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Macaulay2Doc > modules > exterior power of a module

exterior power of a module

The k-th exterior power of a module M is the k-fold tensor product of M together with the equivalence relation:
        m_1 ** m_2 ** .. ** m_k = 0     if m_i = m_j for i != j
        
If M is a free R-module of rank n, then the k-th exterior power of M is a free R-module of rank binomial(n,k). Macaulay2 computes the k-th exterior power of a module M with the command exteriorPower.
i1 : R = ZZ/2[x,y]

o1 = R

o1 : PolynomialRing
i2 : exteriorPower(3,R^6)

      20
o2 = R

o2 : R-module, free
i3 : binomial(6,3)

o3 = 20
Macaulay2 can compute exterior powers of modules that are not free as well.
i4 : exteriorPower(2,R^1)

o4 = 0

o4 : R-module
i5 : I = module ideal (x,y)

o5 = image | x y |

                             1
o5 : R-module, submodule of R
i6 : exteriorPower(2,I)

o6 = cokernel {2} | x y |

                            1
o6 : R-module, quotient of R

See also