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Pseudo-matrices

This chapter deals with pseudo-matrices. We follow the common terminology and conventions introduced in [Coh00], however, we operate on rows, not on columns.

Let $R$ be a Dedekind domain, typically, the maximal order of some number field $K$, further fix some finite dimensional $K$-vectorspace $V$ (with some basis), frequently $K^n$ or the $K$-structure of some extension of $K$. Since in general $R$ is not a PID, the $R$-modules in $V$ are usually not free, but still projective.

Any finitely generated $R$-module $M\subset V$ can be represented as a pseudo-matrix PMat as follows: The structure theory of $R$-modules gives the existence of (fractional) $R$-ideals $\mathfrak A_i$ and elements $\omega_i\in V$ such that $M = \sum \mathfrak A_i \omega_i$ and the sum is direct.

Following Cohen we call modules of the form $\mathfrak A\omega$ for some ideal $\mathfrak A$ and $\omega \in V$ a pseudo element. A system $(\mathfrak A_i, \omega_i)$ is called a pseudo-generating system for $M$ if $\langle \mathfrak A_i\omega_i|i\langle = M$. A pseudo-generating system is called a pseudo-basis if the $\omega_i$ are $K$-linear independent.

A pseudo-matrix $X$ is a tuple containing a vector of ideals $\mathfrak A_i$ ($1\le i\le r$) and a matrix $U\in K^{r\times n}$. The $i$-th row together with the $i$-th ideal defines a pseudo-element, thus an $R$-module, all of them together generate a module $M$.

A pseudo-matrix $X=((\mathfrak A_i)_i, U)$ is said to be in pseudo-hnf if $U$ is essentially upper triangular. Similar to the classical hnf, there is an algorithm that transforms any pseudo-matrix into one in pseudo-hnf while maintaining the module.

Creation

In general to create a PMat one has to specify a matrix and a vector of ideals:

pseudo_matrix — Method
pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldElem}, c::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}

Returns the (row) pseudo matrix representing the $Z_k$-module $\sum c_i m_i$ where $c_i$ are the ideals in $c$ and $m_i$ the rows of $M$.

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pseudo_matrix — Method
pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldOrderElem}, c::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}

Returns the (row) pseudo matrix representing the $Z_k$-module $\sum c_i m_i$ where $c_i$ are the ideals in $c$ and $m_i$ the rows of $M$.

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pseudo_matrix — Method
pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldOrderElem}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}

Returns the free (row) pseudo matrix representing the $Z_k$-module $\sum Z_k m_i$ where $m_i$ are the rows of $M$.

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(Those functions are also available as pseudo_matrix)

Operations

matrix — Method
matrix(M::PMat)

Returns the matrix part of the PMat.

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base_ring — Method
base_ring(M::PMat)

The PMat $M$ defines an $R$-module for some maximal order $R$. This function returns the $R$ that was used to defined $M$.

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base_ring(I::MPolyIdeal)

Return the ambient ring of I.

Examples

julia> R, (x, y) = polynomial_ring(QQ, [:x, :y])
(Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])

julia> I = ideal(R, [x, y])^2
Ideal generated by
  x^2
  x*y
  y^2

julia> base_ring(I)
Multivariate polynomial ring in 2 variables x, y
  over rational field
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base_ring(X::AbsAffineScheme)

On an affine scheme $X/𝕜$ over $𝕜$ this returns the ring $𝕜$.

Examples

julia> X = affine_space(QQ,3)
Affine space of dimension 3
  over rational field
with coordinates [x1, x2, x3]

julia> base_ring(X)
Rational field
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pseudo_hnf — Method
pseudo_hnf(P::PMat)

Transforms $P$ into pseudo-Hermite form as defined by Cohen. Essentially the matrix part of $P$ will be upper triangular with some technical normalisation for the off-diagonal elements. This operation preserves the module.

A optional second argument can be specified as a symbols, indicating the desired shape of the echelon form. Possible are :upperright (the default) and :lowerleft

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pseudo_hnf_with_transform — Method
pseudo_hnf_with_transform(P::PMat)

Transforms $P$ into pseudo-Hermite form as defined by Cohen. Essentially the matrix part of $P$ will be upper triangular with some technical normalisation for the off-diagonal elements. This operation preserves the module. The used transformation is returned as a second return value.

A optional second argument can be specified as a symbol, indicating the desired shape of the echelon form. Possible are :upperright (the default) and :lowerleft

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Examples