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? 
 A: 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.
