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TateOnProducts :: cohomologyTable

cohomologyTable -- compute the the cohomology groups of a (part) of a Tate resolution or sheaf on products of projective spaces

Synopsis

Description

Under the assumption that T is part of a Tate resolution of a sheaf F on a product of two projective space Pn1 x Pn2, the function returns a matrix of cohomology polynomials

i=0|n|   dim Hi(ℙn1×ℙn2,F(c1,c2)) * hi ∈  ℤ[h,k]

for every c=(c1,c2) with a1 ≤c1 ≤b1 and a2 ≤c2 ≤b2. In case T corresponds to an object in the derived category Db(Pn1x Pn2), then hypercohomology polynomials are returned, with the convention that k stands for k=h -1.

If T is not a large enough part of the Tate resolution, such as W below, then the function collects only the contribution of T to the cohomology table of the Tate resolution, according to the formula in Corollary 0.2 of Tate Resolutions on Products of Projective Spaces.

The polynomial for (b1,b2) sits in the north-east corner, the one corresponding to (a1,a2) in the south-west corner.

i1 : n={1,2};kk=ZZ/101;
i3 : (S,E)=setupRings(ZZ/101,n);
i4 : a={1,1}; U=E^{ -a};
i6 : W=(chainComplex {map(E^0,U,0),map(U,E^0,0)})[1]

             1
o6 = 0  <-- E  <-- 0
                    
     -1     0      1

o6 : ChainComplex
i7 : cohomologyTable(W,-{3,3},{3,3})

o7 = | 0 0 0  0 0 0 0 |
     | 0 0 0  0 0 0 0 |
     | 0 0 0  0 0 0 0 |
     | 0 0 0  0 0 0 0 |
     | 0 0 h2 0 0 0 0 |
     | 0 0 0  0 0 0 0 |
     | 0 0 0  0 0 0 0 |

                      7                7
o7 : Matrix (ZZ[h, k])  <--- (ZZ[h, k])
i8 : time T=sloppyTateExtension W
     -- used 4.53428 seconds

      4331      3515      2839      2282      1825      1451      1145      894      687      515      371      251      156      92      68      90      125      132
o8 = E     <-- E     <-- E     <-- E     <-- E     <-- E     <-- E     <-- E    <-- E    <-- E    <-- E    <-- E    <-- E    <-- E   <-- E   <-- E   <-- E    <-- E
                                                                                                                                                                   
     -11       -10       -9        -8        -7        -6        -5        -4       -3       -2       -1       0        1        2       3       4       5        6

o8 : ChainComplex
i9 : cohomologyTable(T,-{3,3},{3,3})

o9 = | 45h 30h 15h 0 15  30  45  |
     | 24h 16h 8h  0 8   16  24  |
     | 9h  6h  3h  0 3   6   9   |
     | 0   0   0   0 0   0   0   |
     | 3h2 2h2 h2  0 h   2h  3h  |
     | 0   0   0   0 0   0   0   |
     | 9h3 6h3 3h3 0 3h2 6h2 9h2 |

                      7                7
o9 : Matrix (ZZ[h, k])  <--- (ZZ[h, k])
i10 : cohomologyTable(T,-{3,4},{3,3})

o10 = | 45h  30h  15h 0 15  30   45   |
      | 24h  16h  8h  0 8   16   24   |
      | 9h   6h   3h  0 3   6    9    |
      | 0    0    0   0 0   0    0    |
      | 3h2  2h2  h2  0 h   2h   3h   |
      | 0    0    0   0 0   0    0    |
      | 9h3  6h3  3h3 0 3h2 6h2  9h2  |
      | 24h3 16h3 8h3 0 8h2 16h2 24h2 |

                       8                7
o10 : Matrix (ZZ[h, k])  <--- (ZZ[h, k])

Ways to use cohomologyTable :