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# regionComplex -- region complex

## Synopsis

• Usage:
regionComplex(T,c,IJK)
• Inputs:
• T, , over the exterior algebra
• c, a list, a (multi) degree
• IJK, , a sequence (I,J,K) of disjoint subsets of \{0..t-1\}
• Outputs:
• , a region complex of T

## Description

We compute the region complex of T as defined in Tate Resolutions on Products of Projective Spaces section 3. Note that different from the paper I,J,K are sublists of 0...t-1 and not subsets of 1..t. In the examples below, only rT2 and rT3 are proper region complexes.

 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 : cohomologyMatrix(T,-{3,3},{3,3}) o8 = | 12h 4 20 36 52 68 84 | | 10h 3 16 29 42 55 68 | | 8h 2 12 22 32 42 52 | | 6h 1 8 15 22 29 36 | | 4h 0 4 8 12 16 20 | | 2h h 0 1 2 3 4 | | 0 2h 4h 6h 8h 10h 12h | 7 7 o8 : Matrix (ZZ[h, k]) <-- (ZZ[h, k]) i9 : c={1,0} o9 = {1, 0} o9 : List i10 : rT0=regionComplex(T,c,({},{0,1},{})); --a single position i11 : cohomologyMatrix(rT0,-{3,3},{3,3}) o11 = | 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 22 0 0 | | 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 | 7 7 o11 : Matrix (ZZ[h, k]) <-- (ZZ[h, k]) i12 : rT1=regionComplex(T,c,({0},{1},{})); --a horizontal half line i13 : cohomologyMatrix(rT1,-{3,3},{3,3}) o13 = | 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 | | 6h 1 8 15 0 0 0 | | 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 | 7 7 o13 : Matrix (ZZ[h, k]) <-- (ZZ[h, k]) i14 : rT2=regionComplex(T,c,({},{0},{})); -- a vertical line i15 : cohomologyMatrix(rT2,-{3,3},{3,3}) o15 = | 0 0 0 0 52 0 0 | | 0 0 0 0 42 0 0 | | 0 0 0 0 32 0 0 | | 0 0 0 0 22 0 0 | | 0 0 0 0 12 0 0 | | 0 0 0 0 2 0 0 | | 0 0 0 0 8h 0 0 | 7 7 o15 : Matrix (ZZ[h, k]) <-- (ZZ[h, k]) i16 : rT3=regionComplex(T,c,({},{},{1})); -- a upper half plane i17 : cohomologyMatrix(rT3,-{3,3},{3,3}) o17 = | 12h 4 20 36 52 68 84 | | 10h 3 16 29 42 55 68 | | 8h 2 12 22 32 42 52 | | 6h 1 8 15 22 29 36 | | 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 | 7 7 o17 : Matrix (ZZ[h, k]) <-- (ZZ[h, k]) i18 : rT4=regionComplex(T,c,({0},{},{1})); --a north east quadrant i19 : cohomologyMatrix(rT4,-{3,3},{3,3}) o19 = | 12h 4 20 36 0 0 0 | | 10h 3 16 29 0 0 0 | | 8h 2 12 22 0 0 0 | | 6h 1 8 15 0 0 0 | | 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 | 7 7 o19 : Matrix (ZZ[h, k]) <-- (ZZ[h, k]) i20 : rT5=regionComplex(T,c,({1},{},{0})); --a south west quadrant i21 : cohomologyMatrix(rT5,-{3,3},{3,3}) o21 = | 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 12 16 20 | | 0 0 0 0 2 3 4 | | 0 0 0 0 8h 10h 12h | 7 7 o21 : Matrix (ZZ[h, k]) <-- (ZZ[h, k])