The Stacks project

Example 62.5.5 (Trace for flat quasi-finite). Let $f : X \to Y$ be a morphism of schemes which is flat, locally quasi-finite, and locally of finite presentation. Then we obtain a canonical positive weighting $w : X \to \mathbf{Z}$ by setting

\[ w(x) = \text{length}_{\mathcal{O}_{X, x}} (\mathcal{O}_{X, x}/\mathfrak m_{f(x)} \mathcal{O}_{X, x}) [\kappa (x) : \kappa (f(x))]_ i \]

See More on Morphisms, Lemma 37.73.7. Thus by Lemmas 62.5.2 and 62.5.3 for $f$ we obtain trace maps

\[ \text{Tr}_{f, K} : f_!f^{-1}K \longrightarrow K \]

functorial for $K$ in $D(Y_{\acute{e}tale}, \Lambda )$ and compatible with arbitrary base change. Note that any base change $f' : X' \to Y'$ of $f$ satisfies the same properties and that $w$ restricts to the canonical weighting for $f'$.


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