diff --git a/experimental/Schemes/LazyGlueing.jl b/experimental/Schemes/LazyGlueing.jl
index 501ac8805c80..a5f0ee2f0327 100644
--- a/experimental/Schemes/LazyGlueing.jl
+++ b/experimental/Schemes/LazyGlueing.jl
@@ -14,6 +14,8 @@
   GD::GlueingDataType
   compute_function::Function
   compute_glueing_domains::Function
+  glueing_domains::Union{Tuple{PrincipalOpenSubset, PrincipalOpenSubset},
+                         Tuple{SpecOpen, SpecOpen}}
   G::AbsGlueing
 
   function LazyGlueing(X::AbsSpec, Y::AbsSpec, 
@@ -40,34 +42,84 @@ function underlying_glueing(G::LazyGlueing)
   return G.G
 end
 
-@attr function glueing_domains(G::LazyGlueing) # TODO: Type annotation
-  if isdefined(G, :compute_glueing_domains)
-    return compute_glueing_domains(G)
+function glueing_domains(G::LazyGlueing) # TODO: Type annotation
+  if !isdefined(G, :glueing_domains)
+    # If there was a function provided to do the extra computation, use it.
+    if isdefined(G, :compute_glueing_domains)
+      G.glueing_domains = G.compute_glueing_domains(G.GD)
+    else
+      # Otherwise default to computation of the whole glueing.
+      G.glueing_domains = glueing_domains(underlying_glueing(G))
+    end
   end
-  # If no extra function was provided, fall back to computing the full glueing.
-  return glueing_domains(underlying_glueing(G))
+  return G.glueing_domains
 end
 
 
 ### Preparations for some sample use cases
-struct RestrictionDataIsomorphism
+mutable struct RestrictionDataIsomorphism
   G::AbsGlueing
   i::AbsSpecMor
   j::AbsSpecMor
+  i_res::SchemeMor
+  j_res::SchemeMor
+  function RestrictionDataIsomorphism(G::AbsGlueing, i::AbsSpecMor, j::AbsSpecMor)
+    return new(G, i, j)
+  end
 end
 
-struct RestrictionDataClosedEmbedding
+mutable struct RestrictionDataClosedEmbedding
   G::AbsGlueing
   X::AbsSpec
   Y::AbsSpec
+  UX::Scheme
+  VY::Scheme
+  function RestrictionDataClosedEmbedding(G::AbsGlueing, X::AbsSpec, Y::AbsSpec)
+    return new(G, X, Y)
+  end
 end
 
 function _compute_restriction(GD::RestrictionDataClosedEmbedding)
-  return restrict(GD.G, GD.X, GD.Y, check=false)
+  _compute_domains(GD) # Fills in the empty fields for the glueing domains
+  return restrict(GD.G, GD.X, GD.Y, GD.UX, GD.VY, check=false)
+end
+
+function _compute_domains(GD::RestrictionDataClosedEmbedding)
+  if isdefined(GD, :UX) && isdefined(GD, :VY)
+    return GD.UX, GD.UY
+  end
+  (U, V) = glueing_domains(GD.G)
+  if U isa PrincipalOpenSubset
+    GD.UX = PrincipalOpenSubset(GD.X, OO(GD.X)(lifted_numerator(complement_equation(U))))
+    GD.VY = PrincipalOpenSubset(GD.Y, OO(GD.Y)(lifted_numerator(complement_equation(V))))
+    return GD.UX, GD.VY
+  elseif U isa SpecOpen
+    GD.UX = intersect(GD.X, U, check=false)
+    GD.VY = intersect(GD.Y, V, check=false)
+    return GD.UX, GD.VY
+  end
 end
 
 function _compute_restriction(GD::RestrictionDataIsomorphism)
-  return restrict(GD.G, GD.i, GD.j, check=false)
+  _compute_domains(GD) # Fills in the empty fields for the glueing domains
+  return restrict(GD.G, GD.i, GD.j, GD.i_res, GD.j_res, check=false)
+end
+
+function _compute_domains(GD::RestrictionDataIsomorphism)
+  if !(isdefined(GD, :i_res) && isdefined(GD, :j_res))
+    U, V = glueing_domains(GD.G)
+
+    GD.i_res = restrict(GD.i, U, preimage(inverse(GD.i), U, check=false), check=false)
+    i_res_inv = restrict(inverse(GD.i), codomain(GD.i_res), U, check=false)
+    set_attribute!(GD.i_res, :inverse, i_res_inv)
+    set_attribute!(i_res_inv, :inverse, GD.i_res)
+
+    GD.j_res = restrict(GD.j, V, preimage(inverse(GD.j), V, check=false), check=false)
+    j_res_inv = restrict(inverse(GD.j), codomain(GD.j_res), V, check=false)
+    set_attribute!(GD.j_res, :inverse, j_res_inv)
+    set_attribute!(j_res_inv, :inverse, GD.j_res)
+  end
+  return codomain(GD.i_res), codomain(GD.j_res)
 end
 
 # Method for compatibility with internal methods of the glueings
diff --git a/src/AlgebraicGeometry/Schemes/Covering/Morphisms/Methods.jl b/src/AlgebraicGeometry/Schemes/Covering/Morphisms/Methods.jl
index 550f74c3eda0..0a7de24e95f3 100644
--- a/src/AlgebraicGeometry/Schemes/Covering/Morphisms/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/Covering/Morphisms/Methods.jl
@@ -33,7 +33,7 @@ function simplify(C::Covering)
     iY, jY = identification_maps(Ysimp)
     G = GD[(X, Y)]
     #new_glueings[(Xsimp, Ysimp)] = restrict(G, jX, jY, check=false)
-    new_glueings[(Xsimp, Ysimp)] = LazyGlueing(Xsimp, Ysimp, _compute_restriction, 
+    new_glueings[(Xsimp, Ysimp)] = LazyGlueing(Xsimp, Ysimp, _compute_restriction, _compute_domains,
                                                RestrictionDataIsomorphism(G, jX, jY)
                                               )
   end
diff --git a/src/AlgebraicGeometry/Schemes/Glueing/Constructors.jl b/src/AlgebraicGeometry/Schemes/Glueing/Constructors.jl
index ffdcf5fe44ce..9662a786fd1a 100644
--- a/src/AlgebraicGeometry/Schemes/Glueing/Constructors.jl
+++ b/src/AlgebraicGeometry/Schemes/Glueing/Constructors.jl
@@ -97,6 +97,14 @@ function restrict(G::AbsGlueing, X::AbsSpec, Y::AbsSpec; check::Bool=true)
   return restrict(underlying_glueing(G), X, Y, check=check)
 end
 
+function restrict(G::AbsGlueing, X::AbsSpec, Y::AbsSpec,
+    Ures::Scheme, Vres::Scheme;
+    check::Bool=true
+  )
+  return restrict(underlying_glueing(G), X, Y, Ures, Vres, check=check)
+end
+
+
 function restrict(G::Glueing, X::AbsSpec, Y::AbsSpec; check::Bool=true)
   U, V = glueing_domains(G)
   f, g = glueing_morphisms(G)
@@ -119,6 +127,30 @@ function restrict(G::SimpleGlueing, X::AbsSpec, Y::AbsSpec; check::Bool=true)
   return SimpleGlueing(X, Y, f_res, g_res, check=check)
 end
 
+function restrict(G::Glueing, X::AbsSpec, Y::AbsSpec,
+    Ures::SpecOpen, Vres::SpecOpen;
+    check::Bool=true
+  )
+  U, V = glueing_domains(G)
+  f, g = glueing_morphisms(G)
+  @check is_closed_embedding(intersect(X, ambient_scheme(U)), ambient_scheme(U)) "the scheme is not a closed in the ambient scheme of the open set"
+  @check is_closed_embedding(intersect(Y, ambient_scheme(V)), ambient_scheme(V)) "the scheme is not a closed in the ambient scheme of the open set"
+  return Glueing(X, Y, restrict(f, Ures, Vres, check=check), restrict(g, Vres, Ures, check=check), check=check)
+end
+
+function restrict(G::SimpleGlueing, X::AbsSpec, Y::AbsSpec,
+    UX::PrincipalOpenSubset, VY::PrincipalOpenSubset;
+    check::Bool=true
+  )
+  U, V = glueing_domains(G)
+  f, g = glueing_morphisms(G)
+  @check is_closed_embedding(intersect(X, ambient_scheme(U)), ambient_scheme(U)) "the scheme is not a closed in the ambient scheme of the open set"
+  @check is_closed_embedding(intersect(Y, ambient_scheme(V)), ambient_scheme(V)) "the scheme is not a closed in the ambient scheme of the open set"
+  f_res = restrict(f, UX, VY, check=check)
+  g_res = restrict(g, VY, UX, check=check)
+  return SimpleGlueing(X, Y, f_res, g_res, check=check)
+end
+
 ########################################################################
 # Identification of glueings under isomorphisms of patches             #
 ########################################################################
@@ -154,6 +186,35 @@ function restrict(G::AbsGlueing, f::AbsSpecMor, g::AbsSpecMor; check::Bool=true)
                 )
 end
 
+function restrict(G::AbsGlueing, f::AbsSpecMor, g::AbsSpecMor,
+    f_res::SchemeMor, g_res::SchemeMor;
+    check::Bool=true
+  )
+  (X1, Y1) = patches(G)
+  X1 === domain(f) || error("maps not compatible")
+  X2 = codomain(f)
+  finv = inverse(f)
+
+  Y1 === domain(g) || error("maps not compatible")
+  Y2 = codomain(g)
+  ginv = inverse(g)
+
+  (h1, h2) = glueing_morphisms(G)
+
+  U2 = codomain(f_res)
+  V2 = codomain(f_res)
+
+  return Glueing(X2, Y2, 
+                 compose(inverse(f_res), 
+                         compose(h1, g_res)
+                        ),
+                 compose(inverse(g_res), 
+                         compose(h2, f_res)
+                        ),
+                 check=check
+                )
+end
+
 ########################################################################
 # Maximal extensions from SimpleGlueings                               #
 ########################################################################
diff --git a/test/AlgebraicGeometry/Schemes/Glueing.jl b/test/AlgebraicGeometry/Schemes/Glueing.jl
index 8ce72dd104f9..7178af84cc79 100644
--- a/test/AlgebraicGeometry/Schemes/Glueing.jl
+++ b/test/AlgebraicGeometry/Schemes/Glueing.jl
@@ -132,3 +132,30 @@ end
   @test underlying_glueing(GG) isa SimpleGlueing
 end
 
+@testset "extra lazy glueing domains" begin
+  P3 = projective_space(QQ, 3)
+  S = homogeneous_coordinate_ring(P3)
+  (x, y, z, w) = gens(S)
+  I = ideal(S, [x*z - y*w, x^4 + y^4 + z^4 + w^4])
+  Y = covered_scheme(P3)
+  II = ideal_sheaf(P3, I)
+  pr = blow_up(II)
+  X = domain(pr)
+
+  for G in values(glueings(oscar.simplified_covering(X)))
+    @test !isdefined(G, :G)
+    glueing_domains(G)
+    @test !isdefined(G, :G)
+    @test isdefined(G, :glueing_domains)
+  end
+
+  for G in values(glueings(oscar.simplified_covering(X)))
+    @test !isdefined(G, :G)
+    underlying_glueing(G)
+    @test isdefined(G, :G)
+    @test isdefined(G, :glueing_domains)
+    @test (G.glueing_domains[1] === glueing_domains(underlying_glueing(G))[1])
+    @test (G.glueing_domains[2] === glueing_domains(underlying_glueing(G))[2])
+  end
+end
+