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schurResolution(...,SyzygyLimit=>...) -- Specifies the number of syzygy modules to be computed

Description

This is an optional argument for the schurResolution routine. It specifies an upper bound for the number of syzygy modules in the equivariant resolution of an equivariant module M to be computed by the routine. If a SyzygyLimit is not specified, then all syzygy modules are computed.

The example below computes the 0-th to 3-rd syzygy modules of the 5-th Veronese embedding of P^2.

i1 : S = schurRing(s,3);
 
i2 : rep = s_{5};
 
i3 : M = {1_S,s_{5},s_{10},s_{15},s_{20},s_{25},s_{30}};
 
i4 : schurResolution(rep,M,SyzygyLimit => 3)

o4 = {{(0, s  )}, {(2, s    + s   )}, {(3, s       + s     + s       + s    
            ()          8,2    6,4          12,2,1    11,4    11,3,1    10,5
     ------------------------------------------------------------------------
     + s       + s       + s    + 2s      + s      + s    + s      + s      +
        10,4,1    10,3,2    9,6     9,5,1    9,4,2    8,7    8,6,1    8,5,2  
     ------------------------------------------------------------------------
     s      + s      + s      + s     )}, {(4, s       + s       + 2s       +
      8,4,3    7,6,2    7,5,3    6,5,4          15,4,1    15,3,2     14,5,1  
     ------------------------------------------------------------------------
     s       + s       + s     + 3s       + 3s       + 2s       + s     +
      14,4,2    14,3,3    13,7     13,6,1     13,5,2     13,4,3    12,8  
     ------------------------------------------------------------------------
     3s       + 4s       + 4s       + s       + s     + 3s       + 5s       +
       12,7,1     12,6,2     12,5,3    12,4,4    11,9     11,8,1     11,7,2  
     ------------------------------------------------------------------------
     5s       + 3s       + 2s       + 3s       + 5s       + 4s       +
       11,6,3     11,5,4     10,9,1     10,8,2     10,7,3     10,6,4  
     ------------------------------------------------------------------------
     3s       + 2s      + 3s      + 4s      + 3s      + s      + 2s      +
       10,5,5     9,9,2     9,8,3     9,7,4     9,6,5    8,8,4     8,7,5  
     ------------------------------------------------------------------------
     s     )}}
      7,7,6

o4 : List

Lowering SyzygyLimit simply chops the output at the requested homological position. For the quadratic Veronese of P^2, asking for only the first syzygy module yields:

i5 : T = schurRing(QQ,t,3);
 
i6 : rep2 = t_{2};
 
i7 : M2 = {1_T,t_{2},t_{4},t_{6},t_{8},t_{10},t_{12}};
 
i8 : schurResolution(rep2,M2,SyzygyLimit => 1)

o8 = {{(0, t  )}, {(2, t   )}}
            ()          2,2

o8 : List
 
i9 : schurResolution(rep2,M2,SyzygyLimit => 2)

o9 = {{(0, t  )}, {(2, t   )}, {(3, t     )}}
            ()          2,2          3,2,1

o9 : List

See also

Functions with optional argument named SyzygyLimit:

Further information

  • Default value: 0
  • Function: schurResolution -- Compute an ``approximate'' equivariant resolution of a module.
  • Option key: SyzygyLimit -- an optional argument

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:6656:0.