Lemma 63.5.2. 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 63.5.1 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 63.4.4 we have f_!\underline{\mathbf{Z}} = (f_{p!}\underline{\mathbf{Z}})^\# where f_{p!}\underline{\mathbf{Z}} is the presheaf constructed in Section 63.4. 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 63.4 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 occurring 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 63.4.2) 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)