The Stacks project

Lemma 47.24.7. Let $A \xrightarrow {a} B \xrightarrow {b} C$ be finite type homomorphisms of Noetherian rings. Then there is a transformation of functors $b^! \circ a^! \to (b \circ a)^!$ which is an isomorphism on $D^+(A)$.

Proof. Choose a polynomial ring $P = A[x_1, \ldots , x_ n]$ over $A$ and a surjection $P \to B$. Choose elements $c_1, \ldots , c_ m \in C$ generating $C$ over $B$. Set $Q = P[y_1, \ldots , y_ m]$ and denote $Q' = Q \otimes _ P B = B[y_1, \ldots , y_ m]$. Let $\chi : Q' \to C$ be the surjection sending $y_ j$ to $c_ j$. Picture

\[ \xymatrix{ & Q \ar[r]_{\psi '} & Q' \ar[r]_\chi & C \\ A \ar[r] & P \ar[r]^\psi \ar[u] & B \ar[u] } \]

By Lemma 47.14.2 for $M \in D(P)$ we have an arrow $\psi ^!(M) \otimes _ B^\mathbf {L} Q' \to (\psi ')^!(M \otimes _ P^\mathbf {L} Q)$ which is an isomorphism whenever $M$ is bounded below. Also we have $\chi ^! \circ (\psi ')^! = (\chi \circ \psi ')^!$ as both functors are adjoint to the restriction functor $D(C) \to D(Q)$ by Section 47.13. Then we see

\begin{align*} b^!(a^!(K)) & = \chi ^!(\psi ^!(K \otimes _ A^\mathbf {L} P)[n] \otimes _ B^\mathbf {L} Q)[m] \\ & \to \chi ^!((\psi ')^!(K \otimes _ A^\mathbf {L} P \otimes _ P^\mathbf {L} Q))[n + m] \\ & = (\chi \circ \psi ')^!(K\otimes _ A^\mathbf {L} Q)[n + m] \\ & = (b \circ a)^!(K) \end{align*}

where we have used in addition to the above More on Algebra, Lemma 15.60.5. $\square$

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