# strand -- take the strand

## Synopsis

• Usage:
strand(T,c,I)
• Inputs:
• T, , over the exterior algebra
• c, a list, a (multi) degree
• I, a list, a sublist of \{0..t-1\} , where t denotes the number of factors
• Outputs:
• , the I-strand of T through c

## Description

We compute the strand of T as defined in Tate Resolutions on Products of Projective Spaces Theorem 0.4. If T is (part of) the Tate resolution of a sheaf $F$, then the I-strand of $T$ through $c$ corresponds to the Tate resolution $R{\pi_J}_*(F(c))$ where $J =\{0,\ldots,t-1\} - I$ is the complement and $\pi_J: \mathbb PP \to \prod_{j \in J} \mathbb P^{n_j}$ denotes the projection.

 i1 : n={1,1}; i2 : (S,E) = productOfProjectiveSpaces n; i3 : T1 = (dual res trim (ideal vars E)^2)[1]; i4 : a=-{2,2};T2=T1**E^{a}[sum a]; i6 : W=beilinsonWindow T2,cohomologyMatrix(W,-2*n,2*n) 15 16 4 o6 = (E <-- E <-- E , | 0 0 0 0 0 |) | 0 0 0 0 0 | 0 1 2 | 0 8 15 0 0 | | 0 4 8 0 0 | | 0 0 0 0 0 | o6 : Sequence i7 : T=tateExtension W; i8 : low = -{2,2};high = {2,2}; i10 : cohomologyMatrix(T,low,high) o10 = | 3 16 29 42 55 | | 2 12 22 32 42 | | 1 8 15 22 29 | | 0 4 8 12 16 | | h 0 1 2 3 | 5 5 o10 : Matrix (ZZ[h, k]) <--- (ZZ[h, k]) i11 : sT1=strand(T,{-1,0},{1}); i12 : cohomologyMatrix(sT1,low,high) o12 = | 0 0 0 0 0 | | 0 0 0 0 0 | | 1 8 15 22 29 | | 0 0 0 0 0 | | 0 0 0 0 0 | 5 5 o12 : Matrix (ZZ[h, k]) <--- (ZZ[h, k]) i13 : sT2=strand(T,{-1,0},{0}); i14 : cohomologyMatrix(sT2,low,high) o14 = | 0 16 0 0 0 | | 0 12 0 0 0 | | 0 8 0 0 0 | | 0 4 0 0 0 | | 0 0 0 0 0 | 5 5 o14 : Matrix (ZZ[h, k]) <--- (ZZ[h, k]) i15 : sT3=strand(T,{-1,0},{0,1}); i16 : cohomologyMatrix(sT3, low,high) o16 = | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 8 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | 5 5 o16 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])