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# tateResolution -- compute the Tate resolution

## Synopsis

• Usage:
T = tateResolution(M,low,high)
phi = tateResolution(A,low,high)
• Inputs:
• M, , multi-graded module representing a sheaf F
• low, a list, a multidegree
• high, a list, a multidegree
• A, , a homomorphism of multi-graded modules from M to N
• Outputs:
• T, , a bounded free complex over the exterior algebra
• phi, , an induced map from T(M) to T(N) over the exterior algebra

## Description

The call

tateResolution(M,low,high)

forms a free subquotient complex the Tate resolution of the sheaf F represented by M in a range that covers all generators corresponding to cohomology groups of twists F(a) of F in the range low <= a <= high, see Tate Resolutions on Products of Projective Spaces.

 i1 : (S,E) = productOfProjectiveSpaces{1,1} o1 = (S, E) o1 : Sequence i2 : low = {-3,-3};high = {3,3}; i4 : T=tateResolution( S^{{1,1}},low, high); i5 : cohomologyMatrix(T,low,high) o5 = | 5h 0 5 10 15 20 25 | | 4h 0 4 8 12 16 20 | | 3h 0 3 6 9 12 15 | | 2h 0 2 4 6 8 10 | | h 0 1 2 3 4 5 | | 0 0 0 0 0 0 0 | | h2 0 h 2h 3h 4h 5h | 7 7 o5 : Matrix (ZZ[h, k]) <-- (ZZ[h, k])

The complex contains some trailing terms and superfluous terms in a wider range, which can be removed using trivial homological truncation.

 i6 : cohomologyMatrix(T,2*low,2*high) o6 = | 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 42k 0 0 | | 0 0 0 0 0 0 0 0 0 0 36k 42k 0 | | 20h 15h 10h 5h 0 5 10 15 20 25 0 0 0 | | 16h 12h 8h 4h 0 4 8 12 16 20 0 0 0 | | 12h 9h 6h 3h 0 3 6 9 12 15 0 0 0 | | 8h 6h 4h 2h 0 2 4 6 8 10 0 0 0 | | 4h 3h 2h h 0 1 2 3 4 5 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 3h2 2h2 h2 0 h 2h 3h 4h 5h 0 0 0 | | 0 0 4h2 2h2 0 2h 4h 6h 8h 10h 0 0 0 | | 0 0 0 3h2 0 3h 6h 9h 12h 15h 0 0 0 | | 0 0 0 0 0 4h 8h 12h 16h 20h 0 0 0 | 13 13 o6 : Matrix (ZZ[h, k]) <-- (ZZ[h, k]) i7 : betti T -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 o7 = total: 84 36 25 40 46 44 35 30 38 56 81 110 141 174 210 -1: 84 36 . . . . . . . . . . . . . 0: . . 25 40 46 44 35 20 10 4 1 . . . . 1: . . . . . . . 10 28 52 80 110 140 170 200 2: . . . . . . . . . . . . 1 4 10 o7 : BettiTally i8 : T'=trivialHomologicalTruncation(T, -sum high,-sum low) 25 40 46 44 35 30 38 56 81 110 141 174 210 o8 = 0 <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- 0 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 o8 : ChainComplex i9 : betti T' -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 o9 = total: 25 40 46 44 35 30 38 56 81 110 141 174 210 0: 25 40 46 44 35 20 10 4 1 . . . . 1: . . . . . 10 28 52 80 110 140 170 200 2: . . . . . . . . . . 1 4 10 o9 : BettiTally i10 : cohomologyMatrix(T',2*low,2*high) o10 = | 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 0 0 0 0 0 0 0 0 0 | | 20h 15h 10h 5h 0 5 10 15 20 25 0 0 0 | | 16h 12h 8h 4h 0 4 8 12 16 20 0 0 0 | | 12h 9h 6h 3h 0 3 6 9 12 15 0 0 0 | | 8h 6h 4h 2h 0 2 4 6 8 10 0 0 0 | | 4h 3h 2h h 0 1 2 3 4 5 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 3h2 2h2 h2 0 h 2h 3h 4h 5h 0 0 0 | | 0 0 4h2 2h2 0 2h 4h 6h 8h 10h 0 0 0 | | 0 0 0 3h2 0 3h 6h 9h 12h 15h 0 0 0 | | 0 0 0 0 0 4h 8h 12h 16h 20h 0 0 0 | 13 13 o10 : Matrix (ZZ[h, k]) <-- (ZZ[h, k])

The call

tateResolution(A,low,high)

where A is a matrix representing the multi-graded module homomorphism from M to N computes the induced map between two free subquotients of Tate resolutions of M and N in the given range.

 i11 : (S,E)=productOfProjectiveSpaces {2,1} o11 = (S, E) o11 : Sequence i12 : low=-{2,1}; high={2,1}; i14 : A=map(S^1, S^{1:{-1,0}}, {{S_0}}) o14 = | x_(0,0) | 1 1 o14 : Matrix S <-- S i15 : M=source A; N=target A; i17 : TA = tateResolution(A, low, high); i18 : TM = tateResolution(M, low, high); i19 : TN = tateResolution(N, low, high); i20 : (source TA == TM, target TA == TN) o20 = (true, true) o20 : Sequence