source: git/Singular/LIB/gradedModules.lib @ 7b4fc8

//////////////////////////////////////////////////////////////////////////
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? (using grlifting4 at the moment)
    grrange(M)         get the row-weightings
    grneg(A)           graded object given by -A
    matrixpres(intvec 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: matrix 
PURPOSE:matrix presentation for direct sum of omega^a[i](i) 
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 resolution of
M=Img(M) and N=Img(N).
EXAMPLE: example grlift; 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
M=Img(M), and N=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;
//Veronese surface

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);

}

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).
NOTE:
               A     f2     f3
0<---M<----F0<----F1<----F2<----F3<----
             |p1   |p2

0<---N<----G0<----G1<----G2<----G3<----
               B(g1)      g2     g3

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!!!  
             
}


/*
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)
NOTE: 
             -f1 0       -f2 0
  (p1 g1)     p2 g2       p3 g3
G0<-----F0+G1<------F1+G2<-------F2+G3<-----

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);
    ASSUME( 0, grtest( grneg( rM[i-1]) ) );
    module C=grconcat( grneg( rM[i-1] ) ,B); // FIXME: '-' is wrong! need a graded-enabled minus (e.g. grneg?)
    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)
{
  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);
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: A graded object
RETURN: -A as graded object
PURPOSE: graded map defined by -A. 
EXAMPLE: example grneg; shows an example
"
{
  ASSUME(0, 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(A);
}




/*
             -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=grlifting4(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);

//    ASSUME( 0, grtest(B) );
//    ASSUME( 0, grtest(-rM[i-1]) );    
    ASSUME( 0, grtest( grneg( rM[i-1]) ) );
    module C=grconcat( grneg( rM[i-1] ) ,B); // FIXME: '-' is wrong! need a graded-enabled minus (e.g. grneg?)
    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); // ????????

}






proc matrixpres(intvec a)
"USAGE:  intvec a.
RETURN: matrix. 
PURPOSE:matrix presentation for direct sum of omega^a[i](i) 
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;

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