# lowerCorner -- compute the lower corner

## Synopsis

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

## Description

Let $k = |deg|$ 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 $m: H -> G$ of the differential.

 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 : betti F -4 -3 -2 -1 0 o4 = total: 70 35 15 5 1 0: 70 35 15 5 1 o4 : BettiTally i5 : tallyDegrees F o5 = (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 o5 : Sequence i6 : deg={2,1} o6 = {2, 1} o6 : List i7 : m=lowerCorner(F,deg); 9 9 o7 : Matrix E <--- E i8 : tally degrees target m, tally degrees source m o8 = (Tally{{-2, -1} => 9}, Tally{{-1, -1} => 6}) {-2, 0} => 3 o8 : Sequence i9 : Fm=(res(coker m,LengthLimit=>7))[sum deg] 9 9 8 15 32 57 91 137 o9 = E <-- E <-- E <-- E <-- E <-- E <-- E <-- E -3 -2 -1 0 1 2 3 4 o9 : ChainComplex i10 : betti Fm -3 -2 -1 0 1 2 3 4 o10 = total: 9 9 8 15 32 57 91 137 0: 9 9 5 1 . . . . 1: . . . 3 7 11 15 19 2: . . 3 11 25 45 71 103 3: . . . . . 1 5 15 o10 : BettiTally i11 : cohomologyMatrix(Fm,-{3,3},{4,4}) 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 | | 6h 3h 0 3 6 9 0 0 | | 2h h 0 1 2 3 0 0 | | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | | 2h3 h3 0 h2 2h2 3h2 0 0 | 8 8 o11 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])

## Ways to use lowerCorner :

• "lowerCorner(ChainComplex,List)"

## For the programmer

The object lowerCorner is .