Banff presentation rebased (oscar-system#3898)

* Make kernel computation more coherent.

* Switch to pushforward in production for prime ideal sheaves.

* Improve heuristics for colength computations.

* Fix heuristic for pullback computations.

* Fix faulty production of ideals.

* Add algorithm keyword and tests.

experimental/Schemes/src/IdealSheaves.jl

@@ -1564,7 +1564,11 @@ function produce_object(F::AbsIdealSheaf, U::AbsAffineScheme)
 end
 
 ### PrimeIdealSheafFromChart
-function produce_object(F::PrimeIdealSheafFromChart, U2::AbsAffineScheme)
+function produce_object(
+    F::PrimeIdealSheafFromChart, U2::AbsAffineScheme; 
+    algorithm::Symbol=:pushforward # Either :pushforward or :pullback
+                                   # This determines how to extend through the gluings.
+  )
   # Initialize some local variables
   X = scheme(F)
   OOX = OO(X)
@@ -1620,14 +1624,29 @@ function produce_object(F::PrimeIdealSheafFromChart, U2::AbsAffineScheme)
     glue = default_covering(X)[W, V2]
     f, g = gluing_morphisms(glue)
     if glue isa SimpleGluing || (glue isa LazyGluing && first(gluing_domains(glue)) isa PrincipalOpenSubset)
-      complement_equation(codomain(g)) in F(W) && return ideal(OO(U2), one(OO(U2))) # We know the ideal is prime. No need to saturate!
-      I2 = F(codomain(g))
-      I = pullback(g)(I2)
-      I = ideal(OO(V2), lifted_numerator.(gens(I)))
-      I = _iterative_saturation(I, lifted_numerator(complement_equation(domain(g))))
-      result = OOX(V2, U2)(ideal(OO(V2), lifted_numerator.(gens(I))))
-      @hassert :IdealSheaves 1 is_one(result) || is_prime(result)
-      return result
+      if algorithm == :pushforward
+        complement_equation(codomain(g)) in F(W) && return ideal(OO(U2), one(OO(U2))) # We know the ideal is prime. No need to saturate!
+        pb_f = pullback(f)::AbsLocalizedRingHom
+        pb_f_res = restricted_map(pb_f)
+        @assert domain(pb_f_res) === ambient_coordinate_ring(V2)
+        Q = preimage(pb_f_res, F(domain(f)))
+        rest = OOX(V2, U2)
+        result = ideal(OO(U2), rest.(gens(Q)))
+        @hassert :IdealSheaves 1 !isone(Q)
+        @hassert :IdealSheaves 1 is_prime(result)
+        set_attribute!(result, :is_prime=>true)
+        return result
+      elseif algorithm == :pullback
+        I2 = F(codomain(g))
+        I = pullback(g)(I2)
+        I = ideal(OO(V2), lifted_numerator.(gens(I)))
+        I = _iterative_saturation(I, lifted_numerator(complement_equation(domain(g))))
+        result = OOX(V2, U2)(ideal(OO(V2), lifted_numerator.(gens(I))))
+        @hassert :IdealSheaves 1 is_one(result) || is_prime(result)
+        return result
+      else
+        error("algorithm not recognized")
+      end
     else
       Z = subscheme(W, F(W))
       pZ = preimage(g, Z, check=false)
@@ -1684,7 +1703,9 @@ function produce_object(I::PullbackIdealSheaf, U::AbsAffineScheme)
   if any(x->x===U, patches(dom))
     f_loc = f_cov[U]
     V = codomain(f_loc)
-    return pullback(f_loc)(original_ideal_sheaf(I)(V))
+    KK = original_ideal_sheaf(I)(V)
+    @assert base_ring(KK) === OO(V)
+    return pullback(f_loc)(KK)
   end
 
   # We are in a chart below a patch in the domain covering
@@ -1785,6 +1806,7 @@ function cheap_sub_ideal(II::PrimeIdealSheafFromChart, U2::AbsAffineScheme)
   if has_ancestor(x->(x===U2), U)
     iso = _flatten_open_subscheme(U, U2)  # an embedding U -> V \subseteq U2
     iso_inv = inverse(iso)
+    P = II(U)
     pb_P = pullback(iso_inv)(P)
     # avoids a saturation by discarding denominators but only produces a subideal
     result = ideal(OO(U2), [g for g in OO(U2).(lifted_numerator.(gens(pb_P))) if !iszero(g)])

experimental/Schemes/src/MorphismFromRationalFunctions.jl

@@ -776,7 +776,7 @@ function _try_randomized_pullback(phi::MorphismFromRationalFunctions, I::AbsIdea
   min_var = minimum([ngens(OO(V)) for V in all_V])
   all_V = [V for V in all_V if ngens(OO(V)) == min_var]
   deg_bound = minimum([maximum([total_degree(lifted_numerator(g)) for g in gens(I(V))]) for V in all_V])
-  all_V = [V for V in all_V if minimum([total_degree(lifted_numerator(g)) for g in gens(I(V))]) == deg_bound]
+  all_V = [V for V in all_V if maximum([total_degree(lifted_numerator(g)) for g in gens(I(V))]) == deg_bound]
   V = first(all_V)
 
   all_U = copy(affine_charts(X))

experimental/Schemes/src/WeilDivisor.jl

@@ -439,16 +439,27 @@ function has_dimension_leq_zero(I::AbsIdealSheaf; covering::Covering=default_cov
   return true
 end
 
-function has_dimension_leq_zero(I::SumIdealSheaf; covering::Covering=default_covering(scheme(I)))
+function has_dimension_leq_zero(I::SumIdealSheaf; 
+    covering::Covering=default_covering(scheme(I)),
+    use_decomposition_info::Bool=true
+  )
   J = summands(I)
-  k = findfirst(x->x isa PrimeIdealSheafFromChart, J)
-  if k !== nothing 
-    P = J[k]
-    U = original_chart(P)
-    if !has_dimension_leq_zero(cheap_sub_ideal(I, U))
-      has_dimension_leq_zero(I(U)) || return false
-    end
-  end
+
+# k = findfirst(x->x isa PrimeIdealSheafFromChart, J)
+# if k !== nothing 
+#   P = J[k]
+#   U = original_chart(P)
+#   if has_decomposition_info(covering) && use_decomposition_info
+#     K = ideal(OO(U), OO(U).(decomposition_info(covering)[U]))
+#     if !has_dimension_leq_zero(cheap_sub_ideal(I, U) + K)
+#       has_dimension_leq_zero(I(U) + K) || return false
+#     end
+#   else
+#     if !has_dimension_leq_zero(cheap_sub_ideal(I, U))
+#       has_dimension_leq_zero(I(U)) || return false
+#     end
+#   end
+# end
 
   common_patches = keys(object_cache(first(J)))
   for JJ in J[2:end]
@@ -461,8 +472,56 @@ function has_dimension_leq_zero(I::SumIdealSheaf; covering::Covering=default_cov
 
   all_patches = patches(covering)
   for U in all_patches
-    if !has_dimension_leq_zero(cheap_sub_ideal(I, U))
-      has_dimension_leq_zero(I(U)) || return false
+    patch_ok = false
+    # go through the object cache of the summands
+    if U in keys(object_cache(I))
+      i = I(U)
+      if has_decomposition_info(covering) && use_decomposition_info
+        K = ideal(OO(U), OO(U).(decomposition_info(covering)[U]))
+        has_dimension_leq_zero(K + i) && (patch_ok = true)
+      else
+        has_dimension_leq_zero(i) && (patch_ok = true)
+      end
+      patch_ok && continue
+    end
+
+    # Do the same for the summands
+    for j in summands(I)
+      if U in keys(object_cache(j))
+        j_loc = j(U)
+        if has_decomposition_info(covering) && use_decomposition_info
+          K = ideal(OO(U), OO(U).(decomposition_info(covering)[U]))
+          has_dimension_leq_zero(K + j_loc) && (patch_ok = true)
+        else
+          has_dimension_leq_zero(j_loc) && (patch_ok = true)
+        end
+        patch_ok && break
+      end
+    end
+    patch_ok && continue
+
+    # repeat with cheap sub-ideals
+    for j in summands(I)
+      j_cheap = cheap_sub_ideal(j, U)
+      if has_decomposition_info(covering) && use_decomposition_info
+        K = ideal(OO(U), OO(U).(decomposition_info(covering)[U]))
+        has_dimension_leq_zero(K + j_cheap) && (patch_ok = true)
+      else
+        has_dimension_leq_zero(j_cheap) && (patch_ok = true)
+      end
+      patch_ok && break
+    end
+    patch_ok && continue
+
+    if has_decomposition_info(covering) && use_decomposition_info
+      K = ideal(OO(U), OO(U).(decomposition_info(covering)[U]))
+      if !has_dimension_leq_zero(cheap_sub_ideal(I, U) + K)
+        has_dimension_leq_zero(I(U) + K) || return false
+      end
+    else
+      if !has_dimension_leq_zero(cheap_sub_ideal(I, U))
+        has_dimension_leq_zero(I(U)) || return false
+      end
     end
   end
   return true
@@ -479,13 +538,53 @@ function _self_intersection(I::AbsIdealSheaf)
   return get_attribute(I, :_self_intersection)::Int
 end
 
+function _is_known_to_be_one(I::AbsIdealSheaf, U::AbsAffineScheme;
+    dec_inf::Vector = elem_type(OO(U))[]
+  )
+  K = ideal(OO(U), dec_inf)
+  if U in keys(object_cache(I))
+    i = I(U)
+    has_attribute(i, :is_one) && get_attribute(i, :is_one) === true && return true
+    one(OO(U)) in gens(i) && return true
+    is_one(K + i) && return true
+  end
+  j = cheap_sub_ideal(I, U)
+  is_one(j + K) && return true
+  return false
+end
+
+function _is_known_to_be_one(I::SumIdealSheaf, U::AbsAffineScheme;
+    dec_inf::Vector = elem_type(OO(U))[]
+  )
+  K = ideal(OO(U), dec_inf)
+  for J in summands(I)
+    if U in keys(object_cache(J))
+      j = J(U)
+      has_attribute(j, :is_one) && get_attribute(j, :is_one) === true && return true
+      one(OO(U)) in gens(j) && return true
+      is_one(K + j) && return true
+    end
+  end
+  if U in keys(object_cache(I))
+    i = I(U)
+    has_attribute(i, :is_one) && get_attribute(i, :is_one) === true && return true
+    one(OO(U)) in gens(i) && return true
+    is_one(K + i) && return true
+  end
+  j = cheap_sub_ideal(I, U)
+  is_one(j + K) && return true
+  return false
+end
+
+
 function colength(I::AbsIdealSheaf; covering::Covering=default_covering(scheme(I)))
   X = scheme(I)
   all_patches = copy(patches(covering))
   patches_done = AbsAffineScheme[]
   patches_todo = AbsAffineScheme[]
   for U in all_patches
-    if U in keys(object_cache(I)) && has_attribute(I(U), :is_one) && is_one(I(U))
+    dec_inf = (has_decomposition_info(covering) ? elem_type(OO(U))[OO(U)(g) for g in decomposition_info(covering)[U]] : elem_type(OO(U))[])
+    if _is_known_to_be_one(I, U; dec_inf)
       push!(patches_done, U)
     else
       push!(patches_todo, U)

src/Rings/mpolyquo-localizations.jl

@@ -1369,15 +1369,18 @@ end
   n = ngens(P)
   imgs_y = t[r+1:(r+n)]
   imgs_x = t[r+n+1:end]
-  I = ideal(A, vcat([one(A) - theta[i]*evaluate(den, imgs_y) for (i, den) in enumerate(denoms)], # Rabinowitsch relations
+  # Sometimes for unnecessarily complicated sets of generators for I the computation 
+  # wouldn't finish. We try to pass to a `small_generating_set` to hopefully reduce the dependency 
+  # on a particular set of generators. 
+  J = ideal(A, vcat([one(A) - theta[i]*evaluate(den, imgs_y) for (i, den) in enumerate(denoms)], # Rabinowitsch relations
                     [theta[i]*evaluate(num, imgs_y) - imgs_x[i] for (i, num) in enumerate(nums)], # Graph relations
-                    [evaluate(g, imgs_y) for g in gens(I)])) # codomain's modulus
+                    [evaluate(g, imgs_y) for g in small_generating_set(I)])) # codomain's modulus
   # We eliminate the Rabinowitsch variables first, the codomain variables second, 
   # and finally get to the domain variables. This elimination should be quicker 
   # than one which does not know the Rabinowitsch property.
   oo = degrevlex(theta)*degrevlex(imgs_y)*degrevlex(imgs_x)
   #oo = lex(theta)*lex(imgs_y)*lex(imgs_x)
-  gb = groebner_basis(I, ordering=oo)
+  gb = groebner_basis(J, ordering=oo)
 
   # TODO: Speed up and use build context.
   res_gens = elem_type(A)[f for f in gb if all(e->all(k->is_zero(e[k]), 1:(n+r)), exponents(f))]

test/AlgebraicGeometry/Schemes/IdealSheaves.jl

@@ -248,3 +248,14 @@ end
   @test 0 in dim.(comp2)
 end
 
+@testset "production of ideals" begin
+  IP2 = projective_space(QQ, [:x, :y, :z])
+  P2 = covered_scheme(IP2)
+  U = first(affine_charts(P2))
+  y, z = gens(OO(U))
+  I = ideal(OO(U), y-z)
+  II = IdealSheaf(P2, U, I)
+  Oscar.produce_object(II, affine_charts(P2)[2]; algorithm=:pullback)
+  Oscar.produce_object(II, affine_charts(P2)[3]; algorithm=:pushforward)
+end
+