The Stacks project

Lemma 62.5.2. Let $f : X \to Y$ be a locally quasi-finite morphism of schemes. Let $w : X \to \mathbf{Z}$ be a weighting of $f$. For any abelian sheaf $\mathcal{F}$ on $Y$ there exists a unique trace map $\text{Tr}_{f, w, \mathcal{F}} : f_!f^{-1}\mathcal{F} \to \mathcal{F}$ having the prescribed behaviour on stalks.

Proof. By Lemma 62.5.1 we have an identification $f_!f^{-1}\mathcal{F} = f_!\underline{\mathbf{Z}} \otimes \mathcal{F}$ compatible with the description of stalks of these sheaves at geometric points. Hence it suffices to produce the map

\[ \text{Tr}_{f, w, \underline{\mathbf{Z}}} : f_!\underline{\mathbf{Z}} \longrightarrow \underline{\mathbf{Z}} \]

having the prescribed behaviour on stalks. By Definition 62.4.4 we have $f_!\underline{\mathbf{Z}} = (f_{p!}\underline{\mathbf{Z}})^\# $ where $f_{p!}\underline{\mathbf{Z}}$ is the presheaf constructed in Section 62.4. Thus it suffices to construct a map

\[ f_{p!}\underline{\mathbf{Z}} \longrightarrow \underline{\mathbf{Z}} \]

of presheaves on $Y_{\acute{e}tale}$. Let $V$ be an object of $Y_{\acute{e}tale}$. Recall from Section 62.4 that

\[ f_{p!}\underline{\mathbf{Z}}(V) = \mathop{\mathrm{colim}}\nolimits _ Z H_ Z(\underline{\mathbf{Z}}) \]

Here the colimit is over the (partially ordered) collection of locally closed subschemes $Z \subset X_ V$ which are finite over $V$. For each such $Z$ we will define a map

\[ H_ Z(\underline{\mathbf{Z}}) \longrightarrow \underline{\mathbf{Z}}(V) \]

compatible with the maps defining the colimit.

Let $Z \subset X_ V$ be locally closed and finite over $V$. Choose an open $U \subset X_ V$ containing $Z$ as a closed subset. An element $s$ of $H_ Z(\underline{\mathbf{Z}})$ is a section $s \in \underline{\mathbf{Z}}(U)$ whose support is contained in $Z$. Let $U_ n \subset U$ be the open and closed subset where the value of $s$ is $n \in \mathbf{Z}$. By the support condition we see that $Z \cap U_ n = U_ n$ for $n \not= 0$. Hence for $n \not= 0$, the open $U_ n$ is also closed in $Z$ (as the complement of all the others) and we conclude that $U_ n \to V$ is finite as $Z$ is finite over $V$. By the very definition of a weighting this means the function $\int _{U_ n \to V} w|_{U_ n}$ is locally constant on $V$ and we may view it as an element of $\underline{\mathbf{Z}}(V)$. Our construction sends $(Z, s)$ to the element

\[ \sum \nolimits _{n \in \mathbf{Z},\ n \not= 0} n \left(\int _{U_ n \to V} w|_{U_ n}\right) \quad \in \quad \underline{\mathbf{Z}}(V) \]

The sum is locally finite on $V$ and hence makes sense; details omitted (in the whole discussion the reader may first choose affine opens and make sure all the schemes occuring in the argument are quasi-compact so the sum is finite). We omit the verification that this construction is compatible with the maps in the colimit and with the restriction mappings defining $f_{p!}\underline{\mathbf{Z}}$.

Let $\overline{y}$ be a geometric point of $Y$ lying over the point $y \in Y$. Taking stalks at $\overline{y}$ the construction above determines a map

\[ (f_!\underline{\mathbf{Z}})_{\overline{y}} = \bigoplus \nolimits _{f(\overline{x}) = \overline{y}} \mathbf{Z} \longrightarrow \mathbf{Z} = \underline{\mathbf{Z}}_{\overline{y}} \]

To finish the proof we will show this map is given by multiplication by $w(\overline{x})$ on the summand corresponding to $\overline{x}$. Namely, pick $\overline{x}$ lying over $\overline{y}$. We can find an étale neighbourhood $(V, \overline{v}) \to (Y, \overline{y})$ such that $X_ V$ contains an open $U$ finite over $V$ such that only the geometric point $\overline{x}$ is in $U$ and not the other geometric points of $X$ lifting $\overline{y}$. This follows from More on Morphisms, Lemma 37.41.3; some details omitted. Then $(U, 1)$ defines a section of $f_!\underline{\mathbf{Z}}$ over $V$ which maps to $1$ in the summand corresponding to $\overline{x}$ and zero in the other summands (see proof of Lemma 62.4.2) and our construction above sends $(U, 1)$ to $\int _{U \to V} w|_ U$ which is constant with value $w(\overline{x})$ in a neighbourhood of $\overline{v}$ as desired. $\square$


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 0GKG. Beware of the difference between the letter 'O' and the digit '0'.