Localizations of modules over computable rings

For localizations of modules, there exists a generic implementation of the common methods such as membership tests, kernel computations, etc. based on the work of Barakat, Posur, et. al; see [Pos18].

Let $R$ be a ring of type <:Ring, $U \subset R$ a multiplicative set of type <:AbsMultSet and $S = R[U^{-1}]$ the localization of $R$ at $U$. Recall that $R$ is computable if one can compute syzygies and lifts over $R$. The results from [Pos18], Theorem 3.9, assert that then also the localization $S$ is computable, provided that there exists a solution to the localization problem (Definition 3.8, [Pos18] and below).

The user who wishes to use the generic code for localizations therefore has to make sure the following two requirements are met:

  1. The code for finitely generated modules and ideals must be functional over $R$, including the computation of coordinates and kernel.

  2. The user has to solve the localization problem by implementing has_nonempty_intersection(U::MultSetType, I::IdealType) for the type MultSetType of multiplicative sets and the type IdealType of ideals in R that they would like to consider.

has_nonempty_intersection(U::AbsMultSet, I::Ideal)

For a finitely generated ideal $I ⊂ R$ and a multiplicative set $U ⊂ R$, this checks whether the intersection $U ∩ I$ is nonempty and returns a triple

(success, f, a).

In the affirmative case, success is true, $f ∈ U ∩ I$ is some element in the intersection and $a ∈ R¹ˣᵏ$ is a Vector{elem_type(R)} such that $f = ∑ᵢ aᵢ⋅gᵢ$ where $gᵢ$ are the elements in gens(I).

When the intersection is empty, this returns (false, f, a) with meaningless values for $f$ and $a$.

Note: When implementing methods of this function, it is recommended to choose $f$ to be the 'least complex' in an appropriate sense for $R$.


Note: In order to clear denominators of row vectors, the generic code uses the method lcm(v::Vector{T}) where T = elem_type(R). If no such method already exists, this has to also be provided; in the worst case by simply returning the product of the denominators.

As soon as the above requirements are met, the methods

   represents_element(u::FreeModElem{T}, M::SubquoModule{T}) where {T<:AbsLocalizedRingElem}
   coordinates(u::FreeModElem{T}, M::SubquoModule{T}) where {T<:AbsLocalizedRingElem}
   kernel(f::FreeModuleHom{DomType, CodType, Nothing}) where {T, DomType<:FreeMod{T}, CodType<:SubquoModule{T}}
   kernel(f::SubQuoHom{DomType, CodType, Nothing}) where {T, DomType<:FreeMod{T}, CodType<:SubquoModule{T}}
   iszero(a::SubquoModuleElem{T}) where {T<:AbsLocalizedRingElem}

will be available for modules over $S$, i.e. for T = elem_type(S). As can easily be seen, having the first three of these methods is already equivalent to $S = R[U^{-1}]$ being computable; hence all higher methods can be derived from these basic ones.

The generic code makes use of a simple caching mechanism for the SubquoModules as follows. For a module $M = (G + N)/N$ with submodules $G, N \subset R^n$ of some free module, the localization $M[U^{-1}]$ over $S = R[U^{-1}]$ has an associated saturated module over $R$:

\[ M' = (G' + N')/N', \quad G' = \{ a \in R^n | \exists u \in U : u \cdot a \in G + N\},\quad N' = \{ b \in R^n | \exists u \in U : u \cdot b \in N\}.\]

While it might be difficult to compute such saturations, we have a generic algorithm to check membership for elements in $M'$ (via represents_element for $M[U^{-1}]$). It is assumed that such membership tests are cheaper for modules over $R$ compared to modules over $S$. For instance in the case where $R$ is a multivariate polynomial ring, once a (relative) groebner basis has been computed for $M$, membership test for $M$ is merely a reduction while for the localization $M[U^{-1}]$ it triggers another groebner basis computation a priori.

But for every element $a \in R^n$ that has already been shown to represent an element in the saturation $M'$, we can cache the results of the computation in an intermediate pre-saturated module $M \subset \tilde M \subset M'$ by adding the necessary generators to $G$ and $N$ for a representation of $a$. Then, checking membership for $a$ a second time will fall back to a membership test in $\tilde M$. For the latter, we assume some caching to already be implemented as, for instance, for the use of groebner bases in the polynomial case.

A sample implementation for various localizations of multivariate polynomial rings can be found in src/Modules/mpoly-localizations.jl. A modified version for localizations of affine algebras which also overwrites some of the generic methods, is in src/Modules/mpolyquo-localizations.jl.