The Stacks project

Lemma 62.5.3. Let $f : X \to Y$ be a locally quasi-finite morphism of schemes. Let $w : X \to \mathbf{Z}$ be a weighting of $f$. The trace maps constructed above have the following properties:

  1. $\text{Tr}_{f, w, \mathcal{F}}$ is functorial in $\mathcal{F}$,

  2. $\text{Tr}_{f, w, \mathcal{F}}$ is compatible with arbitrary base change,

  3. given a ring $\Lambda $ and $K$ in $D(Y_{\acute{e}tale}, \Lambda )$ we obtain $\text{Tr}_{f, w, K} : f_!f^{-1}K \to K$ functorial in $K$ and compatible with arbitrary base change.

Proof. Part (1) either follows from the construction of the trace map in the proof of Lemma 62.5.2 or more simply because the characterization of the map forces it to be true on all stalks. Let

\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } \]

be a cartesian diagram of schemes. Then the function $w' = w \circ g' : X' \to \mathbf{Z}$ is a weighting of $f'$ by More on Morphisms, Lemma 37.73.3. Statement (2) means that the diagram

\[ \xymatrix{ g^{-1}f_!f^{-1}\mathcal{F} \ar[rr]_-{g^{-1}\text{Tr}_{f, w, \mathcal{F}}} \ar@{=}[d] & & g^{-1}\mathcal{F} \ar@{=}[d] \\ f'_!(f')^{-1}g^{-1}\mathcal{F} \ar[rr]^-{\text{Tr}_{f', w', g^{-1}\mathcal{F}}} & & g^{-1}\mathcal{F} } \]

is commutative where the left vertical equality is given by

\[ g^{-1}f_!f^{-1}\mathcal{F} = f'_!(g')^{-1}f^{-1}\mathcal{F} = f'_!(f')^{-1}g^{-1}\mathcal{F} \]

with first equality sign given by Lemma 62.4.10 (base change for lower shriek). The commutativity of this diagram follows from the characterization of the action of our trace maps on stalks and the fact that the base change map of Lemma 62.4.10 respects the descriptions of stalks.

Given parts (1) and (2), part (3) follows as the functors $f^{-1} : D(Y_{\acute{e}tale}, \Lambda ) \to D(X_{\acute{e}tale}, \Lambda )$ and $f_! : D(X_{\acute{e}tale}, \Lambda ) \to D(Y_{\acute{e}tale}, \Lambda )$ are obtained by applying $f^{-1}$ and $f_!$ to any complexes of modules representing the objects in question. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GKH. Beware of the difference between the letter 'O' and the digit '0'.