Lemma 47.9.6 (0BJC)—The Stacks project

Lemma 47.9.6. Let $A$ be a ring and let $I, J \subset A$ be finitely generated ideals. Set $Z = V(I)$ and $Y = V(J)$. Then $Z \cap Y = V(I + J)$ and $R\Gamma _ Y \circ R\Gamma _ Z = R\Gamma _{Y \cap Z}$ as functors $D(A) \to D_{(I + J)^\infty \text{-torsion}}(A)$. For $K \in D^+(A)$ there is a spectral sequence

\[ E_2^{p, q} = H^ p_ Y(H^ q_ Z(K)) \Rightarrow H^{p + q}_{Y \cap Z}(K) \]

as in Derived Categories, Lemma 13.22.2.

Proof. There is a bit of abuse of notation in the lemma as strictly speaking we cannot compose $R\Gamma _ Y$ and $R\Gamma _ Z$. The meaning of the statement is simply that we are composing $R\Gamma _ Z$ with the inclusion $D_{I^\infty \text{-torsion}}(A) \to D(A)$ and then with $R\Gamma _ Y$. Then the equality $R\Gamma _ Y \circ R\Gamma _ Z = R\Gamma _{Y \cap Z}$ follows from the fact that

\[ D_{I^\infty \text{-torsion}}(A) \to D(A) \xrightarrow {R\Gamma _ Y} D_{(I + J)^\infty \text{-torsion}}(A) \]

is right adjoint to the inclusion $D_{(I + J)^\infty \text{-torsion}}(A) \to D_{I^\infty \text{-torsion}}(A)$. Alternatively one can prove the formula using Lemma 47.9.1 and the fact that the tensor product of extended Čech complexes on $f_1, \ldots , f_ r$ and $g_1, \ldots , g_ m$ is the extended Čech complex on $f_1, \ldots , f_ n. g_1, \ldots , g_ m$. The final assertion follows from this and the cited lemma. $\square$

Comments (3)

Comment #3597 by Kestutis Cesnavicius on September 28, 2018 at 04:33

Towards the end of the proof, "Č complex" should be "Čech complex."

Comment #3599 by Kestutis Cesnavicius on September 28, 2018 at 05:09

There is a typo in the statement: $H^p_Y(H^p_Z(...))$ should be $H^p_Y(H^q_Z(...))$ .

Comment #3714 by Johan on October 22, 2018 at 19:30

Thanks and fixes are here.

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