The
k-th exterior power of a module
M is the
k-fold tensor product of
M together with the equivalence relation:$m_1 \otimes m_2 \otimes \cdots \otimes m_k = 0$ 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
|