Eagon northcott complex (#4327)

 Work on hypercomplexes:
    strands of complexes of graded modules for ZZ^m-gradings
    Eagon-Northcott complexes

docs/oscar_references.bib

@@ -989,6 +989,16 @@ @Article{EMSS16
   zbmath        = {6517845}
 }
 
+@Article{EN62,
+  author        = {Eagon, J. and Northcott, D.G.},
+  title         = {Ideals defined by matrices, and a certain complex associated to them},
+  journal       = {Proc. Royal Soc.},
+  volume        = {269},
+  pages         = {188--204},
+  year          = {1962},
+  doi           = {10.1098/rspa.1962.0170}
+}
+
 @Article{ES96,
   author        = {Eisenbud, David and Sturmfels, Bernd},
   title         = {Binomial ideals},

experimental/DoubleAndHyperComplexes/src/DoubleAndHyperComplexes.jl

@@ -1,14 +1,18 @@
 include("Objects/Types.jl")
 include("Objects/Attributes.jl")
 include("Objects/tensor_products.jl")
+include("Objects/tensor_product_functionality.jl")
 include("Objects/tor.jl")
 include("Objects/vector_spaces.jl")
 include("Objects/Constructors.jl")
 include("Objects/total_complexes.jl")
 include("Objects/koszul_complexes.jl")
+include("Objects/new_koszul_complexes.jl")
 include("Objects/degree_zero_complexes.jl")
 include("Objects/new_complex_template.jl")
 include("Objects/linear_strands.jl")
+include("Objects/eagon_northcott_complex.jl")
+include("Objects/induced_ENC.jl")
 
 include("Morphisms/Types.jl")
 include("Objects/cartan_eilenberg_resolution.jl")
@@ -17,6 +21,7 @@ include("Morphisms/ext.jl")
 include("Morphisms/simplified_complexes.jl")
 include("Morphisms/free_resolutions.jl")
 include("Morphisms/strands.jl")
+include("Morphisms/strand_functionality.jl")
 include("Morphisms/koszul_complexes.jl")
 include("Morphisms/morphism_from_maps.jl")
 include("Morphisms/linear_strands.jl")

experimental/DoubleAndHyperComplexes/src/Morphisms/simplified_complexes.jl

@@ -208,9 +208,9 @@ function (fac::SimplifiedChainFactory)(d::AbsHyperComplex, Ind::Tuple)
     for i in 1:m
       w = Sinv[i]
       v = zero(new_dom)
-      for j in 1:length(I)
-        a = w[I[j]]
-        !iszero(a) && (v += a*new_dom[j])
+      for (j, ind) in enumerate(I)
+        success, a = _has_index(w, ind)
+        success && (v += a*new_dom[j])
       end
       push!(img_gens_dom, v)
     end
@@ -401,13 +401,13 @@ function _simplify_matrix!(A::SMat; find_pivot=nothing)
   # Initialize the base change matrices in domain and codomain
   S = sparse_matrix(R, m, m)
   for i in 1:m
-    S[i] = sparse_row(R, [(i, one(R))])
+    S[i] = sparse_row(R, [(i, one(R))]; sort=false)
   end
   Sinv_transp = deepcopy(S)
 
   T = sparse_matrix(R, n, n)
   for i in 1:n
-    T[i] = sparse_row(R, [(i, one(R))])
+    T[i] = sparse_row(R, [(i, one(R))]; sort=false)
   end
   Tinv_transp = deepcopy(T)
 
@@ -479,7 +479,7 @@ function _simplify_matrix!(A::SMat; find_pivot=nothing)
     # been treated.
 
     a_row = deepcopy(A[p])
-    a_row_del = a_row - sparse_row(R, [(q, u)])
+    a_row_del = a_row - sparse_row(R, [(q, u)]; sort=false)
 
     col_entries = Vector{Tuple{Int, elem_type(R)}}()
     for i in 1:m
@@ -488,8 +488,8 @@ function _simplify_matrix!(A::SMat; find_pivot=nothing)
       success, c = _has_index(A, i, q)
       success && push!(col_entries, (i, c::elem_type(R)))
     end
-    a_col = sparse_row(R, col_entries)
-    a_col_del = a_col - sparse_row(R, [(p, u)])
+    a_col = sparse_row(R, col_entries; sort=false)
+    a_col_del = a_col - sparse_row(R, [(p, u)]; sort=false)
 
     uinv = inv(u)
 

experimental/DoubleAndHyperComplexes/src/Morphisms/strand_functionality.jl

@@ -0,0 +1,81 @@
+function (fac::StrandChainFactory)(c::AbsHyperComplex, i::Tuple)
+  M = fac.orig[i]
+  @assert is_graded(M) "module must be graded"
+  R = base_ring(M)
+  kk = coefficient_ring(R)
+  return FreeMod(kk, length(all_exponents(fac.orig[i], fac.d)))
+end
+
+function can_compute(fac::StrandChainFactory, c::AbsHyperComplex, i::Tuple)
+  return can_compute_index(fac.orig, i)
+end
+
+
+function (fac::StrandMorphismFactory)(c::AbsHyperComplex, p::Int, i::Tuple)
+  I = collect(i)
+  Next = I + (direction(c, p) == :chain ? -1 : 1)*[(k==p ? 1 : 0) for k in 1:dim(c)]
+  next = Tuple(Next)
+  orig_dom = fac.orig[i]
+  orig_cod = fac.orig[next]
+  dom = c[i]
+  cod = c[next]
+
+  orig_map = map(fac.orig, p, i)
+
+  # Use a dictionary for fast mapping of the monomials to the 
+  # generators of `cod`.
+  cod_dict = Dict{Tuple{Vector{Int}, Int}, elem_type(cod)}(m=>cod[k] for (k, m) in enumerate(all_exponents(orig_cod, fac.d)))
+  # Hashing of FreeModElem's can not be assumed to be non-trivial. Hence we use the exponents directly.
+  img_gens_res = elem_type(cod)[]
+  R = base_ring(orig_dom)
+  vv = gens(R)
+  for (e, i) in all_exponents(orig_dom, fac.d) # iterate through the generators of `dom`
+    m = prod(x^k for (x, k) in zip(vv, e); init=one(R))*orig_dom[i]
+    v = orig_map(m) # map the monomial
+    # take preimage of the result using the previously built dictionary.
+    # TODO: Iteration over the terms of v is VERY slow due to its suboptimal implementation.
+    # We have to iterate manually. This saves us roughly 2/3 of the memory consumption and 
+    # it also runs three times as fast. 
+    w = zero(cod)
+    for (i, b) in coordinates(v)
+      #g = orig_cod[i]
+      w += sum(c*cod_dict[(n, i)] for (c, n) in zip(AbstractAlgebra.coefficients(b), AbstractAlgebra.exponent_vectors(b)); init=zero(cod))
+    end
+    push!(img_gens_res, w)
+  end
+  return hom(dom, cod, img_gens_res)
+end
+
+function can_compute(fac::StrandMorphismFactory, c::AbsHyperComplex, p::Int, i::Tuple)
+  return can_compute_map(fac.orig, p, i)
+end
+
+### User facing constructor
+function strand(c::AbsHyperComplex{T}, d::Union{Int, FinGenAbGroupElem}) where {T<:ModuleFP}
+  result = StrandComplex(c, d)
+  inc = StrandInclusionMorphism(result)
+  result.inclusion_map = inc
+  return result, inc
+end
+
+
+# TODO: Code duplicated from `monomial_basis`. Clean this up!
+function all_exponents(W::MPolyDecRing, d::FinGenAbGroupElem)
+  D = W.D
+  is_free(D) || error("Grading group must be free")
+  h = hom(free_abelian_group(ngens(W)), W.d)
+  fl, p = has_preimage_with_preimage(h, d)
+  R = base_ring(W)
+  B = Vector{Int}[]
+  if fl
+     k, im = kernel(h)
+     #need the positive elements in there...
+     #Ax = b, Cx >= 0
+     C = identity_matrix(ZZ, ngens(W))
+     A = reduce(vcat, [x.coeff for x = W.d])
+     k = solve_mixed(transpose(A), transpose(d.coeff), C)
+     B = Vector{Int}[k[ee, :] for ee in 1:nrows(k)]
+  end
+  return B
+end
+

experimental/DoubleAndHyperComplexes/src/Morphisms/strands.jl

@@ -1,5 +1,5 @@
 ########################################################################
-# Strands of hyper complexs of graded free modules.
+# Strands of hyper complexes of graded free modules.
 #
 # Suppose S = 𝕜[x₀,…,xₙ] is a standard graded ring and C a 
 # hypercomplex of modules over it with all morphisms of degree 
@@ -14,89 +14,36 @@
 ### Production of the chains
 struct StrandChainFactory{ChainType<:ModuleFP} <: HyperComplexChainFactory{ChainType}
   orig::AbsHyperComplex
-  d::Int
+  d::Union{Int, FinGenAbGroupElem}
 
   function StrandChainFactory(
-      orig::AbsHyperComplex{ChainType}, d::Int
+      orig::AbsHyperComplex{ChainType}, d::Union{Int, FinGenAbGroupElem}
     ) where {ChainType<:ModuleFP}
     return new{FreeMod}(orig, d) # TODO: Specify the chain type better
   end
 end
 
-function (fac::StrandChainFactory)(c::AbsHyperComplex, i::Tuple)
-  M = fac.orig[i]
-  @assert is_graded(M) "module must be graded"
-  R = base_ring(M)
-  @assert is_standard_graded(R) "the base ring must be standard graded"
-  kk = coefficient_ring(R)
-  return FreeMod(kk, length(all_exponents(fac.orig[i], fac.d)))
-end
-
-function can_compute(fac::StrandChainFactory, c::AbsHyperComplex, i::Tuple)
-  return can_compute_index(fac.orig, i)
-end
-
 ### Production of the morphisms 
 struct StrandMorphismFactory{MorphismType<:ModuleFPHom} <: HyperComplexMapFactory{MorphismType}
   orig::AbsHyperComplex
-  d::Int
+  d::Union{Int, FinGenAbGroupElem}
   monomial_mappings::Dict{<:Tuple{<:Tuple, Int}, <:Map}
 
-  function StrandMorphismFactory(orig::AbsHyperComplex, d::Int)
+  function StrandMorphismFactory(orig::AbsHyperComplex, d::Union{Int, FinGenAbGroupElem})
     monomial_mappings = Dict{Tuple{Tuple, Int}, Map}()
     return new{FreeModuleHom}(orig, d, monomial_mappings)
   end
 end
 
-function (fac::StrandMorphismFactory)(c::AbsHyperComplex, p::Int, i::Tuple)
-  I = collect(i)
-  Next = I + (direction(c, p) == :chain ? -1 : 1)*[(k==p ? 1 : 0) for k in 1:dim(c)]
-  next = Tuple(Next)
-  orig_dom = fac.orig[i]
-  orig_cod = fac.orig[next]
-  dom = c[i]
-  cod = c[next]
-
-  orig_map = map(fac.orig, p, i)
-
-  # Use a dictionary for fast mapping of the monomials to the 
-  # generators of `cod`.
-  cod_dict = Dict{Tuple{Vector{Int}, Int}, elem_type(cod)}(m=>cod[k] for (k, m) in enumerate(all_exponents(orig_cod, fac.d)))
-  #cod_dict = Dict{Tuple{Vector{Int}, Int}, elem_type(cod)}(first(exponents(m))=>cod[k] for (k, m) in enumerate(all_monomials(orig_cod, fac.d)))
-  # Hashing of FreeModElem's can not be assumed to be non-trivial. Hence we use the exponents directly.
-  img_gens_res = elem_type(cod)[]
-  R = base_ring(orig_dom)
-  vv = gens(R)
-  for (e, i) in all_exponents(orig_dom, fac.d) # iterate through the generators of `dom`
-    m = prod(x^k for (x, k) in zip(vv, e); init=one(R))*orig_dom[i]
-    v = orig_map(m) # map the monomial
-    # take preimageof the result using the previously built dictionary.
-    # TODO: Iteration over the terms of v is VERY slow due to its suboptimal implementation.
-    # We have to iterate manually. This saves us roughly 2/3 of the memory consumption and 
-    # it also runs three times as fast. 
-    w = zero(cod)
-    for (i, b) in coordinates(v)
-      #g = orig_cod[i]
-      w += sum(c*cod_dict[(n, i)] for (c, n) in zip(coefficients(b), exponents(b)); init=zero(cod))
-    end
-    push!(img_gens_res, w)
-  end
-  return hom(dom, cod, img_gens_res)
-end
-
-function can_compute(fac::StrandMorphismFactory, c::AbsHyperComplex, p::Int, i::Tuple)
-  return can_compute_map(fac.orig, p, i)
-end
-
 ### The concrete struct
 @attributes mutable struct StrandComplex{ChainType, MorphismType} <: AbsHyperComplex{ChainType, MorphismType} 
   internal_complex::HyperComplex{ChainType, MorphismType}
   original_complex::AbsHyperComplex
-  d::Int
+  d::Union{Int, FinGenAbGroupElem}
   inclusion_map::AbsHyperComplexMorphism
 
   function StrandComplex(
-      orig::AbsHyperComplex{ChainType, MorphismType}, d::Int
+      orig::AbsHyperComplex{ChainType, MorphismType}, d::Union{Int, FinGenAbGroupElem}
     ) where {ChainType <: ModuleFP, MorphismType <: ModuleFPHom}
     chain_fac = StrandChainFactory(orig, d)
     map_fac = StrandMorphismFactory(orig, d)
@@ -160,38 +107,3 @@ end
 
 underlying_morphism(phi::StrandInclusionMorphism) = phi.internal_morphism
 
-### User facing constructor
-function strand(c::AbsHyperComplex{T}, d::Int) where {T<:ModuleFP}
-  result = StrandComplex(c, d)
-  inc = StrandInclusionMorphism(result)
-  result.inclusion_map = inc
-  return result, inc
-end
-
-### Some missing methods
-# (Disabled for the moment because the use case was disabled due to slowness)
-#=
-function sparse_matrix(phi::SubQuoHom{<:SubquoModule, <:ModuleFP, Nothing})
-  R = base_ring(domain(phi))
-  m = ngens(domain(phi))
-  n = ngens(codomain(phi))
-  result = sparse_matrix(R, m, n)
-  for (i, g) in enumerate(gens(domain(phi)))
-    result[i] = coordinates(phi(g))
-  end
-  return result
-end
-
-function sparse_matrix(phi::FreeModuleHom{FreeMod{T}, SubquoModule{T}, Nothing}) where {T}
-  V = domain(phi)
-  W = codomain(phi)
-  kk = base_ring(V)
-  m = ngens(V)
-  n = ngens(W)
-  result = sparse_matrix(kk, m, n)
-  for (i, g) in enumerate(gens(V))
-    result[i] = coordinates(phi(g))
-  end
-  return result
-end
-=#

experimental/DoubleAndHyperComplexes/src/Objects/Methods.jl

@@ -340,3 +340,34 @@ function _free_show(io::IO, C::AbsHyperComplex)
   end
 end
 
+### Koszul contraction
+# Given a free `R`-module `F`, a morphism φ: F → R, and an element `v` in ⋀ ᵖ F, 
+# compute the contraction φ(v) ∈ ⋀ ᵖ⁻¹ F.
+# Note: For this internal method φ is only represented as a dense (column) vector.
+# Warning: If the user provides their own parent, they need to make sure that things 
+#          are compatible. As this is an internal function, no sanity checks are done.
+function _contract(
+    v::FreeModElem{T}, phi::Vector{T}; 
+    parent::FreeMod{T}=begin
+      success, F0, p = _is_exterior_power(Oscar.parent(v))
+      @req success "parent is not an exterior power"
+      exterior_power(F0, p-1)
+    end
+  ) where {T}
+  success, F0, p = _is_exterior_power(Oscar.parent(v))
+  @req success "parent is not an exterior power"
+  @assert length(phi) == ngens(F0) "lengths are incompatible"
+  result = zero(parent)
+  n = ngens(F0)
+  for (i, ind) in enumerate(OrderedMultiIndexSet(p, n))
+    is_zero(v[i]) && continue
+    for j in 1:p
+      I = deleteat!(copy(indices(ind)), j)
+      new_ind = OrderedMultiIndex(I, n)
+      result = result + (-1)^j * v[i] * phi[ind[j]] * parent[linear_index(new_ind)]
+    end
+  end
+  return result
+end
+
+

experimental/DoubleAndHyperComplexes/src/Objects/cartan_eilenberg_resolution.jl

@@ -45,7 +45,7 @@ struct CEChainFactory{ChainType} <: HyperComplexChainFactory{ChainType}
   function CEChainFactory(c::AbsHyperComplex; is_exact::Bool=false)
     @assert dim(c) == 1 "complex must be 1-dimensional"
     #@assert has_lower_bound(c, 1) "complex must be bounded from below"
-    return new{chain_type(c)}(c, is_exact, Dict{Int, AbsHyperComplex}(), Dict{Int, AbsHyperComplex}(), Dict{Int, AbsHyperComplexMorphism}())
+    return new{FreeMod}(c, is_exact, Dict{Int, AbsHyperComplex}(), Dict{Int, AbsHyperComplex}(), Dict{Int, AbsHyperComplexMorphism}())
   end
 end
 
@@ -198,12 +198,12 @@ end
     @assert has_lower_bound(c, 1) "complexes must be bounded from below"
     @assert direction(c, 1) == :chain "resolutions are only implemented for chain complexes"
     chain_fac = CEChainFactory(c; is_exact)
-    map_fac = CEMapFactory{MorphismType}() # TODO: Do proper type inference here!
+    map_fac = CEMapFactory{FreeModuleHom}() # TODO: Do proper type inference here!
 
     # Assuming d is the dimension of the new complex
     internal_complex = HyperComplex(2, chain_fac, map_fac, [:chain, :chain]; lower_bounds = Union{Int, Nothing}[0, lower_bound(c, 1)])
     # Assuming that ChainType and MorphismType are provided by the input
-    return new{ChainType, MorphismType}(internal_complex)
+    return new{FreeMod, FreeModuleHom}(internal_complex)
   end
 end