The Stacks project

Lemma 48.21.7. In Situation 48.16.1 let $f : X \to Y$ be a morphism of $\textit{FTS}_ S$. If $f$ is Cohen-Macaulay (More on Morphisms, Definition 37.22.1), then

\[ f^!\mathcal{O}_ Y = \omega _{X/Y}[d] \]

for some coherent $\mathcal{O}_ X$-module $\omega _{X/Y}$ flat over $Y$ where $d$ is the locally constant function on $X$ which gives the relative dimension of $X$ over $Y$.

Proof. The relative dimension $d$ is well defined and locally constant by Morphisms, Lemma 29.29.4. The cohomology sheaves of $f^!\mathcal{O}_ Y$ are coherent by Lemma 48.17.6. We will get flatness of $\omega _{X/Y}$ from Lemma 48.21.4 if we can show the other cohomology sheaves of $f^!\mathcal{O}_ Y$ are zero.

The question is local on $X$, hence we may assume $X$ and $Y$ are affine and the morphism has relative dimension $d$. If $d = 0$, then the result follows directly from Lemma 48.21.6. If $d > 0$, then we may assume there is a factorization

\[ X \xrightarrow {g} \mathbf{A}^ d_ Y \xrightarrow {p} Y \]

with $g$ quasi-finite and flat, see More on Morphisms, Lemma 37.22.8. Then $f^! = g^! \circ p^!$. By Lemma 48.17.3 we see that $p^!\mathcal{O}_ Y \cong \mathcal{O}_{\mathbf{A}^ d_ Y}[-d]$. We conclude by the case $d = 0$. $\square$

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