# Trace map of a finite étale morphism

My question is rather informal.

Let $$f:Y\to X$$ be a finite étale morphism of schemes, and consider the direct image $$f_*$$ and inverse image $$f^{-1}$$ functors on étale sheaves of abelian groups. Then not only $$f^{-1}$$ is left adjoint to $$f_*$$ (this is true for any $$f$$), but also $$f_*$$ is left adjoint to $$f^{-1}$$, and people call the adjunction morphism $$f_*f^{-1}\to \text{id}$$ the $$\textit{trace map}$$ of $$f$$. See for example this section of the Stacks Project.

Why is this called a trace? Does it have anything to do with traces of linear endomorphisms of a vector space?

Let $$\rho : A \to B$$ be a finite flat morphism of rings. The corresponding morphism of schemes $$f : X \to Y$$ is a finite flat and is therefore proper. By Grothendieck duality you have an adjunction $$(f_* , f^!)$$, where $$f_* : \textrm{Qcoh}(X) \to \textrm{Qcoh}(Y)$$ is the direct image and $$f^!$$ its right adjoint. This adjoint can be precised as follows : $$f^!(\mathcal{G}) = \mathcal{H}om_{\mathcal{O}_Y}(f_*\mathcal{O}_X,\mathcal{G})^\sim$$ for $$\mathcal{G} \in \textrm{Qcoh}(Y)$$, where $$(\cdot)^\sim$$ denotes the equivalence between $$\mathcal{O}_X$$-modules and $$f_*\mathcal{O}_X$$-modules over $$Y$$, as $$f$$ is an affine morphism. Now the co-unit $$f_*f^! \mathcal{G} \longrightarrow \mathcal{G}$$ of the adjunction applied to $$\mathcal{O}_Y$$ yields to the map $$\mathrm{Tr}_{\rho} \colon B \longrightarrow A$$ defined as follows : each $$b\in B$$ acts on $$B$$ (viewed as an $$A$$-module through $$\varphi$$) by multiplication. Since $$B$$ is finite flat over $$A$$ and $$A$$ is Noetherian, the $$A$$-module $$B$$ is a locally free $$A$$-module and multiplication by $$b$$ is therefore locally (on an open subset $$D(a) \simeq \mathrm{Spec}(A_a)\subseteq \mathrm{Spec}(A)$$, for an $$a\in A$$ and under some isomorphism $$B_a \simeq A_a^n$$) given by multiplication by a matrix. We define $$\textrm{Tr}_{\rho}(b)$$ to be the trace of this matrix. As the trace of a matrix is independent of the choice of basis this homomorphism of $$A$$-modules is well defined. I think though that the name may even come from the "simplest" case of trace of an element of a field finite extension.