diff --git a/src/Modules/UngradedModules/FreeResolutions.jl b/src/Modules/UngradedModules/FreeResolutions.jl
index b0353b9598b3..8af80f5b6b6e 100644
--- a/src/Modules/UngradedModules/FreeResolutions.jl
+++ b/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
diff --git a/src/Modules/UngradedModules/ModuleGens.jl b/src/Modules/UngradedModules/ModuleGens.jl
index 2c3ba80e5579..445269c65b83 100644
--- a/src/Modules/UngradedModules/ModuleGens.jl
+++ b/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)) # TODO: length doesn't work!
+#   (i, e, c) = first(s)
+#   push_term!(ctx, R(c), e)
+#   return FreeModElem(sparse_row(Qx, [(i, finish(ctx))]))
+# 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(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)) # TODO: length doesn't work!
+#   (i, e, c) = first(s)
+#   push_term!(ctx, R(c), e)
+#   return FreeModElem(sparse_row(Qx, [(i, Qx(finish(ctx)))]))
+# 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(Qx, [(i, Qx(finish(ctx))) for (i, ctx) in cache]), F)
 end
 
 # After creating the required infrastruture in Singular,
diff --git a/src/Modules/UngradedModules/Presentation.jl b/src/Modules/UngradedModules/Presentation.jl
index 5e70d60ef392..210baa120b44 100644
--- a/src/Modules/UngradedModules/Presentation.jl
+++ b/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)
  
diff --git a/src/Modules/mpolyquo.jl b/src/Modules/mpolyquo.jl
index e2aca6b078ef..375cac6eb658 100644
--- a/src/Modules/mpolyquo.jl
+++ b/src/Modules/mpolyquo.jl
@@ -10,18 +10,18 @@
            DomainType<:FreeMod{<:MPolyQuoRingElem},
            CodomainType<:FreeMod{<:MPolyQuoRingElem}
           }
+    @show "this kernel"
   R = base_ring(codomain(f))
-  P = base_ring(R)
-  F = _poly_module(domain(f))
-  M = _as_poly_module(codomain(f))
-  id = _iso_with_poly_module(codomain(f))
-  # Why does img_gens(f) return a list of SubQuoElems???
-  phi = _lifting_iso(codomain(f))
-  g = hom(F, M, phi.(f.(gens(domain(f)))))
-  K, inc = kernel(g)
-  tr =  compose(inc, _poly_module_restriction(domain(f)))
-  KK, inc2 = sub(domain(f), unique!(filter!(!iszero, tr.(gens(K)))))
-  return KK, inc2
+  SR = singular_poly_ring(R)
+  F = domain(f)
+  G = codomain(f)
+  img_gens = images_of_generators(f)
+  MG = ModuleGens(img_gens, G)
+  singular_assure(MG)
+  sing_ker_gens = Singular.syz(singular_generators(MG))
+  @show "done with syzygy computations"
+  KG = ModuleGens(F, sing_ker_gens)
+  return sub(F, oscar_generators(KG))
 end
 
 function Base.in(a::FreeModElem{T}, M::SubModuleOfFreeModule{T}) where {T<:MPolyQuoRingElem}
diff --git a/src/Rings/MPolyMap/MPolyAnyMap.jl b/src/Rings/MPolyMap/MPolyAnyMap.jl
index 873365b9ad5f..03384f28d7ad 100644
--- a/src/Rings/MPolyMap/MPolyAnyMap.jl
+++ b/src/Rings/MPolyMap/MPolyAnyMap.jl
@@ -42,17 +42,37 @@ const _DomainTypes = Union{MPolyRing, MPolyQuoRing}
   coeff_map::U
   img_gens::Vector{V}
   temp_ring           # temporary ring used when evaluating maps
+  variable_indices::Vector{Int} # a table where the i-th entry contains the 
+                                # index of the variable where it's mapped to
+                                # in case the mapping takes such a particularly
+                                # simple form.
 
   function MPolyAnyMap{D, C, U, V}(domain::D,
-                                codomain::C,
-                                coeff_map::U,
-                                img_gens::Vector{V}) where {D, C, U, V}
-      @assert V === elem_type(C)
-      for g in img_gens
-        @assert parent(g) === codomain "elements does not have the correct parent"
-      end
+                                   codomain::C,
+                                   coeff_map::U,
+                                   img_gens::Vector{V}) where {D, C, U, V}
+    @assert V === elem_type(C)
+    for g in img_gens
+      @assert parent(g) === codomain "elements does not have the correct parent"
+    end
     return new{D, C, U, V}(domain, codomain, coeff_map, img_gens)
   end
+  function MPolyAnyMap{D, C, U, V}(domain::D,
+                                   codomain::C,
+                                   coeff_map::U,
+                                   img_gens::Vector{V}) where {D <: Union{<:MPolyRing, <:MPolyQuoRing}, 
+                                                               C <: Union{<:MPolyRing, <:MPolyQuoRing}, 
+                                                               U, V}
+    @assert V === elem_type(C)
+    for g in img_gens
+      @assert parent(g) === codomain "elements does not have the correct parent"
+    end
+    result = new{D, C, U, V}(domain, codomain, coeff_map, img_gens)
+    if all(is_gen(x) for x in img_gens) && allunique(img_gens)
+      result.variable_indices = [findfirst(x==y for y in gens(codomain)) for x in img_gens]
+    end
+    return result
+  end
 end
 
 function MPolyAnyMap(d::D, c::C, cm::U, ig::Vector{V}) where {D, C, U, V}
diff --git a/src/Rings/MPolyMap/MPolyRing.jl b/src/Rings/MPolyMap/MPolyRing.jl
index 4a14a083baae..d248208f2cd9 100644
--- a/src/Rings/MPolyMap/MPolyRing.jl
+++ b/src/Rings/MPolyMap/MPolyRing.jl
@@ -132,17 +132,101 @@ end
 #
 ################################################################################
 
+# Some additional methods needed for the test in the constructor for MPolyAnyMap
+is_gen(x::MPolyQuoRingElem) = is_gen(lift(x))
+is_gen(x::MPolyDecRingElem) = is_gen(forget_grading(x))
+
+function _evaluate_plain(F::MPolyAnyMap{<:MPolyRing, <:MPolyRing}, u)
+  if isdefined(F, :variable_indices)
+    S = codomain(F)::MPolyRing
+    r = ngens(S)
+    ctx = MPolyBuildCtx(S)
+    for (c, e) in zip(AbstractAlgebra.coefficients(u), AbstractAlgebra.exponent_vectors(u))
+      ee = [0 for _ in 1:r]
+      for (i, k) in enumerate(e)
+        ee[F.variable_indices[i]] = k
+      end
+      push_term!(ctx, c, ee)
+    end
+    return finish(ctx)
+  end
+
+  return evaluate(u, F.img_gens)
+end
+
 function _evaluate_plain(F::MPolyAnyMap{<: MPolyRing}, u)
   return evaluate(u, F.img_gens)
 end
 
 # See the comment in MPolyQuo.jl
 function _evaluate_plain(F::MPolyAnyMap{<:MPolyRing, <:MPolyQuoRing}, u)
+  if isdefined(F, :variable_indices)
+    S = base_ring(codomain(F))::MPolyRing
+    r = ngens(S)
+    ctx = MPolyBuildCtx(S)
+    for (c, e) in zip(coefficients(u), exponents(u))
+      ee = [0 for _ in 1:r]
+      for (i, k) in enumerate(e)
+        ee[F.variable_indices[i]] = k
+      end
+      push_term!(ctx, c, ee)
+    end
+    return codomain(F)(finish(ctx))
+  end
   A = codomain(F)
   v = evaluate(lift(u), lift.(_images(F)))
   return simplify(A(v))
 end
 
+function _evaluate_general(F::MPolyAnyMap{<:MPolyRing, <:MPolyRing}, u)
+  if domain(F) === codomain(F) && coefficient_map(F) === nothing
+    return evaluate(map_coefficients(coefficient_map(F), u,
+                                     parent = domain(F)), F.img_gens)
+  else
+    S = temp_ring(F)
+    if S !== nothing
+      if !isdefined(F, :variable_indices) || coefficient_ring(S) !== codomain(F)
+        return evaluate(map_coefficients(coefficient_map(F), u,
+                                         parent = S), F.img_gens)
+      else
+        tmp_poly = map_coefficients(coefficient_map(F), u, parent = S)
+        cod_ring = codomain(F)::MPolyRing
+        r = ngens(cod_ring)
+        ctx = MPolyBuildCtx(cod_ring)
+        for (p, e) in zip(coefficients(tmp_poly), exponents(tmp_poly))
+          ee = [0 for _ in 1:r]
+          for (i, k) in enumerate(e)
+            ee[F.variable_indices[i]] = k
+          end
+          p::MPolyRingElem
+          @assert parent(p) === cod_ring
+          for (c, d) in zip(coefficients(p), exponents(p))
+            push_term!(ctx, c, ee+d)
+          end
+        end
+        return finish(ctx)
+      end
+    else
+      if !isdefined(F, :variable_indices)
+        return evaluate(map_coefficients(coefficient_map(F), u), F.img_gens)
+      else
+        tmp_poly = map_coefficients(coefficient_map(F), u)
+        cod_ring = codomain(F)::MPolyRing
+        r = ngens(cod_ring)
+        ctx = MPolyBuildCtx(cod_ring)
+        for (c, e) in zip(coefficients(tmp_poly), exponents(tmp_poly))
+          ee = [0 for _ in 1:r]
+          for (i, k) in enumerate(e)
+            ee[F.variable_indices[i]] = k
+          end
+          push_term!(ctx, c, ee)
+        end
+        return finish(ctx)
+      end
+    end
+  end
+end
+
 function _evaluate_general(F::MPolyAnyMap{<: MPolyRing}, u)
   if domain(F) === codomain(F) && coefficient_map(F) === nothing
     return evaluate(map_coefficients(coefficient_map(F), u,
@@ -151,7 +235,7 @@ function _evaluate_general(F::MPolyAnyMap{<: MPolyRing}, u)
     S = temp_ring(F)
     if S !== nothing
       return evaluate(map_coefficients(coefficient_map(F), u,
-                                parent = S), F.img_gens)
+                                       parent = S), F.img_gens)
     else
       return evaluate(map_coefficients(coefficient_map(F), u), F.img_gens)
     end
diff --git a/test/Modules/MPolyQuo.jl b/test/Modules/MPolyQuo.jl
index ef654cacf178..4505b9e6c3a0 100644
--- a/test/Modules/MPolyQuo.jl
+++ b/test/Modules/MPolyQuo.jl
@@ -96,7 +96,7 @@ 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)
+  p = free_resolution(M, length = 11)
   @test !iszero(p[10])
 end
 
diff --git a/test/Modules/UngradedModules.jl b/test/Modules/UngradedModules.jl
index c417bbc9368c..b696702483b4 100644
--- a/test/Modules/UngradedModules.jl
+++ b/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