Determines whether the ideal I has sliding depth for k steps
Let K be the Koszul complex on a minimal set of generators of I. We say $I$ has k-sliding depth if for all $i\leq k$ we have $depth(H_{n-codim(I)-i}(K) \geq dim I - i$. Note that if I is perfect then $H_{n-codim(I)}(K)$ is the canonical module, which is Cohen-Macaulay so that I has 0-sliding depth.
i1 : R = QQ[x_1..x_6]; |
i2 : I = minors(2, genericSymmetricMatrix(R,x_1,3))
2 2
o2 = ideal (- x + x x , - x x + x x , - x x + x x , - x x + x x , - x +
2 1 4 2 3 1 5 3 4 2 5 2 3 1 5 3
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2
x x , - x x + x x , - x x + x x , - x x + x x , - x + x x )
1 6 3 5 2 6 3 4 2 5 3 5 2 6 5 4 6
o2 : Ideal of R
|
i3 : c = codim I o3 = 3 |
i4 : m = numgens I o4 = 9 |
i5 : apply (m+1, i-> koszulDepth(i,I))
o5 = {3, 1, 3, 3, 6, 6, 6, 6, 6, 6}
o5 : List
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i6 : hasSlidingDepth(m-c,I) o6 = true |
i7 : I = ideal{x_1*x_2,x_1*x_3,x_2*x_4*x_5,x_1*x_6,x_4*x_6,x_5*x_6};
o7 : Ideal of R
|
i8 : hasSlidingDepth(1,I) o8 = true |
i9 : hasSlidingDepth(2,I) o9 = false |
The object hasSlidingDepth is a method function.