The Stacks project

Lemma 51.6.2. Let $A$ be a Noetherian ring of finite dimension. Let $T \subset \mathop{\mathrm{Spec}}(A)$ be a subset stable under specialization. Let $\{ M_ n\} _{n \geq 0}$ be an inverse system of $A$-modules. Let $i \geq 0$ be an integer. Assume that for every $m$ there exists an integer $m'(m) \geq m$ such that for all $\mathfrak p \in T$ the induced map

\[ H^ i_{\mathfrak p A_\mathfrak p}(M_{k, \mathfrak p}) \longrightarrow H^ i_{\mathfrak p A_\mathfrak p}(M_{m, \mathfrak p}) \]

is zero for $k \geq m'(m)$. Let $m'' : \mathbf{N} \to \mathbf{N}$ be the $2^{\dim (T)}$-fold self-composition of $m'$. Then the map $H^ i_ T(M_ k) \to H^ i_ T(M_ m)$ is zero for all $k \geq m''(m)$.

Proof. We first make a general remark: suppose we have an exact sequence

\[ (A_ n) \to (B_ n) \to (C_ n) \]

of inverse systems of abelian groups. Suppose that for every $m$ there exists an integer $m'(m) \geq m$ such that

\[ A_ k \to A_ m \quad \text{and}\quad C_ k \to C_ m \]

are zero for $k \geq m'(m)$. Then for $k \geq m'(m'(m))$ the map $B_ k \to B_ m$ is zero.

We will prove the lemma by induction on $\dim (T)$ which is finite because $\dim (A)$ is finite. Let $T' \subset T$ be the set of nonminimal primes in $T$. Then $T'$ is a subset of $\mathop{\mathrm{Spec}}(A)$ stable under specialization and the hypotheses of the lemma apply to $T'$. Since $\dim (T') < \dim (T)$ we know the lemma holds for $T'$. For every $A$-module $M$ there is an exact sequence

\[ H^ i_{T'}(M) \to H^ i_ T(M) \to \bigoplus \nolimits _{\mathfrak p \in T \setminus T'} H^ i_{\mathfrak p A_\mathfrak p}(M_\mathfrak p) \]

by Lemma 51.6.1. Thus we conclude by the initial remark of the proof. $\square$

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