The Stacks project

48.23 Cohen-Macaulay schemes

This section is the continuation of Dualizing Complexes, Section 47.20. Duality takes a particularly simple form for Cohen-Macaulay schemes.

Lemma 48.23.1. Let $X$ be a locally Noetherian scheme with dualizing complex $\omega _ X^\bullet $.

  1. $X$ is Cohen-Macaulay $\Leftrightarrow $ $\omega _ X^\bullet $ locally has a unique nonzero cohomology sheaf,

  2. $\mathcal{O}_{X, x}$ is Cohen-Macaulay $\Leftrightarrow $ $\omega _{X, x}^\bullet $ has a unique nonzero cohomology,

  3. $U = \{ x \in X \mid \mathcal{O}_{X, x}\text{ is Cohen-Macaulay}\} $ is open and Cohen-Macaulay.

If $X$ is connected and Cohen-Macaulay, then there is an integer $n$ and a coherent Cohen-Macaulay $\mathcal{O}_ X$-module $\omega _ X$ such that $\omega _ X^\bullet = \omega _ X[-n]$.

Proof. By definition and Dualizing Complexes, Lemma 47.15.6 for every $x \in X$ the complex $\omega _{X, x}^\bullet $ is a dualizing complex over $\mathcal{O}_{X, x}$. By Dualizing Complexes, Lemma 47.20.2 we see that (2) holds.

To see (3) assume that $\mathcal{O}_{X, x}$ is Cohen-Macaulay. Let $n_ x$ be the unique integer such that $H^{n_{x}}(\omega _{X, x}^\bullet )$ is nonzero. For an affine neighbourhood $V \subset X$ of $x$ we have $\omega _ X^\bullet |_ V$ is in $D^ b_{\textit{Coh}}(\mathcal{O}_ V)$ hence there are finitely many nonzero coherent modules $H^ i(\omega _ X^\bullet )|_ V$. Thus after shrinking $V$ we may assume only $H^{n_ x}$ is nonzero, see Modules, Lemma 17.9.5. In this way we see that $\mathcal{O}_{X, v}$ is Cohen-Macaulay for every $v \in V$. This proves that $U$ is open as well as a Cohen-Macaulay scheme.

Proof of (1). The implication $\Leftarrow $ follows from (2). The implication $\Rightarrow $ follows from the discussion in the previous paragraph, where we showed that if $\mathcal{O}_{X, x}$ is Cohen-Macaulay, then in a neighbourhood of $x$ the complex $\omega _ X^\bullet $ has only one nonzero cohomology sheaf.

Assume $X$ is connected and Cohen-Macaulay. The above shows that the map $x \mapsto n_ x$ is locally constant. Since $X$ is connected it is constant, say equal to $n$. Setting $\omega _ X = H^ n(\omega _ X^\bullet )$ we see that the lemma holds because $\omega _ X$ is Cohen-Macaulay by Dualizing Complexes, Lemma 47.20.2 (and Cohomology of Schemes, Definition 30.11.4). $\square$

Lemma 48.23.2. Let $X$ be a locally Noetherian scheme. If there exists a coherent sheaf $\omega _ X$ such that $\omega _ X[0]$ is a dualizing complex on $X$, then $X$ is a Cohen-Macaulay scheme.

Proof. This follows immediately from Dualizing Complexes, Lemma 47.20.3 and our definitions. $\square$

Lemma 48.23.3. In Situation 48.16.1 let $f : X \to Y$ be a morphism of $\textit{FTS}_ S$. Let $x \in X$. If $f$ is flat, then the following are equivalent

  1. $f$ is Cohen-Macaulay at $x$,

  2. $f^!\mathcal{O}_ Y$ has a unique nonzero cohomology sheaf in a neighbourhood of $x$.

Proof. One direction of the lemma follows from Lemma 48.21.7. To prove the converse, we may assume $f^!\mathcal{O}_ Y$ has a unique nonzero cohomology sheaf. Let $y = f(x)$. Let $\xi _1, \ldots , \xi _ n \in X_ y$ be the generic points of the fibre $X_ y$ specializing to $x$. Let $d_1, \ldots , d_ n$ be the dimensions of the corresponding irreducible components of $X_ y$. The morphism $f : X \to Y$ is Cohen-Macaulay at $\eta _ i$ by More on Morphisms, Lemma 37.22.7. Hence by Lemma 48.21.7 we see that $d_1 = \ldots = d_ n$. If $d$ denotes the common value, then $d = \dim _ x(X_ y)$. After shrinking $X$ we may assume all fibres have dimension at most $d$ (Morphisms, Lemma 29.28.4). Then the only nonzero cohomology sheaf $\omega = H^{-d}(f^!\mathcal{O}_ Y)$ is flat over $Y$ by Lemma 48.21.4. Hence, if $h : X_ y \to X$ denotes the canonical morphism, then $Lh^*(f^!\mathcal{O}_ Y) = Lh^*(\omega [d]) = (h^*\omega )[d]$ by Derived Categories of Schemes, Lemma 36.22.8. Thus $h^*\omega [d]$ is the dualizing complex of $X_ y$ by Lemma 48.18.4. Hence $X_ y$ is Cohen-Macaulay by Lemma 48.23.1. This proves $f$ is Cohen-Macaulay at $x$ as desired. $\square$

Remark 48.23.4. In Situation 48.16.1 let $f : X \to Y$ be a morphism of $\textit{FTS}_ S$. Assume $f$ is a Cohen-Macaulay morphism of relative dimension $d$. Let $\omega _{X/Y} = H^{-d}(f^!\mathcal{O}_ Y)$ be the unique nonzero cohomology sheaf of $f^!\mathcal{O}_ Y$, see Lemma 48.21.7. Then there is a canonical isomorphism

\[ f^!K = Lf^*K \otimes _{\mathcal{O}_ X}^\mathbf {L} \omega _{X/Y}[d] \]

for $K \in D^+_\mathit{QCoh}(\mathcal{O}_ Y)$, see Lemma 48.17.9. In particular, if $S$ has a dualizing complex $\omega _ S^\bullet $, $\omega _ Y^\bullet = (Y \to S)^!\omega _ S^\bullet $, and $\omega _ X^\bullet = (X \to S)^!\omega _ S^\bullet $ then we have

\[ \omega _ X^\bullet = Lf^*\omega _ Y^\bullet \otimes _{\mathcal{O}_ X}^\mathbf {L} \omega _{X/Y}[d] \]

Thus if further $X$ and $Y$ are connected and Cohen-Macaulay and if $\omega _ Y$ and $\omega _ X$ denote the unique nonzero cohomology sheaves of $\omega _ Y^\bullet $ and $\omega _ X^\bullet $, then we have

\[ \omega _ X = f^*\omega _ Y \otimes _{\mathcal{O}_ X} \omega _{X/Y}. \]

Similar results hold for $X$ and $Y$ arbitrary finite type schemes over $S$ (i.e., not necessarily separated over $S$) with dualizing complexes normalized with respect to $\omega _ S^\bullet $ as in Section 48.20.

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