CoCalc -- chain_complex

GitHub Repository: sagemath/sagesmc
Path: blob/master/src/sage/homology/chain_complex_morphism.py
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r"""
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Morphisms of chain complexes
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AUTHORS:
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- Benjamin Antieau <[email protected]> (2009.06)
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- Travis Scrimshaw (2012-08-18): Made all simplicial complexes immutable to
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  work with the homset cache.
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This module implements morphisms of chain complexes. The input is a dictionary
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whose keys are in the grading group of the chain complex and whose values are
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matrix morphisms.
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EXAMPLES::
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    from sage.matrix.constructor import zero_matrix
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    sage: S = simplicial_complexes.Sphere(1)
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    sage: S
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    Simplicial complex with vertex set (0, 1, 2) and facets {(1, 2), (0, 2), (0, 1)}
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    sage: C = S.chain_complex()
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    sage: C.differential()
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    {0: [], 1: [ 1  1  0]
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    [ 0 -1 -1]
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    [-1  0  1], 2: []}
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    sage: f = {0:zero_matrix(ZZ,3,3),1:zero_matrix(ZZ,3,3)}
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    sage: G = Hom(C,C)
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    sage: x = G(f)
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    sage: x
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    Chain complex morphism from Chain complex with at most 2 nonzero terms over Integer Ring to Chain complex with at most 2 nonzero terms over Integer Ring
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    sage: x._matrix_dictionary
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    {0: [0 0 0]
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    [0 0 0]
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    [0 0 0], 1: [0 0 0]
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    [0 0 0]
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    [0 0 0]}
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"""
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#*****************************************************************************
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# Copyright (C) 2009 D. Benjamin Antieau <[email protected]>
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#
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#  Distributed under the terms of the GNU General Public License (GPL)
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#
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#    This code is distributed in the hope that it will be useful,
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#    but WITHOUT ANY WARRANTY; without even the implied warranty
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#    of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
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#
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#  See the GNU General Public License for more details; the full text
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#  is available at:
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#
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#                  http://www.gnu.org/licenses/
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#
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#*****************************************************************************
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import sage.matrix.all as matrix
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from sage.structure.sage_object import SageObject
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from sage.rings.integer_ring import ZZ
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def is_ChainComplexMorphism(x):
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    """
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    Returns ``True`` if and only if ``x`` is a chain complex morphism.
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    EXAMPLES::
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        sage: from sage.homology.chain_complex_morphism import is_ChainComplexMorphism
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        sage: S = simplicial_complexes.Sphere(14)
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        sage: H = Hom(S,S)
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        sage: i = H.identity()  # long time (8s on sage.math, 2011)
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        sage: S = simplicial_complexes.Sphere(6)
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        sage: H = Hom(S,S)
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        sage: i = H.identity()
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        sage: x = i.associated_chain_complex_morphism()
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        sage: x # indirect doctest
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        Chain complex morphism from Chain complex with at most 7 nonzero terms over
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        Integer Ring to Chain complex with at most 7 nonzero terms over Integer Ring
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        sage: is_ChainComplexMorphism(x)
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        True
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    """
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    return isinstance(x,ChainComplexMorphism)
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class ChainComplexMorphism(SageObject):
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    """
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    An element of this class is a morphism of chain complexes.
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    """
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    def __init__(self, matrices, C, D, check=True):
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        """
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        Create a morphism from a dictionary of matrices.
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        EXAMPLES::
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            from sage.matrix.constructor import zero_matrix
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            sage: S = simplicial_complexes.Sphere(1)
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            sage: S
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            Simplicial complex with vertex set (0, 1, 2) and facets {(1, 2), (0, 2), (0, 1)}
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            sage: C = S.chain_complex()
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            sage: C.differential()
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            {0: [], 1: [ 1  1  0]
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            [ 0 -1 -1]
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            [-1  0  1], 2: []}
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            sage: f = {0:zero_matrix(ZZ,3,3),1:zero_matrix(ZZ,3,3)}
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            sage: G = Hom(C,C)
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            sage: x = G(f)
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            sage: x
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            Chain complex morphism from Chain complex with at most 2 nonzero terms
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            over Integer Ring to Chain complex with at most 2 nonzero terms over 
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            Integer Ring
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            sage: x._matrix_dictionary
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            {0: [0 0 0]
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            [0 0 0]
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            [0 0 0], 1: [0 0 0]
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            [0 0 0]
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            [0 0 0]}
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        Check that the bug in :trac:`13220` has been fixed::
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            sage: X = simplicial_complexes.Simplex(1)
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            sage: Y = simplicial_complexes.Simplex(0)
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            sage: g = Hom(X,Y)({0:0, 1:0})
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            sage: g.associated_chain_complex_morphism()
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            Chain complex morphism from Chain complex with at most 2 nonzero 
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            terms over Integer Ring to Chain complex with at most 1 nonzero terms 
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            over Integer Ring
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        """
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        if not C.base_ring() == D.base_ring():
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            raise NotImplementedError('morphisms between chain complexes of different'
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                                      ' base rings are not implemented')
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        d = C.degree_of_differential()
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        if d != D.degree_of_differential():
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            raise ValueError('degree of differential does not match')
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        from sage.misc.misc import uniq
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        degrees = uniq(C.differential().keys() + D.differential().keys())
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        initial_matrices = dict(matrices)
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        matrices = dict()
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        for i in degrees:
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            if i - d not in degrees:
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                if not (C.free_module_rank(i) == D.free_module_rank(i) == 0):
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                    raise ValueError('{} and {} are not rank 0 in degree {}'.format(C, D, i))
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                continue
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            try:
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                matrices[i] = initial_matrices.pop(i)
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            except KeyError:
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                matrices[i] = matrix.zero_matrix(C.base_ring(),
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                                                 D.differential(i).ncols(),
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                                                 C.differential(i).ncols(), sparse=True)
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        if check:
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            # all remaining matrices given must be 0x0
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            if not all(m.ncols() == m.nrows() == 0 for m in initial_matrices.values()):
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                raise ValueError('the remaining matrices are not empty')
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            # check commutativity
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            for i in degrees:
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                if i - d not in degrees:
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                    if not (C.free_module_rank(i) == D.free_module_rank(i) == 0):
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                        raise ValueError('{} and {} are not rank 0 in degree {}'.format(C, D, i))
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                    continue
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                if i + d not in degrees:
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                    if not (C.free_module_rank(i+d) == D.free_module_rank(i+d) == 0):
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                        raise ValueError('{} and {} are not rank 0 in degree {}'.format(C, D, i+d))
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                    continue
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                Dm = D.differential(i) * matrices[i]
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                mC = matrices[i+d] * C.differential(i)
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                if mC != Dm:
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                    raise ValueError('matrices must define a chain complex morphism')
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        self._matrix_dictionary = matrices
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        self._domain = C
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        self._codomain = D
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    def __neg__(self):
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        """
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        Returns ``-x``.
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        EXAMPLES::
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            sage: S = simplicial_complexes.Sphere(2)
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            sage: H = Hom(S,S)
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            sage: i = H.identity()
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            sage: x = i.associated_chain_complex_morphism()
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            sage: w = -x
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            sage: w._matrix_dictionary
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            {0: [-1  0  0  0]
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            [ 0 -1  0  0]
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            [ 0  0 -1  0]
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            [ 0  0  0 -1],
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             1: [-1  0  0  0  0  0]
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            [ 0 -1  0  0  0  0]
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            [ 0  0 -1  0  0  0]
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            [ 0  0  0 -1  0  0]
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            [ 0  0  0  0 -1  0]
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            [ 0  0  0  0  0 -1],
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             2: [-1  0  0  0]
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            [ 0 -1  0  0]
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            [ 0  0 -1  0]
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            [ 0  0  0 -1]}
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        """
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        f = dict()
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        for i in self._matrix_dictionary.keys():
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            f[i] = -self._matrix_dictionary[i]
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        return ChainComplexMorphism(f, self._domain, self._codomain)
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    def __add__(self,x):
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        """
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        Returns ``self + x``.
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        EXAMPLES::
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            sage: S = simplicial_complexes.Sphere(2)
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            sage: H = Hom(S,S)
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            sage: i = H.identity()
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            sage: x = i.associated_chain_complex_morphism()
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            sage: z = x+x
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            sage: z._matrix_dictionary
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            {0: [2 0 0 0]
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            [0 2 0 0]
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            [0 0 2 0]
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            [0 0 0 2],
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             1: [2 0 0 0 0 0]
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            [0 2 0 0 0 0]
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            [0 0 2 0 0 0]
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            [0 0 0 2 0 0]
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            [0 0 0 0 2 0]
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            [0 0 0 0 0 2],
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             2: [2 0 0 0]
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            [0 2 0 0]
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            [0 0 2 0]
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            [0 0 0 2]}
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        """
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        if not isinstance(x,ChainComplexMorphism) or self._codomain != x._codomain or self._domain != x._domain or self._matrix_dictionary.keys() != x._matrix_dictionary.keys():
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            raise TypeError("Unsupported operation.")
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        f = dict()
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        for i in self._matrix_dictionary.keys():
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            f[i] = self._matrix_dictionary[i] + x._matrix_dictionary[i]
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        return ChainComplexMorphism(f, self._domain, self._codomain)
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    def __mul__(self,x):
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        """
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        Return ``self * x`` if ``self`` and ``x`` are composable morphisms
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        or if ``x`` is an element of the base ring.
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        EXAMPLES::
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            sage: S = simplicial_complexes.Sphere(2)
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            sage: H = Hom(S,S)
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            sage: i = H.identity()
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            sage: x = i.associated_chain_complex_morphism()
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            sage: y = x*2
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            sage: y._matrix_dictionary
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            {0: [2 0 0 0]
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            [0 2 0 0]
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            [0 0 2 0]
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            [0 0 0 2],
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             1: [2 0 0 0 0 0]
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            [0 2 0 0 0 0]
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            [0 0 2 0 0 0]
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            [0 0 0 2 0 0]
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            [0 0 0 0 2 0]
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            [0 0 0 0 0 2],
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             2: [2 0 0 0]
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            [0 2 0 0]
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            [0 0 2 0]
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            [0 0 0 2]}
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            sage: z = y*y
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            sage: z._matrix_dictionary
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            {0: [4 0 0 0]
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            [0 4 0 0]
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            [0 0 4 0]
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            [0 0 0 4],
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             1: [4 0 0 0 0 0]
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            [0 4 0 0 0 0]
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            [0 0 4 0 0 0]
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            [0 0 0 4 0 0]
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            [0 0 0 0 4 0]
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            [0 0 0 0 0 4],
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             2: [4 0 0 0]
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            [0 4 0 0]
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            [0 0 4 0]
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            [0 0 0 4]}
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        """
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        if not isinstance(x,ChainComplexMorphism) or self._codomain != x._domain:
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            try:
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                y = self._domain.base_ring()(x)
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            except TypeError:
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                raise TypeError("multiplication is not defined")
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            f = dict()
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            for i in self._matrix_dictionary.keys():
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                f[i] = self._matrix_dictionary[i] * y
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            return ChainComplexMorphism(f,self._domain,self._codomain)
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        f = dict()
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        for i in self._matrix_dictionary.keys():
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            f[i] = x._matrix_dictionary[i]*self._matrix_dictionary[i]
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        return ChainComplexMorphism(f,self._domain,x._codomain)
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    def __rmul__(self,x):
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        """
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        Returns ``x * self`` if ``x`` is an element of the base ring.
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        EXAMPLES::
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            sage: S = simplicial_complexes.Sphere(2)
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            sage: H = Hom(S,S)
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            sage: i = H.identity()
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            sage: x = i.associated_chain_complex_morphism()
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            sage: 2*x == x*2
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            True
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            sage: 3*x == x*2
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            False
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        """
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        try:
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            y = self._domain.base_ring()(x)
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        except TypeError:
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            raise TypeError("multiplication is not defined")
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        f = dict()
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        for i in self._matrix_dictionary.keys():
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            f[i] = y * self._matrix_dictionary[i]
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        return ChainComplexMorphism(f,self._domain,self._codomain)
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    def __sub__(self,x):
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        """
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        Return ``self - x``.
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        EXAMPLES::
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            sage: S = simplicial_complexes.Sphere(2)
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            sage: H = Hom(S,S)
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            sage: i = H.identity()
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            sage: x = i.associated_chain_complex_morphism()
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            sage: y = x-x
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            sage: y._matrix_dictionary
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            {0: [0 0 0 0]
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            [0 0 0 0]
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            [0 0 0 0]
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            [0 0 0 0],
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             1: [0 0 0 0 0 0]
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            [0 0 0 0 0 0]
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            [0 0 0 0 0 0]
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            [0 0 0 0 0 0]
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            [0 0 0 0 0 0]
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            [0 0 0 0 0 0],
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             2: [0 0 0 0]
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            [0 0 0 0]
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            [0 0 0 0]
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            [0 0 0 0]}
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        """
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        return self + (-x)
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    def __eq__(self,x):
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        """
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        Return ``True`` if and only if ``self == x``.
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        EXAMPLES::
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            sage: S = SimplicialComplex(is_mutable=False)
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            sage: H = Hom(S,S)
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            sage: i = H.identity()
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            sage: x = i.associated_chain_complex_morphism()
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            sage: x
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            Chain complex morphism from Trivial chain complex over Integer Ring
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            to Trivial chain complex over Integer Ring
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            sage: f = x._matrix_dictionary
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            sage: C = S.chain_complex()
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            sage: G = Hom(C,C)
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            sage: y = G(f)
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            sage: x == y
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            True
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        """
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        return isinstance(x,ChainComplexMorphism) \
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                and self._codomain == x._codomain \
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                and self._domain == x._domain \
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                and self._matrix_dictionary == x._matrix_dictionary
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    def _repr_(self):
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        """
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        Return the string representation of ``self``.
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        EXAMPLES::
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            sage: S = SimplicialComplex(is_mutable=False)
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            sage: H = Hom(S,S)
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            sage: i = H.identity()
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            sage: x = i.associated_chain_complex_morphism()
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            sage: x
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            Chain complex morphism from Trivial chain complex over Integer Ring
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            to Trivial chain complex over Integer Ring
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            sage: x._repr_()
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            'Chain complex morphism from Trivial chain complex over Integer Ring
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            to Trivial chain complex over Integer Ring'
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        """
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        return "Chain complex morphism from {} to {}".format(self._domain, self._codomain)
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