Lemma 47.15.3 (0A7C)—The Stacks project

Lemma 47.15.3. Let $A$ be a Noetherian ring. If $\omega _ A^\bullet $ is a dualizing complex, then the functor

\[ D : K \longmapsto R\mathop{\mathrm{Hom}}\nolimits _ A(K, \omega _ A^\bullet ) \]

is an anti-equivalence $D_{\textit{Coh}}(A) \to D_{\textit{Coh}}(A)$ which exchanges $D^+_{\textit{Coh}}(A)$ and $D^-_{\textit{Coh}}(A)$ and induces an anti-equivalence $D^ b_{\textit{Coh}}(A) \to D^ b_{\textit{Coh}}(A)$. Moreover $D \circ D$ is isomorphic to the identity functor.

Proof. Let $K$ be an object of $D_{\textit{Coh}}(A)$. From Lemma 47.15.2 we see $R\mathop{\mathrm{Hom}}\nolimits _ A(K, \omega _ A^\bullet )$ is an object of $D_{\textit{Coh}}(A)$. By More on Algebra, Lemma 15.98.2 and the assumptions on the dualizing complex we obtain a canonical isomorphism

\[ K = R\mathop{\mathrm{Hom}}\nolimits _ A(\omega _ A^\bullet , \omega _ A^\bullet ) \otimes _ A^\mathbf {L} K \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ A(R\mathop{\mathrm{Hom}}\nolimits _ A(K, \omega _ A^\bullet ), \omega _ A^\bullet ) \]

Thus our functor has a quasi-inverse and the proof is complete. $\square$

Comments (5)

Comment #3623 by Janos Kollar on October 13, 2018 at 11:03

A small remark, but if you say "anti-equivalence" in line 3 then probably should do the same in line 4.

Comment #3726 by Johan on October 23, 2018 at 13:50

Thanks, fixed here.

Comment #8374 by Haohao Liu on May 05, 2023 at 01:46

I am sorry if this question is naïve. Is it trivial that $D$ sends $D^+_{Coh}(A)$ to $D^-_{Coh}(A)$ ? The proof ignores this part.

Comment #8376 by Nicolás on May 05, 2023 at 13:00

Maybe this follows from a spectral sequence argument? Something like $E_2^{p, q} = R\operatorname{Hom}_A(H^{-q}(E), \omega_A^\bullet) \Rightarrow R\operatorname{Hom}_A(E, \omega_A^\bullet).$ Here $\omega_A^\bullet$ has finite injective dimension, hence the $E_2^{p, q}$ is zero for $q$ outside of a fixed interval.

Comment #8980 by Stacks project on April 14, 2024 at 09:43

Directly from the definition, if $K$ has finite injective dimension and if $M$ is bounded below (resp above), then $RHom_A(M, K)$ is bounded above (resp below). Namely, you can represent $M$ by a bounded below (resp above) complex and $K$ by a bounded complex of injectives, and then the $RHom$ is computed by the hom complex which has the requisite boundedness.

Maybe this should be added to Section 15.69?

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2 comment(s) on Section 47.15: Dualizing complexes

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