Subquotients

A subquotient is a submodule of a quotient of a free module. In this section, the expression subquotient refers to a subquotient over a ring of type MPolyRing, MPolyQuoRing, MPolyLocRing, or MPolyQuoLocRing. That is, given a ring $R$ of one of these types, a subquotient $M$ over $R$ is a module of type

\[M = (\text{im } a + \text{im } b)/\text{im } b,\]

where

\[a:R^s ⟶R^p \;\text{ and }\; b:R^t ⟶R^p\]

are two homomorphisms of free $R$-modules with the same codomain. We then refer to

the module $M$ as the subquotient defined by $a$ and $b$,
the codomain $R^p$ as the ambient free module of $M$,
the images of the canonical basis vectors of $R^s$ in $R^p$ as the ambient representatives of the generators of $M$, and
the images of the canonical basis vectors of $R^t$ in $R^p$ as the relations of $M$.

Alternatively, we speak of the subquotient of $\;\text{im } a\;$ by $\;\text{im } b\;$ or the subquotient defined by $A$ and $B$, where $A$ and $B$ are the matrices representing $a$ and $b$, respectively.

Finally, we refer to

the quotient of $R^p$ by the submodule generated by the relations of $M$ as the ambient module of $M$,

and regard $M$ as a submodule of that ambient module, embedded in the natural way.

Note

Recall from the section on free modules that by a free $R$-module we mean a free module of type $R^p$ , where we think of $R^p$ as a free module with a given basis, namely the basis of standard unit vectors. Accordingly, elements of free modules are represented by coordinate vectors, and homomorphisms between free modules by matrices. Here, by convention, vectors are row vectors, and matrices operate by multiplication on the right.

Note

Over a graded ring $R$, we work with graded free modules $R^s$, $R^p$, $R^t$ and graded homomorphisms $a$, $b$. As a consequence, every module involved in the construction of the subquotient defined by $a$ and $b$ carries an induced grading. In particular, the subquotient itself carries an induced grading.

Types

All OSCAR types for the finitely presented modules considered here belong to the abstract type ModuleFP{T}, where T is the element type of the underlying ring. Graded or not, the subquotients belong to the abstract subtype AbstractSubQuo{T} <: ModuleFP{T}, they are modeled as objects of the concrete type SubquoModule{T} <: AbstractSubQuo{T}.

Note

Canonical maps such us the canonical projection onto a quotient module arise in many constructions in commutative algebra. The SubquoModule type is designed so that it allows for the caching of such maps when executing functions. The tensor_product function discussed in this section provides an example.

Constructors

subquotient — Method

subquotient(a::FreeModuleHom, b::FreeModuleHom)

Given homomorphisms a and b between free modules such that codomain(a) === codomain(b), return $(\text{im } a + \text{im } b)/\text{im } b$.

subquotient(F::FreeMod{T}, A::MatElem{T}, B::MatElem{T}) where T

Given matrices A and B with rank F columns, return $(\text{im } a + \text{im } b)/\text{im } b$, where a and b are free module homomorphisms with codomain F represented by A and B.

subquotient(A::MatElem{T}, B::MatElem{T}) where T

Given matrices A and B with the same number of columns, create a free module F whose rank is that number, and return $(\text{im } a + \text{im } b)/\text{im } b$, where a and b are free module homomorphisms with codomain F represented by A and B.

Examples

julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z]);

julia> FR = free_module(R, 1)
Free module of rank 1 over R

julia> AR = R[x; y]
[x]
[y]

julia> BR = R[x^2; y^3; z^4]
[x^2]
[y^3]
[z^4]

julia> MR = SubquoModule(FR, AR, BR)
Subquotient of submodule with 2 generators
  1: x*e[1]
  2: y*e[1]
by submodule with 3 generators
  1: x^2*e[1]
  2: y^3*e[1]
  3: z^4*e[1]

julia> P = ideal(R, [x, y, z]);

julia> U = complement_of_prime_ideal(P);

julia> RL, _ = localization(R, U);

julia> FRL = free_module(RL, 1)
Free module of rank 1 over localization of R at complement of prime ideal (x, y, z)

julia> ARL = RL[x; y]
[x]
[y]

julia> BRL = RL[x^2; y^3; z^4]
[x^2]
[y^3]
[z^4]

julia> MRL = SubquoModule(FRL, ARL, BRL)
Subquotient of submodule with 2 generators
  1: x*e[1]
  2: y*e[1]
by submodule with 3 generators
  1: x^2*e[1]
  2: y^3*e[1]
  3: z^4*e[1]

julia> RQ, _ = quo(R, ideal(R, [2*x^2-y^3, 2*x^2-y^5]));

julia> FRQ = free_module(RQ, 1)
Free module of rank 1 over RQ

julia> ARQ = RQ[x; y]
[x]
[y]

julia> BRQ = RQ[x^2; y^3; z^4]
[x^2]
[y^3]
[z^4]

julia> MRQ = SubquoModule(FRQ, ARQ, BRQ)
Subquotient of submodule with 2 generators
  1: x*e[1]
  2: y*e[1]
by submodule with 3 generators
  1: x^2*e[1]
  2: 2*x^2*e[1]
  3: z^4*e[1]

julia> RQL, _ = localization(RQ, U);

julia> FRQL = free_module(RQL, 1)
Free module of rank 1 over localization of RQ at complement of prime ideal

julia> ARQL = RQL[x; y]
[x]
[y]

julia> BRQL = RQL[x^2; y^3; z^4]
[x^2]
[y^3]
[z^4]

julia> MRQL = SubquoModule(FRQL, ARQL, BRQL)
Subquotient of submodule with 2 generators
  1: x*e[1]
  2: y*e[1]
by submodule with 3 generators
  1: 0
  2: 0
  3: z^4*e[1]

julia> Rg, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]);

julia> F1 = graded_free_module(Rg, [2,2,2]);

julia> F2 = graded_free_module(Rg, [2]);

julia> G = graded_free_module(Rg, [1,1]);

julia> V1 = [y*G[1], (x+y)*G[1]+y*G[2], z*G[2]]
3-element Vector{FreeModElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
 y*e[1]
 (x + y)*e[1] + y*e[2]
 z*e[2]

julia> V2 = [z*G[2]+y*G[1]]
1-element Vector{FreeModElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
 y*e[1] + z*e[2]

julia> a1 = hom(F1, G, V1)
Homogeneous module homomorphism
  from F1
  to G
defined by
  e[1] -> y*e[1]
  e[2] -> (x + y)*e[1] + y*e[2]
  e[3] -> z*e[2]

julia> a2 = hom(F2, G, V2)
Homogeneous module homomorphism
  from F2
  to G
defined by
  e[1] -> y*e[1] + z*e[2]

julia> V = subquotient(a1,a2)
Graded subquotient of graded submodule of G with 3 generators
  1: y*e[1]
  2: (x + y)*e[1] + y*e[2]
  3: z*e[2]
by graded submodule of G with 1 generator
  1: y*e[1] + z*e[2]

julia> A1 = Rg[x y; 2*x^2 3*y^2]
[    x       y]
[2*x^2   3*y^2]

julia> A2 = Rg[x^3 x^2*y; (2*x^2+x*y)*x (2*y^3+y*x^2)]
[          x^3           x^2*y]
[2*x^3 + x^2*y   x^2*y + 2*y^3]

julia> B = Rg[4*x*y^3 (2*x+y)^4]
[4*x*y^3   16*x^4 + 32*x^3*y + 24*x^2*y^2 + 8*x*y^3 + y^4]

julia> F2 = graded_free_module(Rg,[0,0])
Graded free module Rg^2([0]) of rank 2 over Rg

julia> M1 = SubquoModule(F2, A1, B)
Graded subquotient of graded submodule of F2 with 2 generators
  1: x*e[1] + y*e[2]
  2: 2*x^2*e[1] + 3*y^2*e[2]
by graded submodule of F2 with 1 generator
  1: 4*x*y^3*e[1] + (16*x^4 + 32*x^3*y + 24*x^2*y^2 + 8*x*y^3 + y^4)*e[2]