Wrap more cases for syzygies from Singular

[thofma]

From a user on slack, I am just the messenger.

julia> R, (x,y) = polynomial_ring(GF(2), ["x","y"]);

julia> A, _ = quo(R, ideal(R, [x^2+1, y^2+1]));

julia> F = free_module(A,2);

julia> M,_ = sub(F, [F([A(x+1),A(y+1)])]);

julia> FF = free_module(A, ngens(M)); phi = hom(FF, M, gens(M)); K, = kernel(phi); gens(K)
1-element Vector{SubquoModuleElem{MPolyQuoRingElem{FqMPolyRingElem}}}:
 (x*y + x + y + 1)*e[1]

julia> standard_basis(K)
ERROR: wrong rings
Stacktrace:
 [1] error(s::String)
   @ Base ./error.jl:35
 [2] *(M::MonomialOrdering{FqMPolyRing}, N::ModuleOrdering{FreeMod{MPolyQuoRingElem{FqMPolyRingElem}}})
   @ Oscar.Orderings ~/.julia/packages/Oscar/38HwK/src/Rings/orderings.jl:1915
 [3] default_ordering(F::FreeMod{MPolyQuoRingElem{FqMPolyRingElem}})
   @ Oscar ~/.julia/packages/Oscar/38HwK/src/Modules/UngradedModules/Methods.jl:235
 [4] default_ordering(M::SubquoModule{MPolyQuoRingElem{FqMPolyRingElem}})
   @ Oscar ~/.julia/packages/Oscar/38HwK/src/Modules/UngradedModules/SubquoModule.jl:737
 [5] standard_basis(M::SubquoModule{MPolyQuoRingElem{FqMPolyRingElem}})
   @ Oscar ~/.julia/packages/Oscar/38HwK/src/Modules/UngradedModules/SubquoModule.jl:757
 [6] top-level scope
   @ REPL[8]:1

Maybe @wdecker @HechtiDerLachs have an idea or know the person that should have the idea?

Edit: This is with Oscar master version 534513f.

[HechtiDerLachs]

It seems that the user wants to use standard_basis over quotient rings. I am not even sure, this exists. Recently @RafaelDavidMohr carried over some resolutions over quotient rings and I made some first steps to extend the groebner basis functionality for modules over quotient rings in Oscar. However, the latter only became a stub and was not polished to an honest PR.

I think there simply is no groebner- or standard basis functionality for modules over quotient rings at the moment. The computational backend for such modules is implemented via a deflection to the polynomial case; see src/Modules/mpolyquo.jl. This is not very performant, but I just wanted to get some version running back then. And this was used to test the generic interface for the modules.

So you could either say that it is an issue (which it probably is) that we don't have this functionality. Or for the time being the issue is that we do not clearly communicate with the user. For instance, here we should throw a more informative error message explaining that a call to standard_basis is not legitimate here, rather than having something crack down the road.

[iLockya]

This code works on Singular.jl

julia> A,(x,y) = QuotientRing(R, std(Ideal(R, x^2+1,y^2+1)));

julia> F = FreeModule(A,2);

julia> M = Singular.Module(A, F([x+1, y+1]));

julia> K = syz(M);

julia> K
Singular module over Singular polynomial quotient ring (ZZ/2),(x,y),(dp(2),C), with generators:
xy*gen(1)+x*gen(1)+y*gen(1)+gen(1)

julia> std(K)
Singular module over Singular polynomial quotient ring (ZZ/2),(x,y),(dp(2),C), with generators:
xy*gen(1)+x*gen(1)+y*gen(1)+gen(1)

julia> x.parent
Singular polynomial quotient ring (ZZ/2),(x,y),(dp(2),C)

maybe just some simplification of modules?

[HechtiDerLachs]

I'm not too surprised. The functionality is just not integrated in Oscar, yet.

[HechtiDerLachs]

If #4204 is approved and merged, I would remove the "bug"-label here. The issue can probably stay as a request for an enhancement.

[fieker]

On Tue, Oct 15, 2024 at 02:17:55AM -0700, Matthias Zach wrote: I'm not too surprised. The functionality is just not integrated in Oscar, yet.

What is syz(M) supposed to do? Mathematically, I do not think a module has syzygies at all. There should be a command syzygies(Vector of Module Elems) or similar, possibly even liked to nullspace/ kernel of a matrix over polynomials, but never directly to an ideal or module

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[wdecker]

What is syz(M) supposed to do? Mathematically, I do not think a module has syzygies at all. There should be a command syzygies(Vector of Module Elems) or similar, possibly even liked to nullspace/ kernel of a matrix over polynomials, but never directly to an ideal or module

@fieker Of course! But in the example, this is referring to the syz from singular.jl.
In Oscar, so far, we have syzygy_generators which applies to a vector of polynomials. The analogue for modules should be implemented.

[HechtiDerLachs]

The analogue for modules should be implemented.

Done; see #4204.

[jankoboehm]

If #4204 is approved and merged, I would remove the "bug"-label here. The issue can probably stay as a request for an enhancement.

Yes please. We should wrap the cases Singular can do and thing about more generic options. The same might apply for the localizations Singular can do, as well as localizations of quotient rings, depending on how the performance compares to the generic implementations.