The Stacks project

Lemma 48.12.7 (Rigidity). Let $Y$ be a quasi-compact and quasi-separated scheme. Let $f : X \to Y$ be a proper, flat morphism of finite presentation with relative dualizing complex $\omega _{X/Y}^\bullet $ (Remark 48.12.5). There is a canonical isomorphism

48.12.7.1
\begin{equation} \label{duality-equation-pre-rigid} \mathcal{O}_ X = c(L\text{pr}_1^*\omega _{X/Y}^\bullet ) = c(L\text{pr}_2^*\omega _{X/Y}^\bullet ) \end{equation}

and a canonical isomorphism

48.12.7.2
\begin{equation} \label{duality-equation-rigid} \omega _{X/Y}^\bullet = c\left(L\text{pr}_1^*\omega _{X/Y}^\bullet \otimes _{\mathcal{O}_{X \times _ Y X}}^\mathbf {L} L\text{pr}_2^*\omega _{X/Y}^\bullet \right) \end{equation}

where $c$ is the right adjoint of Lemma 48.3.1 for the diagonal $\Delta : X \to X \times _ Y X$.

Proof. Let $a$ be the right adjoint for $f$ as in Lemma 48.3.1. Consider the cartesian square

\[ \xymatrix{ X \times _ Y X \ar[r]_ q \ar[d]_ p & X \ar[d]_ f \\ X \ar[r]^ f & Y } \]

Let $b$ be the right adjoint for $p$ as in Lemma 48.3.1. Then

\begin{align*} \omega _{X/Y}^\bullet & = c(b(\omega _{X/Y}^\bullet )) \\ & = c(Lp^*\omega _{X/Y}^\bullet \otimes _{\mathcal{O}_{X \times _ Y X}}^\mathbf {L} b(\mathcal{O}_ X)) \\ & = c(Lp^*\omega _{X/Y}^\bullet \otimes _{\mathcal{O}_{X \times _ Y X}}^\mathbf {L} Lq^*a(\mathcal{O}_ Y)) \\ & = c(Lp^*\omega _{X/Y}^\bullet \otimes _{\mathcal{O}_{X \times _ Y X}}^\mathbf {L} Lq^*\omega _{X/Y}^\bullet ) \end{align*}

as in (48.12.7.2). Explanation as follows:

  1. The first equality holds as $\text{id} = c \circ b$ because $\text{id}_ X = p \circ \Delta $.

  2. The second equality holds by Lemma 48.12.3.

  3. The third holds by Lemma 48.12.4 and the fact that $\mathcal{O}_ X = Lf^*\mathcal{O}_ Y$.

  4. The fourth holds because $\omega _{X/Y}^\bullet = a(\mathcal{O}_ Y)$.

Equation (48.12.7.1) is proved in exactly the same way. $\square$


Comments (1)

Comment #5384 by Will Chen on

The first sentence of the proof: Should probably say " is the right adjoint for "


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