Example 48.7.3. Let $A \to B$ be a ring map. Let $Y = \mathop{\mathrm{Spec}}(A)$ and $X = \mathop{\mathrm{Spec}}(B)$ and $f : X \to Y$ the morphism corresponding to $A \to B$. As seen in Example 48.3.2 the right adjoint of $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ sends an object $K$ of $D(A) = D_\mathit{QCoh}(\mathcal{O}_ Y)$ to $R\mathop{\mathrm{Hom}}\nolimits (B, K)$ in $D(B) = D_\mathit{QCoh}(\mathcal{O}_ X)$. The trace map is the map
\[ \text{Tr}_{f, K} : R\mathop{\mathrm{Hom}}\nolimits (B, K) \longrightarrow R\mathop{\mathrm{Hom}}\nolimits (A, K) = K \]
induced by the $A$-module map $A \to B$.
Comments (0)