (by default: 75%, 30000).
TODO:    this is experimental at the moment!
EXAMPLE: example grrndmat; shows an example
"
{
  // defaults for sparsepoly
  int p = 75;
  int b = 30000;

  if ( size(#) > 0 )
  {
    ASSUME( 0, (typeof(#[1]) == "int") || (typeof(#[1]) == "bigint") );
    p = #[1];

    if ( size(#) > 1 )
    {
      ASSUME( 0, (typeof(#[2]) == "int") || (typeof(#[2]) == "bigint") );
      b = #[2];
    }
  }

  int n = size(v); // destination: rows!
  int m = size(w); // source: cols

  matrix M[n][m];

  int r,c; intvec ww;

  for( c = m; c > 0; c-- )
  {
    ww = v - intvec(w[c]:n);
    for( r = n; r > 0; r-- )
    {
      if( ww[r] >= 0)
      {
        M[r,c] = sparsepoly(ww[r], ww[r], p, b);
      }

    }
  }

  return(M);
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  print( grrndmat( intvec(0, 1), intvec(1, 2, 3) ) );
}

//


proc KeneshlouMatrixPresentation(intvec a)
"USAGE:  intvec a.
RETURN: graded object
PURPOSE: matrix presentation for direct sum of omega^a[i](i) in form of a graded object
EXAMPLE: example KeneshlouMatrixPresentation; shows an example
{
  int n = size(a)-1;
  //  ring r = 32003,(x(0..n)),dp;
  ASSUME(0, nvars(basering)==(n+1));
  int i,j;

  // find first nonzero exponent a_i
  for(i=1;i<=size(a);i++)
    {
      if(a[i]!=0) {break; };
    }

  // all zeroes?
  if(i>size(a)) {return (grzero()); };

  for(i=2;i<=n;i++)
    {
      if(a[i]!=0) {break; };
    }

  module N;

  if(i>n) 
    { // no middle part
      if(a[1]>0)
        {
          N=grtwist(a[1],0);

          if(a[n+1]>0)
            { N=grsum(N,grtwist(a[n+1],-1));}        
        }  
      else
        { N=grtwist(a[n+1],-1);}
      
      return (N); // grorder(N)); 
    }
  else // i <= n: middle part is present, a_i != 0 
    { // a = a1  ... |  i:2, a_2 ..... i: n, a_n | .... i: n+1a_(n+1)
      j = i - 1;
      module I = maxideal(1); attrib(I,"isHomog", intvec(0)); list L = mres(I, 0); // TODO: use grres() instead!!!
      list kos = grorder(L);
      // make sure that graded maps  are represented by blocks corresponding to the betti diagram?

      def S = grpower(grshift(grobj( kos[j+2], attrib(kos[j+2], "isHomog")), j), a[i]);

      i++;

      for(; i <= n; i++)
        {
          if(a[i]==0) { i++; continue; }
          j = i - 1;
          S = grsum( S, grpower(grshift( grobj( kos[j+2], attrib(kos[j+2], "isHomog")), j), a[i])  );
        }

      // S is the middle (non-zero) part

      if(a[1] > 0 )
        {
          N=grsum(grtwist(a[1],0), S);
        }
      else
        { N = S;}

      if(a[n+1] > 0 )
        { N=grsum(N, grtwist(a[n+1],-1)); }


      return ((N)); //      return (grorder(N)); 
    }
}
example
{ "EXAMPLE:"; echo = 2;
  ring r = 32003,(x(0..4)),dp;

  def N1 = KeneshlouMatrixPresentation(intvec(2,0,0,0,0)); 
  grview(N1);

  def N2 = KeneshlouMatrixPresentation(intvec(0,0,0,0,3));
  grview(N2);

  def N = KeneshlouMatrixPresentation(intvec(2,0,0,0,3));
  grview(N);


  def M1 = KeneshlouMatrixPresentation(intvec(0,1,0,0,0));
  grview(M1);

  def M2 = KeneshlouMatrixPresentation(intvec(0,1,1,0,0));
  grview(M2);

  def M3 = KeneshlouMatrixPresentation(intvec(0,0,0,1,0));
  grview(M3);

  def M = KeneshlouMatrixPresentation(intvec(1,1,1,0,0));
  grview(M);


}

proc grconcat(A,B)
"USAGE: grconcat(A, B), graded objects A and B, dst(A) == dst(B) =: dst
RETURN: graded object
PURPOSE: construct src(A) + src(B) -----> dst  given by (A|B)
EXAMPLE: example grconcat; shows an example
"
{

  ASSUME(1, grtest(A));
  ASSUME(1, grtest(B));
  ASSUME(0, grrange(A)==grrange(B));
 
  intvec v = grrange(A);
  intvec w=grdeg(A),grdeg(B);
  return(grobj(concat(A,B),v,w));
}
example
{ "EXAMPLE:"; echo = 2;
  ring r;
  
  module R=grobj(module([x,y,z]),intvec(0:3));
  grview(R);

  module S=grobj(module([x,0,y],[xy,zy+x2,0]),intvec(0:3));
  grview(S);

  def Q=grconcat(R,S);
  grview(Q);
}


proc grlift(A, B)
"USAGE: grlift(M, N), graded objects M and N
RETURN: transformation matrix (graded object???)
PURPOSE: compute graded matrix which the generators of submodule Im(N) in terms of Im(M).
EXAMPLE: example grlift; shows an example
"
{
  ASSUME(1, grtest(A));
  ASSUME(1, grtest(B));
  ASSUME(0, grrange(A) == grrange(B));

//  matrix T;  module AA = liftstd(A, T); //  AA = module(A*T)
//  matrix U;
  matrix L =lift(A,B/*,U*/);  //  module(B*U) = module(matrix(A)*L)
  
  return(grobj(L, grdeg(A), grdeg(B)));
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;
  module P=grobj(module([xy,0,xz]),intvec(0,1,0));
  grview(P);


  module D=grobj(module([y,0,z],[x2+y2,z,0]),intvec(0,1,0)); 
  grview(D);

  def G=grlift(D,P);
  grview(G);

  ASSUME(0, grisequal( grprod(D, G), P) );  
}

proc grrange(M)
{
//  ASSUME(1, grtest(M)); // Leads to recursive call due to grtest...
  return( attrib(M, "isHomog") );
}

proc grlift0(M, N, alpha1)
"PURPOSE: generic random alpha0 : coker(M) -> coker(N) from random alpha1
NOTE: this proc can work only if some assumptions are fulfilled (due
to Wolfram)! e.g. at the end of a resolution for the source module...
"
{

  ASSUME(1, grtest(M));
  ASSUME(1, grtest(N));

  ASSUME(0, grdeg(M) == grdeg(alpha1) );
  ASSUME(0, grdeg(N) == grrange(alpha1) );
  return(
   grtranspose( grlift( grtranspose( M ),
       grtranspose( grprod( N,  alpha1 ) ) )
       ) ); // alpha0!
    
}
example
{ "EXAMPLE:"; echo = 2;

  ring S = 0, (x(0..3)), dp;
  list kos = grres(grobj(maxideal(1), intvec(0)), 0);
  print( betti(kos), "betti");
  grview(kos);


  // source module:
//  module M = grshift(kos[4], 2); // phi, Syz_3(K(2))
  def M = KeneshlouMatrixPresentation(intvec(0,0,1,0));
//   grview( grres(M, 0) );
  grview(M);

  // destination module:
//   module N = grshift(kos[3], 1); // psi, Syz_2(K(1))
  def N = KeneshlouMatrixPresentation(intvec(0,1,0,0));
//  grview( grres(N, 0) );
  grview(N);

  // random graded of degree 0, homomorphism of free presentations:
  // alpha1: src(M) -> src (N)
  def alpha1 = grrndmap( M, N ); // alpha1
  grview(alpha1);

  // random graded of degree 0, homomorphism of free presentations:
  // alpha0: dst(M) -> dst (N)
  def alpha0 = grlift0(M, N, alpha1);
  grview(alpha0);

}


proc grlifting(M,N)
"USAGE:  graded objects M and N
RETURN: map of chain complexes (as a list)
PURPOSE: construct a map of chain complexes between free resolutions of Img(M) and Img(N).
EXAMPLE: example grlifting; shows an example
"
{  ASSUME(1, grtest(M));
   ASSUME(1, grtest(N));

   list rM=grres(M,0,1);
   list rN=grres(N,0,1);
   int i,j,k;

  for(i=1;i<=size(rM);i++)
  {
    if(size(rM[i])==0){break;}
  }

  for(j=1;j<=size(rN);j++)
  {
    if(size(rN[j])==0){break;}
  }
  int t=min(i,j);

  ASSUME(0, t >= 2);
  
  list P;

  "t: ", t;
  
  P[1]= grrndmap( rM[1], rN[1] ); // alpha1

  if(t==2){return(P[1]);}  
    
  for(k=2; k<=t; k++)
  {
    P[k] = grlift( grprod(P[k-1],rM[k]), rN[k] ); 
     grview(P[k]);  
    
  }
      
  return(P);
    
}
example
{ "EXAMPLE:"; echo = 2;
/*
  ring r=32003,(x,y,z),dp;

  module P=grobj(module([xy,0,xz]),intvec(0,1,0)); 
  grview(P);

  module D=grobj(module([y,0,z],[x2+y2,z,0]),intvec(0,1,0)); 
  grview(D);

  def G=grlifting(D,P);
  grview(G);

  kill r;
  ring r=32003,(x,y,z),dp;

  module D=grobj(module([y,0,z],[x2+y2,z,0], [z3, xy, xy2]),intvec(0,1,0));
  D = grgroebner(D);
  grview( grres(D, 0)); 

  def G=grlifting(D, D);
  grview(G);
*/

  ring S = 0, (x(0..3)), dp;
  list kos = grres(grobj(maxideal(1), intvec(0)), 0);
  print( betti(kos), "betti");
  grview(kos);

//  module M = grshift(kos[4], 2); // phi, Syz_3(K(2))
  def M = KeneshlouMatrixPresentation(intvec(0,0,1,0));
  grview( grres(M, 0) );
  
//   module N = grshift(kos[3], 1); // psi, Syz_2(K(1))
  def N = KeneshlouMatrixPresentation(intvec(0,1,0,0));
  grview( grres(N, 0) );

  grlifting(M, N); // grview(G);


//  def G=grlifting( grgens(M), grgens(N) );  grview(G);


}

proc mappingcone(M,N)
"USAGE:M,N graded objects
RETURN: chain complex (as a list) 
PURPOSE: construct a free resolution of the cokernel of a random map between Img(M), and Img(N).
EXAMPLE: example mappingcone; shows an example
"
{
  ASSUME(1, grtest(M));
  ASSUME(1, grtest(N));

  list P=grlifting(M,N);
  list rM=grres(M,1);
  list rN=grres(N,1);

  int i;
  list T;

  T[1]=grconcat(P[1],rN[2]);

  for(i=2;i<=size(P);i++)
  {
    intvec v=grrange(rM[i]);
    intvec w=grdeg(rN[i+1]);
    int r=size(v);
    int s=size(w);
    module zero = (0:s);

    module A=grconcat(P[i],rN[i+1]);
    module B=grobj(zero,v,w);
    module C=grconcat(-rM[i],B);
    module D=grconcat(grtranspose(C), grtranspose(A));

    T[i]=grtranspose(D);
  }
   return(T);
}
example
{ "EXAMPLE:"; echo = 2;

ring r=32003, (x(0..4)),dp;
def A=KeneshlouMatrixPresentation(intvec(0,0,0,0,3));
def M=grgens(A);
grview(M);

def B=KeneshlouMatrixPresentation(intvec(0,1,0,0,0));
def N=grgens(B);
grview(N);

def R=grlifting(M,N);
grview(R);
def T=mappingcone(M,N);
grview(T);

def U=grtranspose(T[1]);
resolution G=mres(U,0);
print(betti(G),"betti");
ideal I=groebner(flatten(G[2]));
resolution GI=mres(I,0);
print(betti(GI),"betti");
}

// correct
proc grrndmap(def S, def D, list #) 
"USAGE: (S,D), graded objects S and D
RETURN: graded object
PURPOSE: construct a random 0-deg graded homomorphism src(S) -> src(D)
EXAMPLE: example grrndmap; shows an example
"
{

ASSUME(1, grtest(S) );
ASSUME(1, grtest(D) );

// "src: "; grview(S);"dst: "; grview(D);

intvec v = -grdeg(S); // source
intvec w = -grdeg(D); // destination

return (grobj(grrndmat(v, w, #), -w, -v ) );
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module D=grobj(module([y,0,z],[x2+y2,z,0]),intvec(0,1,0));
  grview(D);

  module S=grobj(module([x,0,y],[xy,zy+x2,0]),intvec(0,0,0));
  grview(S);

  def H=grrndmap(D,S);
  grview(H);

}


proc grrndmap2(def D, def S, list #)
"USAGE: (D,S), graded objects S and D
RETURN: graded object
PURPOSE: construct a random 0-deg graded homomorphism between target of D and S.
EXAMPLE: example grrndmap2; shows an example
"
{
  ASSUME(1, grtest(D) );
  ASSUME(1, grtest(S) );
  intvec v = -grrange(D); // source
  intvec w = -grrange(S); // target
  return (grobj(grrndmat(v, w, #), -w, -v ) );
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module D=grobj(module([y,0,z],[x2+y2,z,0]),intvec(0,1,0));
  grview(D);

  module S=grobj(module([x,0,y],[xy,zy+x2,0]),intvec(0,0,0));
  grview(S);

  def G=grrndmap2(D,S);
  grview(G);

}


//                A     f2     f3
// 0<---M<----F0<----F1<----F2<----F3<----
//              |p1   |p2
// 
// 0<---N<----G0<----G1<----G2<----G3<----
//                B(g1)      g2     g3
// 
proc grlifting2(A,B)
"USAGE: (A,B), graded objects A and B (matrices defining maps)
RETURN: map of chain complexes (as a list) 
PURPOSE: construct a map of chain complexes between free resolution of
M=coker(A) and N=coker(B).
EXAMPLE: example grlifting2; shows an example
"
{  ASSUME(1, grtest(A));
   ASSUME(1, grtest(B));

   list rM=grres(A,0);
   list rN=grres(B,0);
   int i,j,k;
   list P;

  // find first zero matrix in rM
  for(i=1;i<=size(rM);i++)
  {
    if(size(rM[i])==0){break;}
  }

  // find first zero matrix in rN
  for(j=1;j<=size(rN);j++)
  {
    if(size(rN[j])==0){break;}
  }

  int t=min(i,j);

  P[1]=grrndmap2(A,B);

  // A(or B)=0 
  if(t==1){return(P[1])};
 
  for(k=2;k<=t;k++)
  {
   def E=grprod(P[k-1],rM[k-1]);
   P[k]=grlift(rN[k-1],E); // ---------->
   /* let yi=(pi)o(fi); to take grlift(gi,yi)
      we should have img(yi) is contained in
      img(gi)=ker(gi-1),i.e (gi-1)oyi=0. we have
      (gi-1)oyi=(gi-1)o(pi)o(fi)=(pi-1)o(fi-1)ofi=0
  */
  }
  return(P);
}
example
{"EXAMPLE:"; echo = 2;

ring r;
module P=grobj(module([xy,0,xz]),intvec(0,1,0));
grview(P);

module D=grobj(module([y,0,z],[x2+y2,z,0]),intvec(0,1,0));
grview(D);

module PP = grpres(P);
grview(PP);

module DD = grpres(D);
grview(DD);


def T=grlifting2(DD,PP); T;

// def Z=grlifting2(P,D); Z; // WRONG!!!  
             
}


/*
//              -f1 0       -f2 0
//   (p1 g1)     p2 g2       p3 g3
// G0<-----F0+G1<------F1+G2<-------F2+G3<-----
proc mappingcone2(A,B)
"USAGE: (A,B), graded objects A and B (matrices defining maps)
RETURN: chain complex (as a list) 
PURPOSE: construct the free resolution of a cokernel of a random map between M=coker(A), and N=coker(B)
EXAMPLE: example mappingcone2;
"

{
  ASSUME(1, grtest(A));
  ASSUME(1, grtest(B));

  list P=grlifting2(A,B);
  list rM=grres(A,1);
  list rN=grres(B,1);

  int i;
  list T;

  T[1]=grconcat(P[1],rN[1]);

  for(i=2;i<=size(P);i++)
  {
    intvec v=grrange(rM[i-1]);
    intvec w=grdeg(rN[i]);
    int r=size(v);
    int s=size(w);
    matrix zero[r][s];

    module A=grconcat(P[i],rN[i]);
    module B=grobj(zero,v,w);

    module C=grconcat( grneg( rM[i-1] ) ,B);
    module D=grconcat(grtranspose(C), grtranspose(A));

    T[i]=grtranspose(D);
  }
  return(T);
}



example
{ "EXAMPLE:"; echo = 2;

ring r=32003,(x(0..4)),dp;
def I=maxideal(1);
module R=grobj(module(I), intvec(0));
resolution FR=mres(R,0);
print(betti(FR,0),"betti");
module K=grobj(module(FR[1]),intvec(-1),intvec(0:5));
grview(K);

module S=grsyz(K);
grview(S);
S;

module SS = grpres(S);

module B=grobj(module([1,0,0],[0,1,0],[0,0,1]),intvec(1,1,1),intvec(1,1,1));
// module B=grobj(module([1],[0,1],[0,0,1],[0,0,0,1], [0,0,0,0,1]),intvec(0:5));
grview(B);
B;
module BB = grpres(B);

def Z=grlifting2(SS,BB);Z;
def G=mappingcone2(SS,BB);G;
}
*/      






proc grlifting3(A,B) 
"TODO: grlifting4 was newer and had more documentation than this proc, but was removed... Please verify and update!
"
{
  ASSUME(1, grtest(A));
  ASSUME(1, grtest(B));


  list rM = grres(A,0,1);

  print( betti(rM), "betti");
  list rN = grres(B,0,1);
  print( betti(rN), "betti");
  
  int i,j,k;

  for(i=1;i<=size(rM);i++)
  {
    if(size(rM[i])==0){break;}
  }

  for(j=1;j<=size(rN);j++)
  {
    if(size(rN[j])==0){break;}
  }
  int t=min(i,j);
  
  list P;

  "t: ", t;
//  grview(rM[t]);  grview(rN[t]);
  
  P[t]= grrndmap2(rM[t],rN[t]);
  grview(P[t]);

  if(t==1){return(P)};

  for(k=t-1; k>=1; k--)
  {
     "k: ", k;
//  grview(rM[k]);  grview(rN[k]);

// def C = grtranspose(rM[k]); def T= grprod(rN[k],P[k+1]);
// def tT = grtranspose(T);

    P[k]= grlift0( rM[k], rN[k], P[k+1] ); // grtranspose(grlift(C,tT));

     grview(P[k]);

   }
   return(P);
     

}



example
{"EXAMPLE:"; echo = 2;

ring r=32003, x(0..4),dp;

def A=grtwist(3,1);
grview(A);

def T=KeneshlouMatrixPresentation(intvec(0,1,0,0,0));
grview(T);

def F=grlifting3(T,A);
grview(F);

def R=KeneshlouMatrixPresentation(intvec(0,0,0,2,0));
def S=KeneshlouMatrixPresentation(intvec(1,2,0,0,0));

def H=grlifting3(R, S);
// grview(H);

// 2nd module does not lie in the first:
// def H=grlifting3(S, R);


//def I=KeneshlouMatrixPresentation(intvec(2,3,0,6,2));
//def J=KeneshlouMatrixPresentation(intvec(4,0,1,2,1));
//def N=grlifting3(I,J); grview(N); 
}

proc grneg(A)
"USAGE: grneg(A), graded object A
RETURN: graded object
PURPOSE: graded map defined by -A 
EXAMPLE: example grneg; shows an example
"
{
  ASSUME(1, grtest(A));
  return( grobj(-A, grrange(A), grdeg(A)) );
}

example
{ "EXAMPLE:"; echo = 2;

   ring r=0,(x,y,z),dp;
   def A=grobj([x2,yz,xyz],intvec(1,1,0));
   grview(A);

   def F=grneg(A);
   grview(F);
}

//             -f1 0        -f2 0
//  (p1 g1)     p2 g2        p3 g3
//G0<-----F0+G1<------F1+G2<-------F2+G3<-----
proc mappingcone3(A,B)
"USAGE: (A,B), graded objects A and B (matrices defining maps)
RETURN: chain complex (as a list) 
PURPOSE: construct a free resolution of the cokernel of a random map between M=coker(A), and N=coker(B)
EXAMPLE: example mappingcone3; shows an example
"
{
  ASSUME(1, grtest(A));
  ASSUME(1, grtest(B));

  list P=grlifting3(A,B);
  list rM=grres(A,0,1);
  list rN=grres(B,0,1);

  int i;
  list T;

  T[1]=grconcat(P[1],rN[1]);

  for(i=2;i<=size(P);i++)
  {
    intvec v= grrange(rM[i-1]);
    intvec w=grdeg(rN[i]);
    int r=size(v);
    int s=size(w);
    matrix zero[r][s];

//    ASSUME( 0, grtest(P[i]) );
//    ASSUME( 0, grtest(rN[i]) );

    module A=grconcat(P[i],rN[i]);
    module B=grobj(zero,v,w);

    module C=grconcat( grneg( rM[i-1] ) ,B);
    module D=grconcat(grtranspose(C), grtranspose(A));

    T[i]=grtranspose(D);
    
    kill A, B, C, D, v, w, r, s, zero;
  }
   return(T);
}



example
{ "EXAMPLE:"; echo = 2;

ring r=32003,x(0..4),dp;

def A=KeneshlouMatrixPresentation(intvec(0,0,0,0,3));
grview(A);

def T= KeneshlouMatrixPresentation(intvec(0,1,0,0,0));
grview(T);

def F=grlifting3(A,T); grview(F);

// BUG in the proc  
def G=mappingcone3(A,T); grview(G);

/*
module W=grtranspose(G[1]);
resolution U=mres(W,0);
print(betti(U,0),"betti"); // ?
ideal P=groebner(flatten(U[2]));
resolution L=mres(P,0);
print(betti(L),"betti");    
*/


def R=KeneshlouMatrixPresentation(intvec(0,0,0,2,0));
grview(R);

def S=KeneshlouMatrixPresentation(intvec(1,2,0,0,0));
grview(S);

def H=grlifting3(R,S); grview(H);

// BUG in the proc
def G=mappingcone3(R,S);


def I=KeneshlouMatrixPresentation(intvec(2,3,0,6,2));
def J=KeneshlouMatrixPresentation(intvec(4,0,1,2,1));
// def N=grlifting3(I,J); 
// 2nd module does not lie in the first:
// def NN=mappingcone3(I,J); // ????????

}






// TODO: Please decide between KeneshlouMatrixPresentation and matrixpres, and replace one with the other!
proc matrixpres(intvec a)
"USAGE:  intvec a
RETURN:  graded object
PURPOSE: matrix presentation for direct sum of omega^a[i](i) in form of a graded object
EXAMPLE: example matrixpres; shows an example
"
{
  int n = size(a)-1;
  //  ring r = 32003,(x(0..n)),dp;
  ASSUME(0, nvars(basering)==(n+1));
  int i,j;

  // find first nonzero exponent a_i
  for(i=1;i<=size(a);i++)
    {
      if(a[i]!=0) {break; };
    }

  // all zeroes?
  if(i>size(a)) {return (grzero()); };
   for(i=2;i<=n;i++)
    {
      if(a[i]!=0) {break; };
    }

  module N;

  if(i>n) 
    { // no middle part
      if(a[1]>0)
        {
          N=grtwist(a[1],-1);

          if(a[n+1]>0)
            { N=grsum(N,grtwist(a[n+1],0));}         
        }  
      else
        { N=grtwist(a[n+1],0);}
      
      return (N); // grorder(N)); 
    }

else // i <= n: middle part is present, a_i != 0 
    { // a = a1  ... |  i:2, a_2 ..... i: n, a_n | .... i: n+1a_(n+1)
      module I = maxideal(1);
      attrib(I,"isHomog", intvec(0));
      list L = mres(I, 0);
      list kos = grorder(L);
      // make sure that graded maps  are represented by blocks corresponding to the betti diagram?
      int j=size(a)-i;
      def S = grpower(grshift(grobj( kos[j+2], attrib(kos[j+2], "isHomog")),j ), a[i]);

      i++;

      for(; i <= n; i++)
        {
          if(a[i]==0) { i++; continue; }
          int j=size(a)-i;
          S = grsum( S, grpower(grshift( grobj( kos[j+2], attrib(kos[j+2], "isHomog")), j), a[i])  );
        }

      // S is the middle (non-zero) part

      if(a[1] > 0 )
        {
          N=grsum(grtwist(a[1],-1), S);
        }
      else
        { N = S;}

      if(a[n+1] > 0 )
        { N=grsum(N, grtwist(a[n+1],0)); }


      return ((N)); //      return (grorder(N)); 
    }
}

example
{"EXAMPLE:"; echo = 2;
ring r = 32003,(x(0..4)),dp;

def R=matrixpres(intvec(1,4,0,0,0));
grview(R);
def S=matrixpres(intvec(0,0,3,0,0));
grview(S);

def N1 = matrixpres(intvec(2,0,0,0,0)); 
grview(N1);

def N2 = matrixpres(intvec(0,0,0,0,3));
grview(N2);

def N = matrixpres(intvec(2,0,0,0,3));
grview(N);


def M1 = matrixpres(intvec(0,1,0,0,0));
grview(M1);

def M2 = matrixpres(intvec(0,1,1,0,0));
grview(M2);

def M3 = matrixpres(intvec(0,0,0,1,0));
grview(M3);

def M = matrixpres(intvec(1,1,1,0,0));
grview(M);
}