# hasSlidingDepth -- Checks if an ideal has the sliding depth property

## Synopsis

• Usage:
b = hasSlidingDepth(k,I)
• Inputs:
• Outputs:
• b, , true if I has sliding depth

## Description

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 ------------------------------------------------------------------------ 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 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