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

Synopsis

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

See also

Ways to use hasSlidingDepth :

For the programmer

The object hasSlidingDepth is a method function.