source: git/Singular/LIB/nchomolog.lib @ 1b2216

/////////////////////////////////////////////////////////////////////////////
version="$Id$";
category="Noncommutative";
info="
LIBRARY:  nchomolog.lib   Procedures for Noncommutative Homological Algebra
AUTHORS:  Viktor Levandovskyy  levandov@math.rwth-aachen.de,
@*        Christian Schilli, christian.schilli@rwth-aachen.de,
@*        Gerhard Pfister, pfister@mathematik.uni-kl.de

OVERVIEW: In this library we present tools of homological algebra for
finitely presented modules over GR-algebras.

PROCEDURES:
 ncExt_R(k,M);      computes presentation of Ext^k(M',R), M module, R basering, M'=coker(M)
 ncHom(M,N);        computes presentation of Hom(M',N'), M,N modules, M'=coker(M), N'=coker(N)
 coHom(A,k);        computes presentation of Hom(R^k,A), A matrix over basering R
 contraHom(A,k);    computes presentation of Hom(A,R^k),     A matrix over basering R
 dmodoublext(M, l); computes presentation of Ext_D^i(Ext_D^i(M,D),D), where D is a basering
 is_cenBimodule(M); checks whether a module presented by M is Artin-centralizing
 is_cenSubbimodule(M); checks whether a subbimodule M is Artin-centralizing
";

LIB "dmod.lib";
LIB "gkdim.lib";
LIB "involut.lib";
LIB "nctools.lib";
LIB "ncalg.lib";
LIB "central.lib";

//  ncExt(k,M,N);            Ext^k(M',N'),   M,N modules, M'=coker(M), N'=coker(N)
//  ncTensorMod(M,N);        Tensor product of modules M'=coker(M), N'=coker(N)
//  ncTor(k,M,N);            Tor_k(M',N'),   M,N modules, M'=coker(M), N'=coker(N)
//  tensorMaps(M,N);       tensor product of  matrices


/* LOG:
5.12.2012, VL: cleanup, is_cenSubbimodule and is_cenBimodule are added for assume checks;
added doc for contraHom and coHom; assume check for ncHom etc.
 */

/* TODO:
add noncomm examples to important precedures ncHom,
 */

proc contraHom(matrix M, int s)
"USAGE:  contraHom(A,k); A matrix, k int
RETURN:  matrix
PURPOSE: compute the matrix of a homomorphism Hom(A,R^k), where R is the basering. Let A be a matrix defining a map F1-->F2 of free R-modules, then the matrix of Hom(F2,R^k)-->Hom(F1,R^k) is computed.
NOTE: if A is matrix of a left (resp. right) R-module homomorphism, then Hom(A,R^k) is a right (resp. left) R-module R-module homomorphism
EXAMPLE: example contraHom; shows an example.
SEE ALSO:
"
{
  // also possible: compute with kontrahom from homolog_lib
  // and warn that the module changes its side
   int n,m=ncols(M),nrows(M);
   int a,b,c;
   matrix R[s*n][s*m];
   for(b=1; b<=m; b++)
   {
      for(a=1; a<=s; a++)
      {
         for(c=1; c<=n; c++)
         {
            R[(a-1)*n+c,(a-1)*m+b] = M[b,c];
         }
      }
   }
   return(R);
}
example
{ "EXAMPLE:"; echo = 2;
  ring A=0,(x,y,z),dp;
  matrix M[3][3]=1,2,3,
               4,5,6,
               7,8,9;
  module cM = contraHom(M,2);
  print(cM);
}

proc coHom(matrix M, int s)
"USAGE:  coHom(A,k); A matrix, k int
PURPOSE: compute the matrix of a homomorphism Hom(R^k,A), where R is the basering. Let A be a matrix defining a map F1-->F2 of free R-modules, then the matrix of Hom(R^k,F1)-->Hom(R^k,F2) is computed.
NOTE: Both A and Hom(A,R^k) are matrices for either left or right R-module homomorphisms
EXAMPLE: example coHom; shows an example.
"
{
   int n,m=ncols(M),nrows(M);
   int a,b,c;
   matrix R[s*m][s*n];
   for(b=1; b<=s; b++)
   {
      for(a=1; a<=m; a++)
      {
         for(c=1; c<=n; c++)
         {
            R[(a-1)*s+b,(c-1)*s+b] = M[a,c];
         }
      }
   }
   return(R);
}
example
{ "EXAMPLE:"; echo = 2;
  ring A=0,(x,y,z),dp;
  matrix M[3][3]=1,2,3,
                 4,5,6,
                 7,8,9;
  module cM = coHom(M,2);
  print(cM);
}


proc ncHom(matrix M, matrix N)
"USAGE:   ncHom(M,N);  M,N modules
COMPUTE: A presentation of Hom(M',N'), M'=coker(M), N'=coker(N)
ASSUME: M' is a left module, N' is a centralizing bimodule
NOTE: ncHom(M,N) is a right module, hence a right presentation matrix
is returned
EXAMPLE: example ncHom; shows examples
"
{
  // assume: M is left module; nothing to check
  // assume: N is centralizing bimodule: to check
  if ( !is_cenBimodule(N) )
  {
    ERROR("Second module in not centralizing.");
  }
  // returns a right presentation matrix (for a right module)
  matrix F  = contraHom(M,nrows(N));
  matrix B  = coHom(N,ncols(M));
  matrix C  = coHom(N,nrows(M));
  def Rbase = basering;
  def Rop   = opposite(Rbase);
  setring Rop;
  matrix Bop = oppose(Rbase, B);
  matrix Cop = oppose(Rbase, C);
  matrix Fop = oppose(Rbase, F);
  matrix Dop = modulo(Fop, Bop);
  matrix Eop = modulo(Dop, Cop);
  setring Rbase;
  matrix E   = oppose(Rop, Eop);
  kill Rop;
  return(E);
}
example
{ "EXAMPLE:"; echo = 2;
  ring A=0,(x,y,z),dp;
  matrix M[3][3]=1,2,3,
                 4,5,6,
                 7,8,9;
  matrix N[2][2]=x,y,
                 z,0;
  module H = ncHom(M,N);
  print(H);
}

proc ncHom_alt(matrix M, matrix N)
{
  // shorter but potentially slower
  matrix F = contraHom(M,nrows(N)); // \varphi^*
  matrix B = coHom(N,ncols(M));     // i
  matrix C = coHom(N,nrows(M));     // j
  matrix D = rightModulo(F,B);      // D
  matrix E = rightModulo(D,C);      // Hom(M,N)
  return(E);
}
example
{ "EXAMPLE:"; echo = 2;
  ring A=0,(x,y,z),dp;
  matrix M[3][3]=1,2,3,
                 4,5,6,
                 7,8,9;
  matrix N[2][2]=x,y,
                 z,0;
  module H = ncHom_alt(M,N);
  print(H);
}

proc ncHom_R(matrix M)
"USAGE:   ncHom_R(M);  M a module
COMPUTE: A presentation of Hom_R(M',R), M'=coker(M)
ASSUME: M' is a left module
NOTE: ncHom_R(M) is a right module, hence a right presentation matrix is returned
EXAMPLE: example ncHom_R;  shows examples
"
{
  // assume: M is left module
  // returns a right presentation matrix
  // for a right module
  matrix F  = transpose(M);
  def Rbase = basering;
  def Rop   = opposite(Rbase);
  setring Rop;
  matrix Fop = oppose(Rbase, F);
  matrix Dop = modulo(Fop, std(0)); //ker Hom(A^n,A) -> Hom(A^m,A)
  matrix Eop = modulo(Dop, std(0)); // its presentation
  setring Rbase;
  matrix E   = oppose(Rop, Eop);
  kill Rop;
  return(E);
}
example
{ "EXAMPLE:"; echo = 2;
  ring A=0,(x,t,dx,dt),dp;
  def W = Weyl(); setring W;
  matrix M[2][2] =
    dt,  dx,
    t*dx,x*dt;
  module H = ncHom_R(M);
  print(H);
  matrix N[2][1] = x,dx;
  H = ncHom_R(N);
  print(H);
}


proc is_cenBimodule(module M)
"USAGE: is_cenBimodule(M);  M module
COMPUTE: 1, if a module, presented by M can be centralizing in the sense of Artin and 0 otherwise
NOTE: only one condition for centralizing factor module can be checked algorithmically
EXAMPLE: example is_cenBimodule;  shows examples
"
{
  // define in a ring R, for a module R: cen(M) ={ m in M: mr = rm for all r in R}
  // according to the definition, M is a centralizing bimodule <=> M is generated by cen(M)
  // if basering R is a G-algebra, then prop 6.4 of BGV indicates it's enough to provide
  // commutation of elements of M with the generators x_i of R
  // prop 6.4 verbatim generalizes to  R = R'/I for a twosided I.
  // is M generates submodule, see the proc is_cenSubbimodule
  // let M be a presentation matrix for P=R*/R*M, then [e_i + M]x_j=x_j[e_i+M]
  // <=> Mx_j - x_jM in M must hold; thus forall j: Mx_j in M; thus M has to be
  // closed from the right, that is to be a two-sided submodule indeed
  // the rest of checks are complicated by now, so do the check only
  // *the algorithm *//
  if (isCommutative() ) { return(int(1));}
  int n = nvars(basering);
  int ans = 0;
  int i,j;
  vector P;
  module N;
  if ( attrib(M,"isSB") != 1)
  {
    N = std(M);
  }
  else
  {
    N = M;
  }
  // N is std(M) now
  for(i=1; i<=ncols(M); i++)
  {
    P = M[i];
    if (P!=0)
    {
      for(j=1; j<=n; j++)
      {
        if ( NF(P*var(j) - var(j)*P, N) != 0)
        {
          return(ans);
        }
      }
    }
  }
  ans = 1;
  return(ans);
}
example
{ "EXAMPLE:"; echo = 2;
  def A = makeUsl2(); setring A;
  poly p = 4*e*f + h^2-2*h; // generator of the center
  matrix M[2][2] = p, p^2-7,0,p*(p+1);
  is_cenBimodule(M); // M is centralizing
  matrix N[2][2] = p, e*f,h,p*(p+1);
  is_cenBimodule(N); // N is not centralizing
}

proc is_cenSubbimodule(module M)
"USAGE: is_cenSubbimodule(M); M module
COMPUTE: 1, if a subbimodule, generated by the columns of M is
centralizing in the sense of Artin and 0 otherwise
EXAMPLE: example is_cenSubbimodule;  shows examples
"
{
  // note: M in R^m is centralizing subbimodule iff it is generated by vectors,
  // each nonconstant component of which is central; 2 check: every entry of the
  // matrix M is central
  if (isCommutative()) { return(int(1));}
  return( inCenter(ideal(matrix(M))) );
}
example
{ "EXAMPLE:"; echo = 2;
  def A = makeUsl2(); setring A;
  poly p = 4*e*f + h^2-2*h; // generator of the center
  matrix M[2][2] = p, p^2-7,0,p*(p+1);
  is_cenSubbimodule(M); // M is centralizing subbimodule
  matrix N[2][2] = p, e*f,h,p*(p+1);
  is_cenSubbimodule(N); // N is not centralizing subbimodule
}


proc ncExt(int i, matrix Ps, matrix Ph)
"USAGE:   Ext(i,M,N);  i int, M,N matrices
COMPUTE: A presentation of Ext^i(M',N');  for M'=coker(M) and N'=coker(N).
ASSUME: M' is a left  module, N' is a centralizing bimodule
NOTE: ncExt(M,N) is a right module, hence a right presentation matrix
is returned
EXAMPLE: example ncExt;  shows examples
"
{
  if ( !is_cenBimodule(Ph) )
  {
    ERROR("Second module in not centralizing.");
  }

  if(i==0) { return(module(ncHom(Ps,Ph))); }
  list Phi   = mres(Ps,i+1);
  module Im  = coHom(Ph,ncols(Phi[i+1]));
  module f   = contraHom(matrix(Phi[i+1]),nrows(Ph));
  module Im1 = coHom(Ph,ncols(Phi[i]));
  module Im2 = contraHom(matrix(Phi[i]),nrows(Ph));
  def Rbase = basering;
  def Rop   = opposite(Rbase);
  setring Rop;
  module fop    = oppose(Rbase,f);
  module Imop   = oppose(Rbase,Im);
  module Im1op  = oppose(Rbase,Im1);
  module Im2op  = oppose(Rbase,Im2);
  module ker_op = modulo(fop,Imop);
  module ext_op = modulo(ker_op,Im1op+Im2op);
  //  ext        = prune(ext);
 // to be discussed and done prune_from_the_left
  setring Rbase;
  module ext = oppose(Rop,ext_op);
  kill Rop;
  return(ext);
}
example
{ "EXAMPLE:"; echo = 2;
  ring R     = 0,(x,y),dp;
  ideal I    = x2-y3;
  qring S    = std(I);
  module M   = [-x,y],[-y2,x];
  module E1  = ncExt(1,M,M);
  E1;
}

proc ncExt_R(int i, matrix Ps)
"USAGE:   ncExt_R(i, M);  i int, M module
COMPUTE:  a presentation of Ext^i(M',R); for M'=coker(M).
RETURN:   right module Ext, a presentation of Ext^i(M',R)
EXAMPLE: example ncExt_R; shows an example
"{
  if (i==0)
  {
    return(ncHom_R(Ps)); // the rest is not needed
  }
  list Phi   = nres(Ps,i+1); // left resolution
  module f   = transpose(matrix(Phi[i+1])); // transp. because of Hom_R
  module Im2 = transpose(matrix(Phi[i]));
  def Rbase = basering;
  def Rop   = opposite(Rbase);
  setring Rop;
  module fop    = oppose(Rbase,f);
  module Im2op  = oppose(Rbase,Im2);
  module ker_op = modulo(fop,std(0));
  module ext_op = modulo(ker_op,Im2op);
  //  ext        = prune(ext);
  // to be discussed and done prune_from_the_left
  // necessary: compute SB!
  // "Computing SB of Ext";
//   option(redSB);
//   option(redTail);
//   ext_op = std(ext_op);
//   int dimop = GKdim(ext_op);
//   printf("Ext has dimension %s",dimop);
//   if (dimop==0)
//   {
//       printf("of K-dimension %s",vdim(ext_op));
//   }
  setring Rbase;
  module ext = oppose(Rop,ext_op); // a right module!
  kill Rop;
  return(ext);
}
example
{ "EXAMPLE:"; echo = 2;
  ring R     = 0,(x,y),dp;
  poly F    = x2-y2;
  def A = annfs(F);  setring A; // A is the 2nd Weyl algebra
  matrix M[1][size(LD)] = LD; // ideal
  print(M);
  print(ncExt_R(1,M)); // hence the Ext^1 is zero
  module E  = ncExt_R(2,M); // define the right module E
  print(E); // E is in the opposite algebra
  def Aop = opposite(A);  setring Aop;
  module Eop = oppose(A,E);
  module T1  = ncExt_R(2,Eop);
  setring A;
  module T1 = oppose(Aop,T1);
  print(T1); // this is a left module Ext^2(Ext^2(M,A),A)
  print(M); // it is known that M holonomic implies Ext^2(Ext^2(M,A),A) iso to M
}

proc nctors(matrix M)
{
  // ext^1_A(adj(M),A)
  def save = basering;
  matrix MM = M;  // left
  def sop = opposite(save);
  setring sop;
  matrix MM  = oppose(save,MM); // right
  MM = transpose(MM); // transposed
  list Phi = nres(MM,2); // i=1
  module f   = transpose(matrix(Phi[2])); // transp. because of Hom_R
  module Im2 = transpose(matrix(Phi[1]));
  setring save;
  module fop    = oppose(sop,f);
  module Im2op  = oppose(sop,Im2);
  module ker_op = modulo(fop,std(0));
  module ext_op = modulo(ker_op,Im2op);
  //  matrix E = ncExt_R(1,MM);
  //  setring save;
  //  matrix E = oppose(sop,E);
  return(ext_op);
}

proc altExt_R(int i, matrix Ps, map Invo)
  // TODO!!!!!!!!
  // matrix Ph
  // work thru Involutions;
{
  if(i==0)
  { // return the formal adjoint
    matrix Ret   = transpose(Ps);
    matrix Retop = involution(Ret, Invo);
    //    "Computing prune of Hom";
    //    Retop = prune(Retop);
    //    Retop = std(Retop);
    return(Retop);
  }
  list Phi   = mres(Ps,i+1);
  //  module Im  = coHom(Ph,ncols(Phi[i+1]));
  module f   = transpose(matrix(Phi[i+1]));
  f = involution(f, Invo);
  //= contraHom(matrix(Phi[i+1]),nrows(Ph));
  //  module Im1 = coHom(Ph,ncols(Phi[i]));
  module Im2 = transpose(matrix(Phi[i]));
  Im2 = involution(Im2, Invo);
  //contraHom(matrix(Phi[i]),nrows(Ph));
  module ker_op = modulo(f,std(0));
  module ext_op = modulo(ker_op,Im2);
  //  ext        = prune(ext);
 // to be discussed and done prune_from_the_left
  // optionally: compute SB!
  //  "Computing prune of Ext";
  ext_op = std(ext_op);
  int dimop = GKdim(ext_op);
  printf("Ext has dimension %s",dimop);
  if (dimop==0)
  {
      printf("of K-dimension %s",vdim(ext_op));
  }
  module ext = involution(ext_op, Invo); // what about transpose?
  return(ext);
}
example
{ "EXAMPLE:"; echo = 2;
  ring R     = 0,(x,y),dp;
  ideal I    = x2-y3;
  qring S    = std(I);
  module M   = [-x,y],[-y2,x];
  module E1  = ncExt(2,M,M);
  E1;
}

proc tensorMaps(matrix M, matrix N)
{
   int r = ncols(M);
   int s = nrows(M);
   int p = ncols(N);
   int q = nrows(N);
   int a,b,c,d;
   matrix R[s*q][r*p];
   for(b=1;b<=p;b++)
   {
      for(d=1;d<=q;d++)
      {
         for(a=1;a<=r;a++)
         {
            for(c=1;c<=s;c++)
            {
               R[(c-1)*q+d,(a-1)*p+b]=M[c,a]*N[d,b];
            }
         }
      }
   }
   return(R);
}

proc ncTensorMod(matrix Phi, matrix Psi)
{
   int s=nrows(Phi);
   int q=nrows(Psi);
   matrix A=tensorMaps(unitmat(s),Psi);  //I_s tensor Psi
   matrix B=tensorMaps(Phi,unitmat(q));  //Phi tensor I_q
   matrix R=concat(A,B);                 //sum of A and B
   return(R);
}


proc ncTor(int i, matrix Ps, matrix Ph)
{
  if(i==0) { return(module(ncTensorMod(Ps,Ph))); }
                               // the tensor product
  list Phi   = mres(Ph,i+1);     // a resolution of Ph
  module Im  = tensorMaps(unitmat(nrows(Phi[i])),Ps);
  module f   = tensorMaps(matrix(Phi[i]),unitmat(nrows(Ps)));
  module Im1 = tensorMaps(unitmat(ncols(Phi[i])),Ps);
  module Im2 = tensorMaps(matrix(Phi[i+1]),unitmat(nrows(Ps)));
  module ker = modulo(f,Im);
  module tor = modulo(ker,Im1+Im2);
  //  tor        = prune(tor);
  return(tor);
}


static proc Hochschild()
{
  ring A    = 0,(x,y),dp;
  ideal I   = x2-y3;
  qring B   = std(I);
  module M  = [-x,y],[-y2,x];
  ring C    = 0,(x,y,z,w),dp; // x->z, y->w
  ideal I   = x2-y3,z3-w2;
  qring Be  = std(I);   //the enveloping algebra
  matrix AA[1][2]  = x-z,y-w;  //the presentation of the algebra B as Be-module
  module MM = imap(B,M);
  module E = ncExt(1,AA,MM);
  print(E);  //the presentation of the H^1(A,M)

ring A          = 0,(x,y),dp;
ideal I         = x2-y3;
qring B         = std(I);
ring C          = 0,(x,y,z,w),dp;
ideal I         = x2-y3,z3-w2;
qring Be        = std(I);   //the enveloping algebra
matrix AA[1][2] = x-z,y-w;  //the presentation of B as Be-module
matrix AAA[1][2] = z,w; // equivalent? pres. of B
print(ncExt(1,AA,AA));  //the presentation of the H^1(A,A)
print(ncExt(1,AAA,AAA));
}

static proc Lie()
{
// consider U(sl2)* U(sl2)^opp;
LIB "ncalg.lib";
ring A = 0,(e,f,h,H,F,E),Dp; // any degree ordering
int N = 6; // nvars(A);
matrix @D[N][N];
@D[1,2] = -h;
@D[1,3] = 2*e;
@D[2,3] = -2*f;
@D[4,5] = 2*F;
@D[4,6] = -2*E;
@D[5,6] = H;
 def AA = nc_algebra(1,@D); setring AA;
ideal Q = E,F,H;
poly Z = 4*e*f+h^2-2*h; // center
poly Zo = 4*F*E+H^2+2*H;  // center opposed
ideal Qe = Z,Zo;
//qring B = twostd(Qe);
//ideal T = e-E,f-F,h-H;
//ideal T2 = e-H,f-F,h-E;
//Q = twostd(Q); // U is U(sl2) as left U(sl2)* U(sl2)^opp -- module
matrix M[1][3] = E,F,H;
module X0 = ncExt(0,M,M);
print(X0);

module X1 = ncExt(1,M,M);
print(X1);
module X2 = ncExt(2,M,M); // equal to Tor^Z_1(K,K)
print(X2);

// compute  Tor^Z_1(K,K)
ring r = 0,(z),dp;
ideal i = z;
matrix I[1][1]=z;
Tor(1,I,I);
}


proc AllExts(module N, list #)
  // computes and shows everything
  // assumes we are in the opposite
  // and N is dual of some M
  // if # is given, map Invo and Ext_Invo are used
{
  int UseInvo = 0;
  int sl = size(#);
  if (sl >0)
  {
    ideal I = ideal(#[1]);
    map Invo = basering, I;
    UseInvo  = 1;
    "Using the involution";
  }
  int nv = nvars(basering);
  int i,d;
  module E;
  list EE;
  print("--- module:"); print(matrix(N));
  for (i=1; i<=nv; i++)
  {
    if (UseInvo)
    {
      E = altExt_R(i,N,Invo);
    }
    else
    {
      E = ncExt_R(i,N);
    }
    printf("--- Ext %s",i);
    print(matrix(E));
    EE[i] = E;
  }
  return(E);
}

proc dmodualtest(module M, int n)
{
  // computes the "dual" of the "dual" of a d-mod M
  // where n is the half-number of vars of Weyl algebra
  // assumed to be basering
  // returns the difference between M and Ext^n_D(Ext^n_D(M,D),D)
  def save = basering;
  setring save;
  module Md = ncExt_R(n,M); // right module
  // would be nice to use "prune"!
  // NO! prune performs left sided operations!!!
  //  Md = prune(Md);
  //  print(Md);
  def saveop = opposite(save);
  setring saveop;
  module Mdop = oppose(save,Md); // left module
  // here we're eligible to use prune
  Mdop = prune(Mdop);
  module Mopd = ncExt_R(n,Mdop); // right module
  setring save;
  module M2 = oppose(saveop,Mopd);  // left module
  M2 = prune(M2); // eligible since M2 is a left mod
  M2 = groebner(M2);
  ideal tst = M2 - M;
  tst = groebner(tst);
  return(tst);
}
example
{ "EXAMPLE:"; echo = 2;
  ring R   = 0,(x,y),dp;
  poly F   = x3-y2;
  def A    = annfs(F);
  setring A;
  dmodualtest(LD,2);
}


proc dmodoublext(module M, list #)
"USAGE:   dmodoublext(M [,i]);  M module, i optional int
COMPUTE:  a presentation of Ext^i(Ext^i(M,D),D) for basering D
RETURN:   left module
NOTE: by default, i is set to the integer part of the half of number of variables of D
@* for holonomic modules over Weyl algebra, the double ext is known to be holonomic left module
EXAMPLE: example dmodoublext; shows an example
"
{
  // assume: basering is a Weyl algebra?
  def save = basering;
  setring save;
  // if a list is nonempty and contains an integer N, n = N; otherwise n = nvars/2
  int n;
  if (size(#) > 0)
  {
    //    if (typeof(#) == "int")
    //    {
      n = int(#[1]);
      //    }
//     else
//     {
//       ERROR("the optional argument expected to have type int");
//     }
  }
  else
  {
    n = nvars(save); n = n div 2;
  }
  // returns Ext^i_D(Ext^i_D(M,D),D), that is
  // computes the "dual" of the "dual" of a d-mod M (for n = nvars/2)
  module Md = ncExt_R(n,M); // right module
  // no prune yet!
  def saveop = opposite(save);
  setring saveop;
  module Mdop = oppose(save,Md); // left module
  // here we're eligible to use prune
  Mdop = prune(Mdop);
  module Mopd = ncExt_R(n,Mdop); // right module
  setring save;
  module M2 = oppose(saveop,Mopd);  // left module
  kill saveop;
  M2 = prune(M2); // eligible since M2 is a left mod
  def M3;
  if (nrows(M2)==1)
  {
    M3 = ideal(M2);
  }
  else
  {
    M3 = M2;
  }
  M3 = groebner(M3);
  return(M3);
}
example
{ "EXAMPLE:"; echo = 2;
  ring R   = 0,(x,y),dp;
  poly F   = x3-y2;
  def A    = annfs(F);
  setring A;
  dmodoublext(LD);
  LD;
  // fancier example:
  setring A;
  ideal I = Dx*(x2-y3),Dy*(x2-y3);
  I = groebner(I);
  print(dmodoublext(I,1));
  print(dmodoublext(I,2));
}

static proc part_Ext_R(matrix M)
{
  // if i==0
    matrix Ret = transpose(Ps);
    def Rbase = basering;
    def Rop   = opposite(Rbase);
    setring Rop;
    module Retop = oppose(Rbase,Ret);
    module Hm = modulo(Retop,std(0)); // right kernel of transposed
    //    "Computing prune of Hom";
    //    Retop = prune(Retop);
    //    Retop = std(Retop);
    setring Rbase;
    Ret = oppose(Rop, Hm);
    kill Rop;
    return(Ret);
// some checkz:
//  setring Rbase;
  // ker_op is the right Kernel of f^t:
  //  module ker = oppose(Rop,ker_op);
  //  print(f*ker);
//  module ext = oppose(Rop,ext_op);
}