# upperCorner -- compute the upper corner

## Synopsis

• Usage:
m=upperCorner(F,d)
• Inputs:
• F, , over the exterior algebra
• d, a list, a (multi)-degree
• Outputs:
• , a submatrix of the differential $F_k -> F_{k-1}$

## Description

Let $k = -|d|$ be the total degree and $G \subset F_k$ the summand spanned by the generators of $F_k$ in degree d, $H \subset F_{k-1}$ the summand spanned by generators of degree d' with $0 \le d'-d \le n$. The function returns the corresponding submatrix $m: G -> H$ of the differential.

So the source will be generated in a single degree, and the target will be generated in multiple degrees. The names comes from the fact that when we resolve this map, this map creates the "upper corner" in the corner complex.

 i1 : n={1,2}; i2 : (S,E) = productOfProjectiveSpaces n o2 = (S, E) o2 : Sequence i3 : F=dual res((ker transpose vars E)**E^{{ 2,3}},LengthLimit=>4) 70 35 15 5 1 o3 = E <-- E <-- E <-- E <-- E -4 -3 -2 -1 0 o3 : ChainComplex i4 : cohomologyMatrix(F,-{3,3},{4,4}) o4 = | 0 0 0 15 0 0 0 0 | | 0 0 0 10 20 0 0 0 | | 0 0 0 6 12 18 0 0 | | 0 0 0 3 6 9 12 0 | | 0 0 0 1 2 3 4 5 | | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | 8 8 o4 : Matrix (ZZ[h, k]) <--- (ZZ[h, k]) i5 : betti F -4 -3 -2 -1 0 o5 = total: 70 35 15 5 1 0: 70 35 15 5 1 o5 : BettiTally i6 : tallyDegrees F o6 = (Tally{{-1, -3} => 20}, Tally{{-1, -2} => 12}, Tally{{-1, -1} => 6}, {-2, -2} => 18 {-2, -1} => 9 {-2, 0} => 3 {-3, -1} => 12 {-3, 0} => 4 {0, -2} => 6 {-4, 0} => 5 {0, -3} => 10 {0, -4} => 15 ------------------------------------------------------------------------ Tally{{-1, 0} => 2}, Tally{{0, 0} => 1}) {0, -1} => 3 o6 : Sequence i7 : deg={2,1} o7 = {2, 1} o7 : List i8 : m=upperCorner(F,deg); 30 9 o8 : Matrix E <--- E i9 : tally degrees target m, tally degrees source m o9 = (Tally{{-2, -2} => 18}, Tally{{-2, -1} => 9}) {-3, -1} => 12 o9 : Sequence i10 : Fm=(res(coker m,LengthLimit=>4))[sum deg+1] 30 9 2 3 8 o10 = E <-- E <-- E <-- E <-- E -4 -3 -2 -1 0 o10 : ChainComplex i11 : betti Fm -4 -3 -2 -1 0 o11 = total: 30 9 2 3 8 0: 30 9 . . . 1: . . 2 1 . 2: . . . . 1 3: . . . 2 7 o11 : BettiTally i12 : cohomologyMatrix(Fm,-{3,3},{4,4}) o12 = | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 18 0 0 | | 0 0 0 0 0 9 12 0 | | 0 h2 0 h 2h 0 0 0 | | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | | 0 0 0 h3 2h3 0 0 0 | 8 8 o12 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])

## Ways to use upperCorner :

• "upperCorner(ChainComplex,List)"

## For the programmer

The object upperCorner is .