The Stacks project

Lemma 51.6.1. Let $A$ be a Noetherian ring. Let $T \subset \mathop{\mathrm{Spec}}(A)$ be a subset stable under specialization. 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 for every $A$-module $M$ there is an exact sequence

\[ 0 \to \mathop{\mathrm{colim}}\nolimits _{Z, f} H^1_ f(H^{i - 1}_ Z(M)) \to H^ i_{T'}(M) \to H^ i_ T(M) \to \bigoplus \nolimits _{\mathfrak p \in T \setminus T'} H^ i_{\mathfrak p A_\mathfrak p}(M_\mathfrak p) \]

where the colimit is over closed subsets $Z \subset T$ and $f \in A$ with $V(f) \cap Z \subset T'$.

Proof. For every $Z$ and $f$ the spectral sequence of Dualizing Complexes, Lemma 47.9.6 degenerates to give short exact sequences

\[ 0 \to H^1_ f(H^{i - 1}_ Z(M)) \to H^ i_{Z \cap V(f)}(M) \to H^0_ f(H^ i_ Z(M)) \to 0 \]

We will use this without further mention below.

Let $\xi \in H^ i_ T(M)$ map to zero in the direct sum. Then we first write $\xi $ as the image of some $\xi ' \in H^ i_ Z(M)$ for some closed subset $Z \subset T$, see Lemma 51.5.3. Then $\xi '$ maps to zero in $H^ i_{\mathfrak p A_\mathfrak p}(M_\mathfrak p)$ for every $\mathfrak p \in Z$, $\mathfrak p \not\in T'$. Since there are finitely many of these primes, we may choose $f \in A$ not contained in any of these such that $f$ annihilates $\xi '$. Then $\xi '$ is the image of some $\xi '' \in H^ i_{Z'}(M)$ where $Z' = Z \cap V(f)$. By our choice of $f$ we have $Z' \subset T'$ and we get exactness at the penultimate spot.

Let $\xi \in H^ i_{T'}(M)$ map to zero in $H^ i_ T(M)$. Choose closed subsets $Z' \subset Z$ with $Z' \subset T'$ and $Z \subset T$ such that $\xi $ comes from $\xi ' \in H^ i_{Z'}(M)$ and maps to zero in $H^ i_ Z(M)$. Then we can find $f \in A$ with $V(f) \cap Z = Z'$ and we conclude. $\square$

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