unmixedLayer -- computes the $i$th unmixed layer of $I$.

Usage:

unmixedLayer(I,i)
Inputs:
- I, an ideal, a homogeneous ideal of the polynomial ring $S=K[x_1,\ldots,x_n]$, with $K$ a field
- i, an integer, an integer greater or equal than -1
Outputs:
- U, a module, the $i$th unmixed layer of $I$

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 $I^{<d>}=S$. The $i$th unmixed layer of $I$ is defined as $U_i(I)=I^{<i>}/I^{<i>}$.

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 : unmixedLayer(J,7)

o4 = subquotient (| -x_2y_1+x_1y_2 -x_3y_1+x_1y_3 -x_4y_1+x_1y_4 -x_5y_1+x_1y_5 -x_6y_1+x_1y_6 -x_7y_1+x_1y_7 -x_8y_1+x_1y_8 -x_9y_1+x_1y_9 -x_10y_1+x_1y_10 -x_7y_6+x_6y_7 -x_9y_8+x_8y_9 -x_10y_8+x_8y_10 -x_10y_9+x_9y_10 |, | -x_2y_1+x_1y_2 -x_3y_1+x_1y_3 -x_4y_1+x_1y_4 -x_5y_1+x_1y_5 -x_6y_1+x_1y_6 -x_7y_1+x_1y_7 -x_8y_1+x_1y_8 -x_9y_1+x_1y_9 -x_10y_1+x_1y_10 -x_7y_6+x_6y_7 -x_9y_8+x_8y_9 -x_10y_8+x_8y_10 -x_10y_9+x_9y_10 |)

                               1
o4 : S-module, subquotient of S

Ways to use unmixedLayer:

unmixedLayer(Ideal,ZZ)

For the programmer

The object unmixedLayer is a method function.

The source of this document is in SCMAlgebras.m2:481:0.

unmixedLayer -- computes the $i$th unmixed layer of $I$.

Description

See also

Ways to use unmixedLayer:

For the programmer