isSCM -- checks whether a module or an ideal is sequentially Cohen-Macaulay

Synopsis

Usage:

isSCM M

isSCM I
Inputs:
- M, a module, a finitely generated graded $S$-module, $S=K[x_1,\ldots,x_n]$, with $K$ a field or an ideal $I\subset S$
Outputs:
- b, a Boolean value, whether the module $M$ or the graded algebra $S/I$ is sequentially Cohen-Macaulay

Description

Given a finitely generated graded $S$-module $M$, this method checks if the $i$th module of deficiency of $M$, $\omega^{i}(M)$, if non-zero, is Cohen-Macaulay of dimension $i$. For a homogeneous ideal $I\subset S$, the function checks if $\mathrm{depth} S/{I^{<i>}} \geq i+1$, where $I^{<i>}$ is the $i$th filter ideal.

i1 : S=QQ[x_1..x_5];

i2 : M=coker matrix{{x_1*x_2,x_3*x_4,0,0},{0,x_1*x_5,x_2*x_4,0}};

i3 : isSCM M

o3 = false

i4 : S = QQ[x_1..x_10,y_1..y_10];

i5 : 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}};

i6 : J=ideal(for e in E list x_(e#0)*y_(e#1)-x_(e#1)*y_(e#0));

o6 : Ideal of S

i7 : isSCM J

o7 = true

Ways to use isSCM:

isSCM(Ideal)
isSCM(Module)

For the programmer

The object isSCM is a method function.