The Stacks project

Example 62.5.7 (Trace for quasi-finite over normal). Let $Y$ be a geometrically unibranch and locally Noetherian scheme, for example $Y$ could be a normal variety. Let $f : X \to Y$ be a locally quasi-finite morphism of schemes. Then there exists a positive weighting $w : X \to \mathbf{Z}$ for $f$ which is roughly defined by sending $x$ to the “generic separable degree” of $\mathcal{O}_{X, x}^{sh}$ over $\mathcal{O}_{Y, f(x)}^{sh}$. See More on Morphisms, Lemma 37.73.8. Thus by Lemmas 62.5.2 and 62.5.3 for $f$ and $w$ we obtain trace maps

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

functorial for $K$ in $D(Y_{\acute{e}tale}, \Lambda )$ and compatible with arbitrary base change. However, in this case, given a base change $f' : X' \to Y'$ of $f$ the restriction of $w$ to $X'$ in general does not have a “natural” interpretation in terms of the morphism $f'$.


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