Remark

An element $z$ of $T^n(M)$ is called a (contravariant) skew-symmetric tensor of order $n$ if $\sigma.z = \varepsilon_\sigma z$ for every permutation $\sigma \in S_n$ (cf. § 6, no. 3, Remark); these elements form a sub-A-module $A'_n(M)$ of $T^n(M)$. For all $z \in T^n(M)$, we write $a.z = \sum_{\sigma \in S_n} \varepsilon_\sigma(\sigma.z)$ and call $a.z$ the skew-symmetrization of $z$; as $\varepsilon_{\sigma\tau} = \varepsilon_\sigma \varepsilon_\tau$, it is immediately seen that $a.z$ is a skew-symmetric tensor and therefore $z \mapsto a.z$ is an endomorphism of $T^n(M)$ whose image $A''_n(M)$ is contained in $A'_n(M)$; in general $A''_n(M) \neq A'_n(M)$ (Exercise 8). If $z$ is a skew-symmetric tensor, then $a.z = n!z$; hence, when $n!$ is invertible in $A$, the endomorphism $z \mapsto (n!)^{-1}a.z$ is a projector of $T^n(M)$ whose image is

Moreover the kernel of this projector is just $\mathfrak{J}_n''$; for attention may obviously be confined to the case where $n \geqslant 2$, hence 2 (a divisor of $n!$) is invertible in $A$ and $x \otimes x = 2^{-1}(x \otimes x + x \otimes x)$; therefore $\mathfrak{J}_n''$ is generated by the elements $z + \rho.z$, where $\rho$ is a transposition exchanging two consecutive numbers in $\{1, n\}$; moreover, for two permutations $\sigma, \tau$ in $S_n$, we can write

whence it follows that $z - \varepsilon_\sigma(\sigma.z) \in \mathfrak{J}_n''$ for all $z \in T^n(M)$ and $\sigma \in S_n$. Therefore (always assuming $n!$ is invertible in $A$), it is seen that

for all $z \in T^n(M)$, which establishes our assertion.

When $n!$ is invertible in $A$, the submodules $A'_n(M)$ and $\mathfrak{J}_n''$ of $T^n(M)$ are therefore supplementary and the restriction to $A'_n(M)$ of the canonical homomorphism $T^n(M) \to \bigwedge^n(M) = T^n(M)/\mathfrak{J}_n''$ is an A-module isomorphism, which allows us in the case in question to identify skew-symmetric tensors of order $n$ with the elements of the $n$-th exterior power of $M$. Note here also that this identification is not compatible with multiplication, the product in $T(M)$ of two skew-symmetric tensors not being skew-symmetric in general.