# bgg -- make a linear free complex from a module over an exterior algebra or a symmetric algebra

## Synopsis

• Usage:
LP = bgg P
RM = bgg(M,LengthLimit=>4)
• Inputs:
• P, , module over an exterior algebra E
• M, , module over an symmetric algebra S
• Optional inputs:
• LengthLimit => ..., default value null
• Outputs:
• LP, , over a symmetric algebra
• RM, , over a exterior algebra

## Description

If P is an E-module, then LP becomes a linear complex of free S-modules, where (S,E) is the Koszul pair corresponding to a product of projective spaces. Similarly, if M is an S-module, them RM becomes a linear free complex over the exterior algebra E of length bounded by the LengthLimit.

The complex LP is that produced from P by the Bernstein-Gel'fand-Gel'fand functor called L in our paper Tate Resolutions on Products of Projective Spaces. Similarly, the complex RM produced from M is a bounded piece of the infinite complex of the Bernstein-Gel'fand-Gel'fand functor called R in loc.cit. L and R form a pair of adjoint functors.

 i1 : (S,E) = productOfProjectiveSpaces{1,2} o1 = (S, E) o1 : Sequence i2 : P = prune truncate({1,2},E^1)**E^{{1,2}}; i3 : LP = bgg P 1 5 6 o3 = S <-- S <-- S -2 -1 0 o3 : ChainComplex i4 : netList apply(toList(min LP..max LP), i-> decompose ann HH_i LP) +------------------------------------+------------------------+ o4 = |ideal (x , x , x , x , x )| | | 1,2 1,1 1,0 0,1 0,0 | | +------------------------------------+------------------------+ |ideal (x , x ) |ideal (x , x , x )| | 0,1 0,0 | 1,2 1,1 1,0 | +------------------------------------+------------------------+ |ideal () | | +------------------------------------+------------------------+ i5 : M = prune HH_0 LP o5 = cokernel {1, 1} | -x_(1,2) | {1, 1} | x_(1,0) | {1, 1} | -x_(1,1) | 3 o5 : S-module, quotient of S i6 : betti res M 0 1 o6 = total: 3 1 2: 3 1 o6 : BettiTally i7 : high = {3,3} o7 = {3, 3} o7 : List i8 : cohomologyMatrix(M, -high, high) o8 = | 45h 30h 15h 0 15 30 45 | | 24h 16h 8h 0 8 16 24 | | 9h 6h 3h 0 3 6 9 | | 0 0 0 0 0 0 0 | | 3h2 2h2 h2 0 h 2h 3h | | 0 0 0 0 0 0 0 | | 9h3 6h3 3h3 0 3h2 6h2 9h2 | 7 7 o8 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
 i9 : M=module ideal vars S o9 = image | x_(0,0) x_(0,1) x_(1,0) x_(1,1) x_(1,2) | 1 o9 : S-module, submodule of S i10 : RM = bgg(M,LengthLimit=>3) 70 35 15 5 o10 = E <-- E <-- E <-- E -4 -3 -2 -1 o10 : ChainComplex i11 : betti RM -4 -3 -2 -1 o11 = total: 70 35 15 5 0: 70 35 15 5 o11 : BettiTally i12 : tallyDegrees RM o12 = (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}) {0, -1} => 3 o12 : Sequence