add most of the direct_sum new stuff

src/AbstractAlgebra.jl

@@ -735,6 +735,10 @@
 export can_solve
 export can_solve_with_solution
 export can_solve_with_solution_and_kernel
+export canonical_injection
+export canonical_injections
+export canonical_projection
+export canonical_projections
 export canonical_unit
 export change_base_ring
 export change_coefficient_ring
@@ -851,6 +855,7 @@
 export hnf_via_popov
 export hnf_via_popov_with_transform
 export hnf_with_transform
+export hom
 export hooklength
 export ideal
 export identity_map
@@ -1162,6 +1167,12 @@
    Generic.YoungTableau(part, fill)
 end
 
+################################################################################
+#
+#   generic stubs for sub, quo, ...
+#
+################################################################################
+
 @doc raw"""
     sub(m::Module{T}, subs::Vector{<:Generic.Submodule{T}}) where T <: RingElement
 
@@ -1177,6 +1188,14 @@
    Generic.sub(m, subs)
 end
 
+function canonical_injections(D)
+  return [canonical_injection(D, i) for i=1:_number_of_direct_product_factors(D)]
+end
+
+function canonical_projections(D)
+  return [canonical_projections(D, i) for i=1:_number_of_direct_product_factors(D)]
+end
+
 export Generic
 
 ###############################################################################

src/fundamental_interface.jl

@@ -304,3 +304,32 @@ function sub! end
 Return `b*c`, possibly modifying the object `a` in the process.
 """
 function mul! end
+
+@doc raw"""
+    canonical_injection(D, i)
+
+Return the i-th canonical injection into the direct sum or product objects `D`.
+"""
+function canonical_injection end
+
+@doc raw"""
+    canonical_projection(D, i)
+
+Return the i-th canonical projection into the direct sum or product objects `D`.
+"""
+function canonical_projection end
+
+@doc raw"""
+    _number_of_direct_product_factors(D)
+
+Return the number of factors/ summands in the direct product/ sum object `D`
+"""
+function _number_of_direct_product_factors end
+
+@doc raw"""
+    hom(D, C, data)
+
+Return the homomorphism from the domain `D` into the codomain `C` defined by the data.
+"""
+function hom end
+

src/generic/DirectSum.jl

@@ -152,6 +152,13 @@
    return DirectSumModuleElem{T}(D, matv)
 end
 
+function AbstractAlgebra.canonical_injection(A::DirectSumModule, i::Int)
+  B = summands(A)[i]
+  return hom(B, A, [direct_sum_injection(i, A, x) for x = gens(B)])
+end
+
+AbstractAlgebra._number_of_direct_product_factors(A::DirectSumModule) = length(summands(A))
+
 function direct_sum_projection(D::DirectSumModule{T}, i::Int, v::AbstractAlgebra.FPModuleElem{T}) where {T <: RingElement}
    # Find starting point of the given module in the large vectors
    S = summands(D)
@@ -164,6 +171,10 @@
    return elem_type(m)(m, matv)
 end
 
+function AbstractAlgebra.canonical_projection(A::DirectSumModule, i::Int)
+  return hom(A, summands(A)[i], [direct_sum_projection(A, i, x) for x = gens(A)])
+end
+
 function direct_sum(m::Vector{<:AbstractAlgebra.FPModule{T}}) where T <: RingElement
    length(m) == 0 && error("Cannot take a direct sum of an empty vector of modules")
    # Check base rings are the same
@@ -239,3 +250,16 @@
 
    return ModuleHomomorphism(D, A, transpose(hvcat(Tuple([length(SD) for i = 1:length(SA)]), map(x->transpose(x.matrix), m)...)))
 end
+
+function AbstractAlgebra.hom(A::AbstractAlgebra.Generic.DirectSumModule{T}, B::AbstractAlgebra.Generic.DirectSumModule{T}, M::Matrix{<:Map{<:AbstractAlgebra.FPModule{T}, <:AbstractAlgebra.FPModule{T}}}) where {T}
+  pro = canonical_projections(A)
+  im = canonical_injections(B)
+  s = hom(A, B, [zero(B) for i = 1:dim(A)])
+  for i=1:length(pro)
+    for j=1:length(im)
+      s += pro[i]*M[i,j]*im[j]
+    end
+  end
+  return s
+end
+