Lemma 47.25.7 (0E9P)—The Stacks project

Lemma 47.25.7. Let $R \to A$ be a finite type homomorphism of Noetherian rings. Let $\mathfrak q \subset A$ be a prime ideal lying over $\mathfrak p \subset R$. Then

\[ H^ i(\omega _{A/R}^\bullet )_\mathfrak q \not= 0 \Rightarrow - d \leq i \]

where $d$ is the dimension of the fibre of $\mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(R)$ over $\mathfrak p$ at the point $\mathfrak q$.

Proof. Choose a factorization $R \to P \to A$ with $P = R[x_1, \ldots , x_ n]$ as in Section 47.24 so that $\omega _{A/R}^\bullet = R\mathop{\mathrm{Hom}}\nolimits (A, P)[n]$. We have to show that $R\mathop{\mathrm{Hom}}\nolimits (A, P)_\mathfrak q$ has vanishing cohomology in degrees $< n - d$. By Lemma 47.13.3 this means we have to show that $\mathop{\mathrm{Ext}}\nolimits _ P^ i(P/I, P)_{\mathfrak r} = 0$ for $i < n - d$ where $\mathfrak r \subset P$ is the prime corresponding to $\mathfrak q$ and $I$ is the kernel of $P \to A$. We may rewrite this as $\mathop{\mathrm{Ext}}\nolimits _{P_\mathfrak r}^ i(P_\mathfrak r/IP_\mathfrak r, P_\mathfrak r)$ by More on Algebra, Lemma 15.65.4. Thus we have to show

\[ \text{depth}_{IP_\mathfrak r}(P_\mathfrak r) \geq n - d \]

by Lemma 47.11.1. By Lemma 47.11.5 we have

\[ \text{depth}_{IP_\mathfrak r}(P_\mathfrak r) \geq \dim ((P \otimes _ R \kappa (\mathfrak p))_\mathfrak r) - \dim ((P/I \otimes _ R \kappa (\mathfrak p))_\mathfrak r) \]

The two expressions on the right hand side agree by Algebra, Lemma 10.116.4. $\square$

The Stacks project

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