Variety of submodules of a finitely presented module
$begingroup$
Assume that $A$ is a graded $k$-algebra with $A_d=k$ in every degree $d$. Let $M$ be an $A$-module with finite free presentation $F_1to F_0to Mto 0$. I want to undertand the collection of finitely presented submodules $Nsubseteq M$ of prescribed dimension.
So let $G_1to G_0to Nto 0$ be the presentation of $N$. An inclusion $Nsubseteq M$ is determined by a commutative square
$$begin{array}{} G_1 & xrightarrow{g}& G_0\llap{scriptstyle i_1}downarrow &&downarrowrlap{scriptstyle i_0}\F_1&xrightarrow[f]{}&F_0end{array}$$
with both vertical maps injections. Since $A$ is one-dimensional in every degree, these maps can be represented by $k$-valued matrices (where the grading might determine certain entries to be zero).
So I end up with a quadratic equation $i_0 g=fi_1$ in the entries of these matrices. The solution set $Z(i_0 g-fi_1)$ is an algebraic variety.
What would be a good strategy to find the solutions of this equation system, provided that I know it only has finitely many solutions (up to a $GL_text{something}$-action)?
linear-algebra abstract-algebra algorithms modules numerical-linear-algebra
asked Jan 18 at 16:57
BubayaBubaya
452212
$endgroup$
add a comment |
$begingroup$
Assume that $A$ is a graded $k$-algebra with $A_d=k$ in every degree $d$. Let $M$ be an $A$-module with finite free presentation $F_1to F_0to Mto 0$. I want to undertand the collection of finitely presented submodules $Nsubseteq M$ of prescribed dimension.
So let $G_1to G_0to Nto 0$ be the presentation of $N$. An inclusion $Nsubseteq M$ is determined by a commutative square
$$begin{array}{} G_1 & xrightarrow{g}& G_0\llap{scriptstyle i_1}downarrow &&downarrowrlap{scriptstyle i_0}\F_1&xrightarrow[f]{}&F_0end{array}$$
with both vertical maps injections. Since $A$ is one-dimensional in every degree, these maps can be represented by $k$-valued matrices (where the grading might determine certain entries to be zero).
So I end up with a quadratic equation $i_0 g=fi_1$ in the entries of these matrices. The solution set $Z(i_0 g-fi_1)$ is an algebraic variety.
What would be a good strategy to find the solutions of this equation system, provided that I know it only has finitely many solutions (up to a $GL_text{something}$-action)?
linear-algebra abstract-algebra algorithms modules numerical-linear-algebra
asked Jan 18 at 16:57
BubayaBubaya
452212
$endgroup$
add a comment |
$begingroup$
Assume that $A$ is a graded $k$-algebra with $A_d=k$ in every degree $d$. Let $M$ be an $A$-module with finite free presentation $F_1to F_0to Mto 0$. I want to undertand the collection of finitely presented submodules $Nsubseteq M$ of prescribed dimension.
So let $G_1to G_0to Nto 0$ be the presentation of $N$. An inclusion $Nsubseteq M$ is determined by a commutative square
$$begin{array}{} G_1 & xrightarrow{g}& G_0\llap{scriptstyle i_1}downarrow &&downarrowrlap{scriptstyle i_0}\F_1&xrightarrow[f]{}&F_0end{array}$$
with both vertical maps injections. Since $A$ is one-dimensional in every degree, these maps can be represented by $k$-valued matrices (where the grading might determine certain entries to be zero).
So I end up with a quadratic equation $i_0 g=fi_1$ in the entries of these matrices. The solution set $Z(i_0 g-fi_1)$ is an algebraic variety.
What would be a good strategy to find the solutions of this equation system, provided that I know it only has finitely many solutions (up to a $GL_text{something}$-action)?
linear-algebra abstract-algebra algorithms modules numerical-linear-algebra
asked Jan 18 at 16:57
BubayaBubaya
452212
$endgroup$
Assume that $A$ is a graded $k$-algebra with $A_d=k$ in every degree $d$. Let $M$ be an $A$-module with finite free presentation $F_1to F_0to Mto 0$. I want to undertand the collection of finitely presented submodules $Nsubseteq M$ of prescribed dimension.
So let $G_1to G_0to Nto 0$ be the presentation of $N$. An inclusion $Nsubseteq M$ is determined by a commutative square
$$begin{array}{} G_1 & xrightarrow{g}& G_0\llap{scriptstyle i_1}downarrow &&downarrowrlap{scriptstyle i_0}\F_1&xrightarrow[f]{}&F_0end{array}$$
with both vertical maps injections. Since $A$ is one-dimensional in every degree, these maps can be represented by $k$-valued matrices (where the grading might determine certain entries to be zero).
So I end up with a quadratic equation $i_0 g=fi_1$ in the entries of these matrices. The solution set $Z(i_0 g-fi_1)$ is an algebraic variety.
What would be a good strategy to find the solutions of this equation system, provided that I know it only has finitely many solutions (up to a $GL_text{something}$-action)?
linear-algebra abstract-algebra algorithms modules numerical-linear-algebra
linear-algebra abstract-algebra algorithms modules numerical-linear-algebra
asked Jan 18 at 16:57
BubayaBubaya
452212
asked Jan 18 at 16:57
BubayaBubaya
452212
asked Jan 18 at 16:57
BubayaBubaya
452212
asked Jan 18 at 16:57
BubayaBubaya
452212
asked Jan 18 at 16:57
BubayaBubaya
452212
452212
add a comment |
add a comment |
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