Example 63.5.5 (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 37.75.7. Thus by Lemmas 63.5.2 and 63.5.3 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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