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# BGGL -- Exterior module to linear complex

- Usage:
    `L = BGGL(P,S)`
- Inputs:
  - P, a module, module over an exterior algebra
  - S, a ring, polynomial ring on the same number of vars
- Outputs:
  - L, a complex, linear chain complex over S corresponding to P

## Description

Implements the left adjoint BGG functor. The more primitive function SymExt does this only for linearly presented exterior modules.

    i1 : E = ZZ/101[a,b,c,d, SkewCommutative => true]

    o1 = E

    o1 : PolynomialRing, 4 skew commutative variable(s)

    i2 : P = E^1/ideal(a*b,c)

    o2 = cokernel | ab c |

                                1
    o2 : E-module, quotient of E

    i3 : betti freeResolution(P, LengthLimit => 5)

                0 1 2 3  4  5
    o3 = total: 1 2 4 7 11 16
             0: 1 1 1 1  1  1
             1: . 1 3 6 10 15

    o3 : BettiTally

    i4 : hf(0..3, P)

    o4 = {1, 3, 2, 0}

    o4 : List

    i5 : S = ZZ/101[x,y,z,w]

    o5 = S

    o5 : PolynomialRing

    i6 : betti BGGL(P,S)

                -2 -1 0
    o6 = total:  2  3 1
             0:  2  3 1

    o6 : BettiTally

## Caveat

This script really belongs in the BGG package.

## For the programmer

The object BGGL is a function closure.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/CompleteIntersectionResolutions.m2:4575:0.