Adds free resolutions over quotient rings by using Singular.sres (#…

…4134)

* adapt module gens to quotient rings

* adjusts signatures of `prune_with_map` and `_presentation_minimal`

* compute free resolutions over quotients with sres
---------

Co-authored-by: Rafael Mohr <Rafael.Mohr@lip6.fr>
Co-authored-by: HechtiDerLachs <zach@mathematik.uni-kl.de>
Co-authored-by: Lars Göttgens <lars.goettgens@gmail.com>

Project.toml

@@ -40,7 +40,7 @@ Polymake = "0.11.20"
 Random = "1.6"
 RandomExtensions = "0.4.3"
 Serialization = "1.6"
-Singular = "0.23.4"
+Singular = "0.23.8"
 TOPCOM_jll = "0.17.8"
 UUIDs = "1.6"
 cohomCalg_jll = "0.32.0"

src/Modules/UngradedModules/FreeResolutions.jl

@@ -230,15 +230,15 @@ function _extend_free_resolution(cc::Hecke.ComplexOfMorphisms, idx::Int)
 end
 
 @doc raw"""
-    free_resolution(M::SubquoModule{<:MPolyRingElem}; 
-        ordering::ModuleOrdering = default_ordering(M),
-        length::Int = 0, algorithm::Symbol = :fres
-      )
+    free_resolution(M::SubquoModule{T}; 
+        length::Int=0,
+        algorithm::Symbol = T <:MPolyRingElem ? :fres : :sres) where {T <: Union{MPolyRingElem, MPolyQuoRingElem}}
 
 Return a free resolution of `M`.
 
 If `length != 0`, the free resolution is only computed up to the `length`-th free module.
-Current options for `algorithm` are `:fres`, `:nres`, and `:mres`.
+Current options for `algorithm` are `:fres`, `:nres`, and `:mres` for modules over
+polynomial rings and `:sres` for modules over quotients of polynomial rings.
 
 !!! note
     The function first computes a presentation of `M`. It then successively computes
@@ -255,6 +255,10 @@ Current options for `algorithm` are `:fres`, `:nres`, and `:mres`.
     [EMSS16](@cite). Typically, this is more efficient than the approaches above, but the 
     resulting resolution is far from being minimal.
 
+!!! note
+    If `M` is a module over a quotient of a polynomial ring then the `length` keyword must
+    be set to a nonzero value.
+
 # Examples
 ```jldoctest
 julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z])
@@ -392,16 +396,20 @@ julia> matrix(map(FM3, 1))
 
 ```
 
-**Note:** Over rings other than polynomial rings, the method will default to a lazy, 
+**Note:** Over rings other than polynomial rings or quotients of polynomial rings, the method will default to a lazy, 
 iterative kernel computation.
 """
-function free_resolution(M::SubquoModule{<:MPolyRingElem}; 
-                         ordering::ModuleOrdering = default_ordering(M),
-                         length::Int=0, algorithm::Symbol=:fres)
+function free_resolution(M::SubquoModule{T}; 
+                         length::Int=0,
+                         algorithm::Symbol = T <:MPolyRingElem ? :fres : :sres) where {T <: Union{MPolyRingElem, MPolyQuoRingElem}}
 
   coefficient_ring(base_ring(M)) isa AbstractAlgebra.Field ||
       error("Must be defined over a field.")
 
+  if T <: MPolyQuoRingElem
+    !iszero(length) || error("Specify a length up to which a free resolution should be computed")
+  end
+
   cc_complete = false
 
   #= Start with presentation =#
@@ -435,6 +443,9 @@ function free_resolution(M::SubquoModule{<:MPolyRingElem};
   elseif algorithm == :nres
     gbpres = singular_kernel_entry
     res = Singular.nres(gbpres, length)
+  elseif algorithm == :sres && T <: MPolyQuoRingElem
+    gbpres = Singular.std(singular_kernel_entry)
+    res = Singular.sres(gbpres, length)
   else
     error("Unsupported algorithm $algorithm")
   end

src/Modules/UngradedModules/ModuleGens.jl

@@ -261,22 +261,49 @@ end
 
 Convert a Singular vector to a free module element.
 """
-function (F::FreeMod)(s::Singular.svector)
-  pos = Int[]
-  values = []
+function (F::FreeMod{<:MPolyRingElem})(s::Singular.svector)
   Rx = base_ring(F)
-  R = base_ring(Rx)
-  for (i, e, c) = s
-    f = Base.findfirst(==(i), pos)
-    if f === nothing
-      push!(values, MPolyBuildCtx(base_ring(F)))
-      f = length(values)
-      push!(pos, i)
+  R = coefficient_ring(Rx)
+  ctx = MPolyBuildCtx(Rx)
+
+  # shortcut in order not to allocate the dictionary
+  if isone(length(s))
+    (i, e, c) = first(s)
+    push_term!(ctx, R(c), e)
+    return FreeModElem(sparse_row(Rx, [(i, finish(ctx))]), F)
+  end
+
+  cache = IdDict{Int, typeof(ctx)}()
+  for (i, e, c) in s
+    ctx = get!(cache, i) do
+      MPolyBuildCtx(Rx)
+    end
+    push_term!(ctx, R(c), e)
+  end
+  return FreeModElem(sparse_row(Rx, [(i, finish(ctx)) for (i, ctx) in cache]), F)
+end
+
+function (F::FreeMod{<:MPolyQuoRingElem})(s::Singular.svector)
+  Qx = base_ring(F)::MPolyQuoRing
+  Rx = base_ring(Qx)::MPolyRing
+  R = coefficient_ring(Rx)
+  ctx = MPolyBuildCtx(Rx)
+
+  # shortcut in order not to allocate the dictionary
+  if isone(length(s))
+    (i, e, c) = first(s)
+    push_term!(ctx, R(c), e)
+    return FreeModElem(sparse_row(Qx, [(i, Qx(finish(ctx)))]), F)
+  end
+
+  cache = IdDict{Int, typeof(ctx)}()
+  for (i, e, c) in s
+    ctx = get!(cache, i) do
+      MPolyBuildCtx(Rx)
     end
-    push_term!(values[f], R(c), e)
+    push_term!(ctx, R(c), e)
   end
-  pv = Tuple{Int, elem_type(Rx)}[(pos[i], base_ring(F)(finish(values[i]))) for i=1:length(pos)]
-  return FreeModElem(sparse_row(base_ring(F), pv), F)
+  return FreeModElem(sparse_row(Qx, [(i, Qx(finish(ctx))) for (i, ctx) in cache]), F)
 end
 
 # After creating the required infrastruture in Singular,

src/Modules/UngradedModules/Presentation.jl

@@ -532,7 +532,7 @@ function prune_with_map(M::ModuleFP)
   return N, b
 end
 
-function prune_with_map(M::ModuleFP{T}) where {T<:MPolyRingElem{<:FieldElem}} # The case that can be handled by Singular
+function prune_with_map(M::ModuleFP{T}) where {T<:Union{MPolyRingElem, MPolyQuoRingElem}} # The case that can be handled by Singular
 
   # Singular presentation
   pm = presentation(M)
@@ -577,7 +577,8 @@ function prune_with_map(M::ModuleFP{T}) where {T<:MPolyRingElem{<:FieldElem}} #
 end
 
 function _presentation_minimal(SQ::ModuleFP{T};
-                               minimal_kernel::Bool=true) where {T<:MPolyRingElem{<:FieldElem}}
+                               minimal_kernel::Bool=true) where {T <: Union{MPolyRingElem, MPolyQuoRingElem}}
+  R = base_ring(SQ)
 
   R = base_ring(SQ)
 

test/Modules/MPolyQuo.jl

@@ -96,8 +96,8 @@ end
   A2 = FreeMod(A, 2)
   v = [x*A2[1] + y*A2[2], z*A2[1] + (x-1)*A2[2]]
   M, _ = quo(A2, v)
-  p = free_resolution(M)
-  @test !iszero(p[10])
+  p = free_resolution(M, length = 11)
+  @test p[10] isa FreeMod
 end
 
 @testset "free resolutions II" begin
@@ -106,8 +106,8 @@ end
   A, _ = quo(R, I)
   A1 = FreeMod(A, 1)
   M, _ = quo(A1, [y*A1[1], z*A1[1]])
-  p = free_resolution(M)
-  @test iszero(p[10])
+  p = free_resolution(M, length = 11)
+  @test p[10] isa FreeMod
 end
 
 @testset "kernels of FreeMod -> SubquoModule" begin

test/Modules/UngradedModules.jl

@@ -275,6 +275,14 @@ end
   @test relations(C) == [zero(F)]
   @test domain(isom) == F
   @test codomain(isom) == C
+
+  R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);
+  A, p = quo(R, ideal(R, x^5))
+  M1 = identity_matrix(A, 2)
+  M2 = A[-x-y 2*x^2+x; z^4 0; 0 z^4; 8*x^3*y - 4*x^3 - 4*x^2*y + 2*x^2 + 2*x*y - x - y x; x^4 0]
+  M = SubquoModule(M1, M2)
+  fr = free_resolution(M, length = 9)
+  @test all(iszero, homology(fr)[2:end])
 end
 
 @testset "Prune With Map" begin