LoadPackage("qpa");

DirectSumOfMaps := function(maps)
  local M, N, p, q, i, result;

  if IsEmpty(maps) then
    return fail;
  fi;

  M := DirectSumOfQPAModules(List(maps, f -> Source(f)));
  N := DirectSumOfQPAModules(List(maps, f -> Range(f)));

  p := DirectSumProjections(M);
  q := DirectSumInclusions(N);

  result := ZeroMapping(M, N);

  for i in [1..Length(maps)] do
    result := result + p[i]*maps[i]*q[i];
  od;

  return result;
end;

StrongTiltingModule := function(A)
  local n, i, j, S, F, L, s, r, t, q, g, d, T;

  if not IsRadicalSquareZeroAlgebra(A) then
    Error("this is not a radical-square-zero algebra,n");
  fi;

  S := SimpleModules(A);
  n := Length(S);

  Perform(S, s -> ProjDimensionOfModule(s, n));

  F := DirectSumOfQPAModules(Filtered(S, s -> HasProjDimension(s)));
  if F = fail then
    F := ZeroModule(A);
  fi;

  L := [];
  for s in S do
    if HasProjDimension(s) then
      Add(L, IdentityMapping(s));
    else
      r := KernelInclusion(ProjectiveCover(s));
      t := TraceOfModule(F, Source(r));
      q := IdentityMapping(Range(r));
      Add(L, MorphismOnCoKernel(t*r, r, t, q));
    fi;
  od;

  T := [];
  for i in [1..n] do
    if HasProjDimension(S[i]) then
      q := CoKernelProjection(InjectiveEnvelope(S[i]));
      d := DimensionVector(Range(q));
      g := [];
      for j in [1..n] do
        Append(g, ListWithIdenticalEntries(d[j], L[j]));
      od;
      if IsEmpty(g) then
        Add(T, Source(q));
      else
        Add(T, Source(PullBack(q, DirectSumOfMaps(g))[1]));
      fi;
    else
      Add(T, Source(L[i]));
    fi;
  od;

  return DirectSumOfQPAModules(T);
end;