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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