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VirtualResolutions :: virtualOfPair

virtualOfPair -- creates a virtual resolution from a free resolution by keeping only summands of specified degrees

Synopsis

Description

Given an ideal, a module, or a free resolution, this function keeps only the summands in the minimal graded free resolution generated in degrees in L. If the list L contains only one element which is in the multigraded regularity of M plus the dimension vector, the output will be the virtual resolution of a pair as defined in Section 1 of [BES]. See Algorithm 3.4 of [BES,arXiv:1703.07631] for further details.

For example, consider the ideal of three points in 1×ℙ1.

i1 : X = toricProjectiveSpace(1)**toricProjectiveSpace(1);
i2 : S = ring X; B = ideal X;

o3 : Ideal of S
i4 : J = saturate(intersect(
           ideal(x_1 - 1*x_0, x_3 - 4*x_2),
           ideal(x_1 - 2*x_0, x_3 - 5*x_2),
           ideal(x_1 - 3*x_0, x_3 - 6*x_2)),
           B)

                                     3      2          2    3        2  
o4 = ideal (3x x  + x x  - x x , 120x  - 74x x  + 15x x  - x , 120x x  -
              0 2    1 2    0 3      2      2 3      2 3    3      1 2  
     ------------------------------------------------------------------------
                    2       2     2       2                  2      3  
     34x x x  - 2x x  + 3x x , 40x x  + 6x x  - 13x x x  - 3x x , 6x  -
        1 2 3     0 3     1 3     1 2     0 3      0 1 3     1 3    0  
     ------------------------------------------------------------------------
        2         2    3
     11x x  + 6x x  - x )
        0 1     0 1    1

o4 : Ideal of S

We can now compute its minimal free resolution and a virtual resolution. One can show that (2,0) is in the multigraded regularity of this example. Thus, since we want to compute a virtual resolution we apply virtualOfPair to the element (3,1) since (3,1)=(2,0)+(1,1) and (1,1) is the dimension vector for 1×ℙ1.

i5 : minres = res J;
i6 : vres = virtualOfPair(J,{{3,1}}) --(3,1) = (2,0) + (1,1)

      1      3      2
o6 = S  <-- S  <-- S  <-- 0
                           
     0      1      2      3

o6 : ChainComplex

Finally, we check that the result is indeed virtual.

i7 : isVirtual(J,B,vres)

o7 = true

Caveat

Given an element of the multigraded regularity, one must add the dimension vector of the product of projective spaces for this to return a virtual resolution.

Ways to use virtualOfPair :