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tateResolution -- finite piece of the Tate resolution

Description

This function takes as input a presentation matrix m of a finitely generated graded S-module M an exterior algebra E and two integers l and h. If r is the regularity of M, then this function computes the piece of the Tate resolution from cohomological degree l to cohomological degree max(r+2,h). For instance, for the homogeneous coordinate ring of a point in the projective plane: 
i1 : S = ZZ/32003[x_0..x_2]; 
i2 : E = ZZ/32003[e_0..e_2, SkewCommutative=>true];
 
i3 : m = matrix{{x_0,x_1}};

             1      2
o3 : Matrix S  <-- S
 
i4 : regularity coker m

o4 = 0
 
i5 : T = tateResolution(m,E,-2,4)

      1      1      1      1      1      1      1
o5 = E  <-- E  <-- E  <-- E  <-- E  <-- E  <-- E
                                                
     0      1      2      3      4      5      6

o5 : Complex
 
i6 : betti T

            0 1 2 3 4 5 6
o6 = total: 1 1 1 1 1 1 1
        -4: 1 1 1 1 1 1 1

o6 : BettiTally
 
i7 : T.dd_1

o7 = {-4} | e_2 |

             1      1
o7 : Matrix E  <-- E

See also

Ways to use tateResolution:

  • tateResolution(Matrix,PolynomialRing,ZZ,ZZ)

For the programmer

The object tateResolution is a method function.

The source of this document is in /__w/M2/M2/macaulay2-1.26.05+ds/M2/Macaulay2/packages/BGG.m2:753:0.