Remark 45.14.6 (Uniqueness of trace maps). Assume given data (D0), (D1), and (D2') satisfying axioms (A1) – (A7). Let $X$ be a smooth projective scheme over $k$ which is nonempty and equidimensional of dimension $d$. Combining what was said in the proofs of Lemma 45.14.5 and Homology, Lemma 12.17.5 we see that

defines a perfect duality between $H^ i(X)$ and $H^{2d - i}(X)(d)$ for all $i$. In particular, the linear map $\int _ X = \lambda : H^{2d}(X)(d) \to F$ of axiom (A6) is unique! We will call the linear map $\int _ X$ the trace map of $X$ from now on.

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