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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