Implement flattenings of graded polynomial rings #2906

HechtiDerLachs · 2023-10-09T17:02:29Z

The difference to what we have is that we get graded rings again and they have the correct default ordering.

…ient rings.

lgoettgens · 2023-10-10T23:53:15Z

src/Rings/MPolyMap/AffineAlgebras.jl

+                                                       U1 <: MPolyRingElem{T},
+                                                       U2 <: MPolyRingElem{T}, # types in domain an codomain might differ: one might be decorated, the other not.
+                                                       DT <: Union{MPolyRing{T}, MPolyQuoRing{U1}},
+                                                       CT <: Union{MPolyRing{T}, MPolyQuoRing{U2}}}


Suggested change

U1 <: MPolyRingElem{T},

U2 <: MPolyRingElem{T}, # types in domain an codomain might differ: one might be decorated, the other not.

DT <: Union{MPolyRing{T}, MPolyQuoRing{U1}},

CT <: Union{MPolyRing{T}, MPolyQuoRing{U2}}}

DT <: Union{MPolyRing{T}, MPolyQuoRing{<:MPolyRingElem{T}}},

CT <: Union{MPolyRing{T}, MPolyQuoRing{<:MPolyRingElem{T}}}}

For the case of different Us, there is no need to specify them concretely.

Yes, seems reasonable. I just preferred to stay close to what I found from before, also to not confuse the original authors unnecessarily much in case they want to look and confirm. Either way is fine with me, but personally I wouldn't bother to change this.

HechtiDerLachs

I left some comments and questions about the code and would be happy to discuss these at some point.

HechtiDerLachs · 2023-10-11T10:25:15Z

src/Modules/ModulesGraded.jl

+# TODO: implement further base changes. But for this we need more functionality 
+# for potential change of grading groups. See the open pr #2677.
+function change_base_ring(


This actually became quite relevant for me in this PR: change_base_ring for graded modules and translating the gradings. For special cases where I know the gradings don't change, you can customize this. But in general, I was forced to introduce an internal helper routine below. Maybe this can one day be used in a front-end method as a workhorse for the case of preserved gradings.

HechtiDerLachs · 2023-10-11T10:26:50Z

src/Modules/ModulesGraded.jl

+  return MS, map
+end
+
+function _regularity_bound(M::SubquoModule)


I will need this for the algorithm to compute derived pushforwards which I described in my article here.

HechtiDerLachs · 2023-10-11T10:30:36Z

src/Modules/flattenings.jl

+  return F_flat, iso, iso_inv
+end
+
+function flatten(


I pursued the following approach: There is one user facing method flatten that can be called on any reasonable object. The user does not need to care about which flattening is being used and where it is cached, etc. Internally, a RingFlattening of R is created and stored in the ring R. This one is called by default in the flatten methods and it maintains all associated flattened objects like ideals, modules, etc. The internal user/programmer can or rather should use a specified flattening and not the cached one when coding.

Does that make sense?

In other words: flat_counterparts is some dictionary which encodes the correspondence between non-flattened and flattened objects -- to me the idea makes sense (provided this does not eat up too much space)

HechtiDerLachs · 2023-10-11T10:32:06Z

src/Rings/MPolyMap/flattenings.jl

@@ -35,6 +37,30 @@
                                                    )
  end

+  function RingFlattening(


There's a lot of code duplication here for the graded cases. But I didn't find how to avoid this, really. Maybe the people involved in the development of graded stuff have some ideas?

HechtiDerLachs · 2023-10-11T10:33:53Z

src/Rings/MPolyMap/flattenings.jl

+  if !haskey(flat_counterparts(phi), x)
+    result = phi.orig_to_flat(x)
+    flat_counterparts(phi)[x] = result
+    return result
+  end
+ return flat_counterparts(phi)[x]::elem_type(codomain(phi))


I'm not 100% sure that doing this is really performant. And maybe we should also find a way to turn it off, e.g. using keyword arguments. Apart from that: @fingolfin , do you think this flat_counterpart architecture that I set up here is a safe and correct application of the WeakKeyIdDicts? I will link again to this below.

HechtiDerLachs · 2023-10-11T10:35:50Z

src/Rings/MPolyMap/flattenings.jl

@@ -263,6 +411,30 @@ function map_from_coefficient_ring_to_flattening(phi::RingFlattening)
  return phi.base_ring_to_flat
 end

+### flat counterparts of algebraic objects defined over the domain


Here I explain the philosophy of caching the flat_counterparts. One issue might be that everything with cached flat counterpart must be "mathematically immutable". I think that's more or less safe in Oscar, but one probably has to be careful with things like in-place operations on matrices. Does anyone see problems here?

HechtiDerLachs · 2023-10-11T10:37:06Z

src/Rings/mpoly-localizations.jl

@@ -3185,3 +3185,24 @@ end

 dim(R::MPolyLocRing{<:Field, <:FieldElem, <:MPolyRing, <:MPolyElem, <:MPolyComplementOfPrimeIdeal}) = nvars(base_ring(R)) - dim(prime_ideal(inverted_set(R)))

+########################################################################
+# Localizations of graded rings                                        #
+########################################################################


I was happily surprised that localizing graded rings does not seem to pose any problems so far. The code is sufficiently generic so that it just works. But I am wondering whether there are some more functions which we should provide for such localizations, see below.

HechtiDerLachs · 2023-10-11T10:37:36Z

test/Rings/MPolyAnyMap/AffineAlgebras.jl

@@ -66,3 +66,12 @@
    @test f(x) == x
  end
 end
+
+@testset "cross-type kernels" begin


This is the test corresponding to my fix of the AffAlgHom.

HechtiDerLachs · 2023-10-11T10:38:59Z

test/Rings/MPolyAnyMap/flattenings.jl

+
+  M, _ = quo(F, [u^5*F[1], v*F[1]])
+  M_flat, iso, iso_inv = Oscar.flatten(M)
+  res = free_resolution(M)


@jankoboehm : Could you maybe have a look at this and tell me whether I implemented this custom free resolution in accordance with how you guys designed this? Maybe just run the code of this test in the REPL once, or something?

HechtiDerLachs · 2023-10-18T16:19:03Z

I will add a cached=true/false option to the mapping of elements. Otherwise, there seem to be no real objections to merging this from @fingolfin 's side, if I understand correctly.

afkafkafk13

Overall this looks good. I left you a few minor remarks in the code.

Just a remark on wording:
In commutative algebra 'flat' is of course a property of a module,
here you use it in the sense of 'flattened'. You might want to mention that somewhere in the file.
(By the way: I fully accept this choice for better readability of the code)

afkafkafk13 · 2023-10-21T20:43:50Z

src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Objects/Attributes.jl

 julia> homogeneous_coordinates_on_affine_cone(P)
-3-element Vector{MPolyQuoRingElem{QQMPolyRingElem}}:
+3-element Vector{MPolyQuoRingElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
 x


Not concerning this PR, but more generally: we need to rethink the printing here. The way it is now, it does not have any added value for the user, whereas it is a lot better in line 399.

We cannot change the printing of julia Arrays.

afkafkafk13 · 2023-10-23T13:10:41Z

src/Modules/flattenings.jl

+  return F_flat, iso, iso_inv
+end
+
+function flatten(


In other words: flat_counterparts is some dictionary which encodes the correspondence between non-flattened and flattened objects -- to me the idea makes sense (provided this does not eat up too much space)

afkafkafk13 · 2023-10-23T16:03:10Z

test/Rings/MPolyAnyMap/flattenings.jl

+  R, (x, y) = polynomial_ring(QQ, [:x, :y], cached=false)
+  S, (u, v) = grade(polynomial_ring(R, [:u, :v], cached=false)[1])
+


Does this creation of S also work with graded_polynomial_ring?
If so, it might be appropriate to do this. (This is the case I expect)
If not, @fingolfin would certainly like to hear about the problem.

simonbrandhorst

Looks good to me.

simonbrandhorst · 2023-10-24T11:37:38Z

src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Objects/Attributes.jl

 julia> homogeneous_coordinates_on_affine_cone(P)
-3-element Vector{MPolyQuoRingElem{QQMPolyRingElem}}:
+3-element Vector{MPolyQuoRingElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
 x


We cannot change the printing of julia Arrays.

HechtiDerLachs added 4 commits October 9, 2023 19:01

Implement flattenings of graded polynomial rings over various coeffic…

f99241e

…ient rings.

Implement free graded resolutions via flattenings.

16c8949

Clean up code and use caching in the flattening.

8cac4b5

Fix kernels of ring homomorphisms.

73a820a

HechtiDerLachs force-pushed the graded_flattenings branch from 5e1706f to 73a820a Compare October 10, 2023 13:30

lgoettgens reviewed Oct 10, 2023

View reviewed changes

HechtiDerLachs commented Oct 11, 2023

View reviewed changes

Repair doctests.

1ceef4b

afkafkafk13 reviewed Oct 23, 2023

View reviewed changes

Merge branch 'master' into graded_flattenings

8e8898b

simonbrandhorst approved these changes Oct 25, 2023

View reviewed changes

simonbrandhorst merged commit 4eec6b5 into oscar-system:master Oct 25, 2023
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Implement flattenings of graded polynomial rings #2906

Implement flattenings of graded polynomial rings #2906

HechtiDerLachs commented Oct 9, 2023

lgoettgens Oct 10, 2023

HechtiDerLachs Oct 11, 2023

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HechtiDerLachs commented Oct 18, 2023

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		R, (x, y) = polynomial_ring(QQ, [:x, :y], cached=false)
		S, (u, v) = grade(polynomial_ring(R, [:u, :v], cached=false)[1])

Implement flattenings of graded polynomial rings #2906

Implement flattenings of graded polynomial rings #2906

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HechtiDerLachs commented Oct 9, 2023

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HechtiDerLachs commented Oct 18, 2023

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