CoCalc -- chain_complex

GitHub Repository: sagemath/sagelib
Path: blob/master/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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This module implements morphisms of chain complexes. The input is a dictionary whose
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keys are in the grading group of the chain complex and whose values are 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]}
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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 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):
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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]}
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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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        if C._grading_group != ZZ:
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            raise NotImplementedError, "Chain complex morphisms are not implemented over gradings other than ZZ."
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        d = C._degree
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        if d != D._degree:
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            raise ValueError, "Chain complex morphisms are not defined for chain complexes of different degrees."
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        if d != -1 and d != 1:
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            raise NotImplementedError, "Chain complex morphisms are not implemented for degrees besides -1 and 1."
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        dim_min = min(min(C.differential().keys()),min(D.differential().keys()))
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        dim_max = max(max(C.differential().keys()),max(D.differential().keys()))
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        if not C.base_ring()==D.base_ring():
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            raise NotImplementedError, "Chain complex morphisms between chain complexes of different base rings are not implemented."
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        for i in range(dim_min,dim_max):
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            try:
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                matrices[i]
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            except KeyError:
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                matrices[i] = matrix.zero_matrix(C.base_ring(),D.differential()[i].ncols(),C.differential()[i].ncols(),sparse=True)
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            try:
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                matrices[i+1]
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            except KeyError:
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                matrices[i+1] = matrix.zero_matrix(C.base_ring(),D.differential()[i+1].ncols(),C.differential()[i+1].ncols(),sparse=True)
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            if d==-1:
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                if (i+1) in C.differential().keys() and (i+1) in D.differential().keys():
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                    if not matrices[i]*C.differential()[i+1]==D.differential()[i+1]*matrices[i+1]:
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                        raise ValueError, "Matrices must define a chain complex morphism."
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                elif (i+1) in C.differential().keys():
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                    if not matrices[i]*C.differential()[i+1].is_zero():
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                        raise ValueError, "Matrices must define a chain complex morphism."
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                elif (i+1) in D.differential().keys():
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                    if not D.differential()[i+1]*matrices[i+1].is_zero():
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                        raise ValueError, "Matrices must define a chain complex morphism."
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            else:
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                if i in C.differential().keys() and i in D.differential().keys():
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                    if not matrices[i+1]*C.differential()[i]==D.differential()[i]*matrices[i]:
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                        raise ValueError, "Matrices must define a chain complex morphism."
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                elif i in C.differential().keys():
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                    if not matrices[i+1]*C.differential()[i].is_zero():
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                        raise ValueError, "Matrices must define a chain complex morphism."
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                elif i in D.differential().keys():
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                    if not D.differential()[i]*matrices[i].is_zero():
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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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        Returns self*x if self and x are composable morphisms 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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        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: 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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        Returns True if and only if self==x.
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        EXAMPLES::
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            sage: S = SimplicialComplex(3)
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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 Chain complex with at most 0 nonzero terms over Integer Ring to Chain complex with at most 0 nonzero terms 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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        if not isinstance(x,ChainComplexMorphism) or self._codomain != x._codomain or self._domain != x._domain or self._matrix_dictionary != x._matrix_dictionary:
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            return False
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        else:
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            return True
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    def _repr_(self):
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        """
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        Returns the string representation of self.
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        EXAMPLES::
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            sage: S = SimplicialComplex(3)
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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 Chain complex with at most 0 nonzero terms over Integer Ring to Chain complex with at most 0 nonzero terms over Integer Ring
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            sage: x._repr_()
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            'Chain complex morphism from Chain complex with at most 0 nonzero terms over Integer Ring to Chain complex with at most 0 nonzero terms over Integer Ring'
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        """
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        return "Chain complex morphism from " + self._domain._repr_() + " to " + self._codomain._repr_()
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