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filterIdeal -- computes the $i$th filter ideal of $I$.

Synopsis

Description

Let $I\subset S$ be a homogeneous ideal, with $d=\dim S/I$, and let $I=\displaystyle\bigcap_{j=1}^r Q_j$ be the minimal primary decomposition of $I$. For all $1\leq j\leq r$, let $P_j = \sqrt{Q_j}$ be the radical of $Q_j$. For all $-1\leq i\leq d$, the $i$th filter ideal of $I$ is $$I^{<i>} = \bigcap_{\dim S/{P_j}>i} Q_{j},$$ where $I^{<-1>}=I$ and $I^{<d>}=S$.

i1 : S = QQ[x_1..x_10,y_1..y_10];
 
i2 : E = {{1,2},{1,3},{1,4},{1,5},{1,6},{1,7},{1,8},{1,9},{1,10},{6,7},{8,9},{8,10},{9,10}};
 
i3 : J=ideal(for e in E list x_(e#0)*y_(e#1)-x_(e#1)*y_(e#0));

o3 : Ideal of S
 
i4 : filterIdeal(J,5)

o4 = ideal (- x y  + x y , - x y  + x y , - x y  + x y , - x y  + x y , -
               2 1    1 2     3 1    1 3     4 1    1 4     5 1    1 5   
     ------------------------------------------------------------------------
     x y  + x y , - x y  + x y , - x y  + x y , - x y  + x y , - x  y  +
      6 1    1 6     7 1    1 7     8 1    1 8     9 1    1 9     10 1  
     ------------------------------------------------------------------------
     x y  , - x y  + x y , - x y  + x y , - x  y  + x y  , - x  y  + x y  )
      1 10     7 6    6 7     9 8    8 9     10 8    8 10     10 9    9 10

o4 : Ideal of S

See also

Ways to use filterIdeal:

For the programmer

The object filterIdeal is a method function.