Gröbner basis for modules

Describe the bug While computing some Gröbner bases for modules I encountered some differences between documentation and behaviour in Oscar. It seems like the Gröbner basis is not computed with respect to the requested module ordering. In the example below it looks like during the computation of groebner_basis and leading_module the ordering invlex(M)*deglex(R) is replaced by lex(M)*deglex(R) and therefore does not return a correct result.

To Reproduce

julia> R,_ = polynomial_ring(QQ, 1, :t);

julia> M = free_module(R, 3);

julia> ord = invlex(M)*deglex(R);

julia> g1 = 2M[1] + M[3];

julia> leading_term(g1, ordering=ord)
e[3]

julia> g2 = M[1] + M[2] + M[3];

julia> leading_term(g2, ordering=ord)
e[3]

julia> U,_ = sub(M, [g1, g2])
(Submodule with 2 generators
  1: 2*e[1] + e[3]
  2: e[1] + e[2] + e[3]
represented as subquotient with no relations, Hom: U -> M)

julia> G = groebner_basis(U, ordering=ord)
Gröbner basis with elements
e[3] + 2*e[2]
e[3] + e[2] + e[1]
 with respect to the ordering
invlex([gen(1), gen(2), gen(3)])*deglex([t1])

julia> LG,_ = sub(M, [leading_term(G[i], ordering=ord) for i=1:length(G)])
(Submodule with 2 generators
  1: e[3]
  2: e[3]
represented as subquotient with no relations, Hom: LG -> M)

julia> LU = leading_module(U, ord)
Submodule with 2 generators
  1: 2*e[2]
  2: e[1]
represented as subquotient with no relations

julia> LG == LU
false

Expected behavior The expected result can currently be computed by the following, mathmatically wrong code by replacing invlex(M)*deglex(R) with lex(M)*deglex(R).

julia> G1 = groebner_basis(U, ordering=lex(M)*deglex(R))
Gröbner basis with elements
-e[1] + e[2]
e[1] + e[2] + e[3]
 with respect to the ordering
lex([gen(1), gen(2), gen(3)])*deglex([t1])

julia> LG1,_ = sub(M, [leading_term(G1[i], ordering=ord) for i=1:length(G)])
(Submodule with 2 generators
  1: e[2]
  2: e[3]
represented as subquotient with no relations, Hom: LG1 -> M)

julia> LU1 = leading_module(U, lex(M)*deglex(R))
Submodule with 2 generators
  1: e[2]
  2: e[3]
represented as subquotient with no relations

julia> LG1==LU1
true

Version Information

julia> Oscar.versioninfo(full=true)
OSCAR version 1.3.0-DEV - #master, b3ab222d48 -- 2024-11-12 10:38:02 +0100
  combining:
    AbstractAlgebra.jl   v0.43.10
    GAP.jl               v0.12.0
    Hecke.jl             v0.34.6
    Nemo.jl              v0.47.3
    Polymake.jl          v0.11.22
    Singular.jl          v0.23.10
  building on:
    FLINT_jll               v300.100.300+0
    GAP_jll                 v400.1300.102+2
    Singular_jll            v404.0.606+0
    libpolymake_julia_jll   v0.13.0+0
    libsingular_julia_jll   v0.46.0+0
    polymake_jll            v400.1300.2+0
See `]st -m` for a full list of dependencies.

Julia Version 1.7.3
Commit 742b9abb4d (2022-05-06 12:58 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: Intel(R) Core(TM) i5-8265U CPU @ 1.60GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-12.0.1 (ORCJIT, skylake)
Official https://julialang.org/ release

Additional context The same problem also occurs with the orderings deglex(R)*lex(M) and deglex(R)*invlex(M).

oscar-system / Oscar.jl

Gröbner basis for modules #4303