Lemma 47.25.2. Let $\varphi : R \to A$ be a finite type map of Noetherian rings.

1. If $\varphi$ is perfect, then

1. $\omega _{A/R}^\bullet$ is in $D^ b_{\textit{Coh}}(A)$,

2. $\omega _{A/R}^\bullet$ has finite tor dimension over $R$, and

3. $A \to R\mathop{\mathrm{Hom}}\nolimits _ A(\omega _{A/R}^\bullet , \omega _{A/R}^\bullet )$ is an isomorphism.

2. If $\varphi$ is flat, then (1)(a), (1)(b), and (1)(c) hold and

1. $\omega _{A/R}^\bullet$ is $R$-perfect (More on Algebra, Definition 15.83.1), and

2. for every map $R \to k$ to a field the base change $\omega _{A/R}^\bullet \otimes _ A^\mathbf {L} (A \otimes _ R k)$ is a dualizing complex for $A \otimes _ R k$.

Proof. Assume $R \to A$ is perfect. Choose $R \to P \to A$ as in the definition of $\varphi ^!$. Then $A$ is perfect as a $P$-modue (More on Algebra, Lemma 15.82.2). This shows that $\omega _{A/R}^\bullet$ is in $D^ b_{\textit{Coh}}(A)$ by Lemma 47.24.2. This proves (1)(a). To show $\omega _{A/R}^\bullet$ has finite tor dimension as a complex of $R$-modules, observe that $\omega _{A/R}^\bullet = \varphi ^!(R) = R\mathop{\mathrm{Hom}}\nolimits (A, P)[n]$ maps to $R\mathop{\mathrm{Hom}}\nolimits _ P(A, P)[n]$ in $D(P)$, which is perfect in $D(P)$ (More on Algebra, Lemma 15.74.15), hence has finite tor dimension in $D(R)$ as $R \to P$ is flat. This proves (1)(b). The object $R\mathop{\mathrm{Hom}}\nolimits _ A(\omega _{A/R}^\bullet , \omega _{A/R}^\bullet )$ of $D(A)$ maps in $D(P)$ to

\begin{align*} R\mathop{\mathrm{Hom}}\nolimits _ P(\omega _{A/R}^\bullet , R\mathop{\mathrm{Hom}}\nolimits (A, P)[n]) & = R\mathop{\mathrm{Hom}}\nolimits _ P(R\mathop{\mathrm{Hom}}\nolimits _ P(A, P)[n], P)[n] \\ & = R\mathop{\mathrm{Hom}}\nolimits _ P(R\mathop{\mathrm{Hom}}\nolimits _ P(A, P), P) \end{align*}

This is equal to $A$ by the already used More on Algebra, Lemma 15.74.15. This proves (1)(c).

Assume $\varphi$ is flat. Then $R \to A$ is a perfect ring map (More on Algebra, Lemma 15.82.4) and we see that (1)(a), (1)(b), and (1)(c) hold. Of course, then $\omega _{A/R}^\bullet$ is $R$-perfect by (1)(a) and (1)(b) and the definitions. Let $R \to k$ be as in (2)(b). By Lemma 47.25.1 there is an isomorphism

$\omega _{A/R}^\bullet \otimes _ A^\mathbf {L} (A \otimes _ R k) \cong \omega ^\bullet _{A \otimes _ R k/k}$

and the right hand side is a dualizing complex by Lemma 47.24.3. This finishes the proof. $\square$

Comment #8352 by Bogdan on

It seems that (1) and (2) hold as long as $\varphi$ is finite type and of finite Tor dimension.

Comment #8959 by on

Thanks for this remark. At the moment we have only defined relatively perfect complexes in the flat case, see Remark 36.35.14 for why. Note that with option (B) in that remark what you say would be correct. So I have simply restated this lemma in a way avoiding the notion of "relatively perfect" in the non-flat case. See here.

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