I am working on old qualifying problems involving tensor products. I am stuck on a statement about invertible elements in an exterior algebra and was wondering if this was a well known fact in a book somewhere. I think most of this notation is standard from dummite and foote but the notes I have been using are slightly different than what I have seen in textbooks so far.
Let $V$ be a finite dimensional vector space over a field $F$.
Let $T(V) = \oplus_{k=0}^{\infty} T^{k}(V)$ where $T^k(V) = V \otimes V \otimes \ldots \otimes V$ is tensor product of $k$ modules.
Let $\wedge V $ denote the exterior algebra of the $F$-module $V$, that is the quotient of the tensor algebra $T(V)$ by the ideal $A(M)$ generated by all $v \otimes v$ for $v \in V$.
Let $x \in \wedge V$. So that $x = \sum_{k\geq 0} x_k$ where each $x \in \wedge^k V = T^k(V)/A^k(V)$
How do you prove that $x$ is invertible if and only if $x_0 \neq 0$