The Stacks project

Lemma 12.17.5. Let $F$ be a field. Let $\mathcal{C}$ be the category of graded $F$-vector spaces viewed as a monoidal category as in Example 12.17.4. If $V$ in $\mathcal{C}$ has a left dual $W$, then $\sum _ n \dim _ F V^ n < \infty $ and the map $\epsilon $ defines nondegenerate pairings $W^{-n} \times V^ n \to F$.

Proof. As unit we take By Categories, Definition 4.43.5 we have maps

\[ \eta : \mathbf{1} \to V \otimes W\quad \epsilon : W \otimes V \to \mathbf{1} \]

Since $\mathbf{1} = F$ placed in degree $0$, we may think of $\epsilon $ as a sequence of pairings $W^{-n} \times V^ n \to F$ as in the statement of the lemma. Choose bases $\{ e_{n, i}\} _{i \in I_ n}$ for $V^ n$ for all $n$. Write

\[ \eta (1) = \sum e_{n, i} \otimes w_{-n, i} \]

for some elements $w_{-n, i} \in W^{-n}$ almost all of which are zero! The condition that $(\epsilon \otimes 1) \circ (1 \otimes \eta )$ is the identity on $W$ means that

\[ \sum \nolimits _{n, i} \epsilon (w, e_{n, i})w_{-n, i} = w \]

Thus we see that $W$ is generated as a graded vector space by the finitely many nonzero vectors $w_{-n, i}$. The condition that $(1 \otimes \epsilon ) \circ (\eta \otimes 1)$ is the identity of $V$ means that

\[ \sum \nolimits _{n, i} e_{n, i}\ \epsilon (w_{-n, i}, v) = v \]

In particular, setting $v = e_{n, i}$ we conclude that $\epsilon (w_{-n, i}, e_{n, i'}) = \delta _{ii'}$. Thus we find that the statement of the lemma holds and that $\{ w_{-n, i}\} _{i \in I_ n}$ is the dual basis for $W^{-n}$ to the chosen basis for $V^ n$. $\square$

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