TEST(Cokernel, Diagonal)
{
AbelianGroup Y(3, 0);
MatrixQ f = {{0, 0, 0}, {0, 2, 0}, {0, 0, 3}};
MatrixQList to_C;
MatrixQList from_C;
GroupWithMorphisms C =
compute_cokernel(3, f, Y, MatrixQRefList(), MatrixQRefList());
EXPECT_EQ(1, C.group.free_rank());
ASSERT_EQ(1, C.group.tor_rank());
EXPECT_EQ(1, C.group(0));
}
GroupWithMorphisms compute_image(const std::size_t p, const MatrixQ& f,
const AbelianGroup& X, const AbelianGroup& Y)
{
MatrixQRefList to_X;
MatrixQList from_X = {MatrixQ::identity(f.width())};
GroupWithMorphisms K = compute_kernel(p, f, X, Y, to_X, ref(from_X));
MatrixQList to_X_2 = {MatrixQ::identity(f.width())};
MatrixQList from_X_2 = {f};
GroupWithMorphisms img =
compute_cokernel(p, f, Y, ref(to_X_2), ref(from_X_2));
return img;
}
GroupWithMorphisms compute_image(const mod_t p, const MatrixQ& f,
const AbelianGroup& X, const AbelianGroup& Y)
{
MatrixQRefList to_X_dummy;
MatrixQList from_X = {MatrixQ::identity(f.width())};
GroupWithMorphisms K = compute_kernel(p, f, X, Y, to_X_dummy, ref(from_X));
MatrixQList to_X_2 = {MatrixQ::identity(X.rank())};
MatrixQList from_X_2 = {f,MatrixQ::identity(X.rank())};//hacky, since id:X->X doesn't vanish on K.
//but due to the implementation, the columns of this will contain
//representatives in X for the generators of img.
GroupWithMorphisms img =
compute_cokernel(p, K.maps_from[0], X, ref(to_X_2), ref(from_X_2));
return img;
}
GroupWithMorphisms compute_kernel(const std::size_t p, const MatrixQ& f,
const AbelianGroup& X, const AbelianGroup& Y,
const MatrixQRefList& to_X_ref,
const MatrixQRefList& from_X_ref)
{
MatrixQ f_rel_Y(f.height(), f.width() + Y.tor_rank());
f_rel_Y(0, 0, f.height(), f.width()) = f;
f_rel_Y(0, f.width(), Y.tor_rank(), Y.tor_rank()) = Y.torsion_matrix(p);
MatrixQ rel_x_lift(f.width() + Y.tor_rank(), X.tor_rank());
rel_x_lift(0, 0, X.tor_rank(), X.tor_rank()) = X.torsion_matrix(p);
// rel_x_lift(f.width(), 0, Y.tor_rank(), X.tor_rank()) = -lift of f\circ
// rel_x over rel_Y.
// Can be computed by multiplying the columns of f with the orders of X,
// and dividing the rows by the orders of Y.
for (std::size_t i = 0; i < Y.tor_rank(); ++i) {
for (std::size_t j = 0; j < X.tor_rank(); ++j) {
long order_diff = static_cast<long>(X(j) - Y(i));
rel_x_lift(f.width() + i, j) = -f(i, j) * p_pow_q(p, order_diff);
}
}
// build to_X_rel_Y, from_X_rel_Y.
MatrixQList to_X_rel_Y;
for (MatrixQ& g_to_X : to_X_ref) {
to_X_rel_Y.emplace_back(f.width() + Y.tor_rank(), g_to_X.width());
to_X_rel_Y.back()(0, 0, f.width(), g_to_X.width()) = g_to_X;
MatrixQ fg = f * g_to_X;
for (std::size_t i = 0; i < Y.tor_rank(); ++i) {
for (std::size_t j = 0; j < g_to_X.width(); ++j) {
to_X_rel_Y.back()(f.width() + i, j) = -fg(i, j) / p_pow_z(p, Y(i));
}
}
}
MatrixQList from_X_rel_Y;
for (MatrixQ& g_from_X : from_X_ref) {
from_X_rel_Y.emplace_back(g_from_X.height(), f.width() + Y.tor_rank());
from_X_rel_Y.back()(0, 0, g_from_X.height(), f.width()) = g_from_X;
}
MatrixQRefList to_X_rel_Y_ref = ref(to_X_rel_Y);
MatrixQRefList from_X_rel_Y_ref = ref(from_X_rel_Y);
to_X_rel_Y_ref.emplace_back(rel_x_lift);
MatrixQRefList to_Y;
MatrixQRefList from_Y;
smith_reduce_p(p, f_rel_Y, to_X_rel_Y_ref, from_X_rel_Y_ref, to_Y, from_Y);
std::size_t rank_diff;
for (rank_diff = 0; rank_diff < std::min(f_rel_Y.height(), f_rel_Y.width());
++rank_diff) {
if (f_rel_Y(rank_diff, rank_diff) == 0) break;
}
// next, restrict attention to the entries corresponding to zero columns of
// f_rel_Y:
// for rel_x_lift as well as the entries of to_X_rel_Y, take the submatrices
// formed
// by the corresponding rows. For from_X_rel_Y, take the submatrices formed
// by
// the corresponding columns.
MatrixQ rel_K = rel_x_lift(rank_diff, 0, rel_x_lift.height() - rank_diff,
rel_x_lift.width());
MatrixQList to_free_K;
for (MatrixQ& g_to_X_rel_Y : to_X_rel_Y) {
to_free_K.emplace_back(g_to_X_rel_Y(
rank_diff, 0, g_to_X_rel_Y.height() - rank_diff, g_to_X_rel_Y.width()));
}
MatrixQList from_free_K;
for (MatrixQ& g_from_X_rel_Y : from_X_rel_Y) {
from_free_K.emplace_back(
g_from_X_rel_Y(0, rank_diff, g_from_X_rel_Y.height(),
g_from_X_rel_Y.width() - rank_diff));
}
MatrixQRefList to_free_K_ref = ref(to_free_K);
MatrixQRefList from_free_K_ref = ref(from_free_K);
AbelianGroup free_K(rel_K.height(), 0);
// then, compute the cokernel of the new rel_x_lift with the respective
// to_Y, from_Y.
return compute_cokernel(p, rel_K, free_K, to_free_K_ref, from_free_K_ref);
}
