polishing and size

Author: jankoboehm

docs/src/CommutativeAlgebra/homological_algebra.md

@@ -17,12 +17,16 @@ supporting computations in homological algebra.
 ```@docs
 prune_with_map(M::ModuleFP)
 ```
-## Finiteness as a set
+## Finiteness and cardinality as a set
 
 ```@docs
 is_finite(M::SubquoModule{T}) where {T<:Union{ZZRingElem, FieldElem}}
 ```
 
+```@docs
+size(M::SubquoModule{T}) where {T<:Union{ZZRingElem, FieldElem}}
+```
+
 ## Presentations
 
 ```@docs

src/Modules/UngradedModules/FreeResolutions.jl

@@ -502,39 +502,11 @@ function free_resolution(M::SubquoModule{T};
   return FreeResolution(cc)
 end
 
-# kept for the comments
-# function free_resolution(M::SubquoModule{T}) where {T<:RingElem}
-#   # This generic code computes a free resolution in a lazy way.
-#   # We start out with a presentation of M and implement 
-#   # an iterative fill function to compute every higher term 
-#   # on request.
-#   R = base_ring(M)
-#   p = presentation(M)
-#   p.fill = function(C::Hecke.ComplexOfMorphisms, k::Int)
-#     # TODO: Use official getter and setter methods instead 
-#     # of messing manually with the internals of the complex.
-#     for i in first(chain_range(C)):k-1
-#       N = domain(map(C, i))
-
-#       if iszero(N) # Fill up with zero maps
-#         C.complete = true
-#         phi = hom(N, N, elem_type(N)[]; check=false)
-#         pushfirst!(C.maps, phi)
-#         continue
-#       end
-
-#       K, inc = kernel(map(C, i))
-#       nz = findall(!is_zero, gens(K))
-#       F = FreeMod(R, length(nz))
-#       phi = hom(F, C[i], iszero(length(nz)) ? elem_type(C[i])[] : inc.(gens(K)[nz]); check=false)
-#       pushfirst!(C.maps, phi)
-#     end
-#     return first(C.maps)
-#   end
-#   return p
-# end
-
 function free_resolution(M::SubquoModule{T}; length::Int=0) where {T<:RingElem}
+  # This generic code computes a free resolution in a lazy way.
+  # We start out with a presentation of M and implement
+  # an iterative fill function to compute every higher term
+  # on request.
   R = base_ring(M)
   p = presentation(M)
   p.fill = function(C::Hecke.ComplexOfMorphisms, k::Int)

src/Modules/UngradedModules/Presentation.jl

@@ -706,7 +706,7 @@ end
 @doc raw"""
     is_finite(M::SubquoModule{T}) where {T<:Union{ZZRingElem, FieldElem}}
 
-Determine whether the finitely presented module `M` over `ZZ` or a field is finite as a set.
+Determine whether the finitely presented module `M` over `ZZRing` or a `Field` is finite as a set.
 
 This is done by computing a minimal presentation.
 
@@ -782,3 +782,62 @@ function is_finite(M::SubquoModule{T}) where {T<:Union{ZZRingElem, FieldElem}}
     error("The base ring $(typeof(R)) is not supported.")
   end
 end
+
+@doc raw"""
+    size(M::SubquoModule{T}) where {T<:Union{ZZRingElem, FieldElem}}
+
+Compute the cardinality of the finitely presented module `M` over `ZZRing` or a `Field` as a set.
+Returns `PosInf` if the module is infinite.
+
+This is done by computing a minimal presentation.
+
+# Examples
+```jldoctest
+julia> R = ZZ;
+
+julia> F = free_module(FreeMod, R, 2);
+
+julia> M = cokernel(hom(F, F, matrix(ZZ, [2 0; 0 3])));
+
+julia> size(M)
+6
+
+julia> N = cokernel(hom(F, F, matrix(ZZ, [1 0; 0 0])));
+
+julia> size(N)
+infinity
+
+julia> K, a = finite_field(7, "a");
+
+julia> G = free_module(FreeMod, K, 3);
+
+julia> H = free_module(FreeMod, K, 1);
+
+julia> L = cokernel(hom(H, G, matrix(K, [1 0 0])));
+
+julia> size(L)
+49
+```
+"""
+function size(M::SubquoModule{T}) where {T<:Union{ZZRingElem, FieldElem}}
+  M_prime, _ = prune_with_map_atomic(M)
+  R = base_ring(M_prime)
+  pres = presentation(M_prime)
+  rel_matrix = matrix(map(pres, 1))
+  nc, nr = size(rel_matrix, 2), size(rel_matrix, 1)
+  has_free_generator = any(j -> all(i -> iszero(rel_matrix[i, j]), 1:nr), 1:nc)
+  if isa(R, ZZRing)
+    if has_free_generator
+      return PosInf()
+    end
+    return prod(abs(rel_matrix[i,i]) for i in 1:min(nr,nc))
+  elseif isa(R, Field)
+    if is_finite(R)
+      q = order(R)
+      dim = ngens(M_prime)
+      return q^dim
+    else
+      return nc == 0 ? 1 : PosInf()
+    end
+  end
+end

test/Modules/UngradedModules.jl

@@ -1731,4 +1731,28 @@ end
     @test_throws AssertionError graded_free_module(ZZ, [1])
   end
 
+  @testset "size of modules" begin
+    R = ZZ
+    F = free_module(FreeMod, R, 2)
+    M = cokernel(hom(F, F, matrix(ZZ, [2 0; 0 3])))
+    @test size(M) == 6
+
+    N = cokernel(hom(F, F, matrix(ZZ, [7 0; 0 0])))
+    @test size(N) == PosInf()
+
+    K, a = finite_field(7, "a")
+    G = free_module(FreeMod, K, 3)
+    H = free_module(FreeMod, K, 1)
+    L = cokernel(hom(H, G, matrix(K, [1 0 0])))
+    @test size(L) == 49
+
+    K, a = finite_field(7, "a")
+    G = free_module(FreeMod, K, 3)
+    L = cokernel(hom(G, G, matrix(K, [1 0 0; 0 1 0; 0 0 1])))
+    @test size(L) == 1
+
+    G = free_module(FreeMod, QQ, 3)
+    L = cokernel(hom(G, G, matrix(QQ, [1 0 0; 0 1 0; 0 0 1])))
+    @test size(L) == 1
+  end
 end