The Stacks project

Remark 48.17.5 (Local description upper shriek). In Situation 48.16.1 let $f : X \to Y$ be a morphism of $\textit{FTS}_ S$. Using the lemmas above we can compute $f^!$ locally as follows. Suppose that we are given affine opens

\[ \xymatrix{ U \ar[r]_ j \ar[d]_ g & X \ar[d]^ f \\ V \ar[r]^ i & Y } \]

Since $j^! \circ f^! = g^! \circ i^!$ (Lemma 48.16.3) and since $j^!$ and $i^!$ are given by restriction (Lemma 48.17.1) we see that

\[ (f^!E)|_ U = g^!(E|_ V) \]

for any $E \in D^+_\mathit{QCoh}(\mathcal{O}_ X)$. Write $U = \mathop{\mathrm{Spec}}(A)$ and $V = \mathop{\mathrm{Spec}}(R)$ and let $\varphi : R \to A$ be the finite type ring map corresponding to $g$. Choose a presentation $A = P/I$ where $P = R[x_1, \ldots , x_ n]$ is a polynomial algebra in $n$ variables over $R$. Choose an object $K \in D^+(R)$ corresponding to $E|_ V$ (Derived Categories of Schemes, Lemma 36.3.5). Then we claim that $f^!E|_ U$ corresponds to

\[ \varphi ^!(K) = R\mathop{\mathrm{Hom}}\nolimits (A, K \otimes _ R^\mathbf {L} P)[n] \]

where $R\mathop{\mathrm{Hom}}\nolimits (A, -) : D(P) \to D(A)$ is the functor of Dualizing Complexes, Section 47.13 and where $\varphi ^! : D(R) \to D(A)$ is the functor of Dualizing Complexes, Section 47.24. Namely, the choice of presentation gives a factorization

\[ U \rightarrow \mathbf{A}^ n_ V \to \mathbf{A}^{n - 1}_ V \to \ldots \to \mathbf{A}^1_ V \to V \]

Applying Lemma 48.17.3 exactly $n$ times we see that $(\mathbf{A}^ n_ V \to V)^!(E|_ V)$ corresponds to $K \otimes _ R^\mathbf {L} P[n]$. By Lemmas 48.9.5 and 48.17.4 the last step corresponds to applying $R\mathop{\mathrm{Hom}}\nolimits (A, -)$.

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