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Let $X$ be smooth algebraic variety over $\mathbb C$ and $D_1, D_2$ are snc divisors such that $D_1\cup D_2$ is also snc.

Is it true that there is an exact sequence

$$H^*(X, D_1\cup D_2)\to H^*(X, D_1)\to H^*(D_2, D_1\cap D_2)\to H^{*+1}(X, D_1\cup D_2)\to\dots$$

The motivation is that it is standard long exact sequence for the pair $(Y, D_2)$ where $Y=(X, D_1)$.

If it is not true, then what else exact sequence there is to compute $H^*(X, D_1\cup D_2)$?

The cohomology is singular cohomology with rational coefficients.

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  • $\begingroup$ No, what you have is a spectral sequence. See Deligne, Théorie de Hodge II, §3. $\endgroup$
    – abx
    Oct 26, 2023 at 14:54
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    $\begingroup$ @abx There is a spectral sequence, but this exact sequence is also fine. $\endgroup$
    – Will Sawin
    Oct 26, 2023 at 14:58

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This works for any two closed sets. The relative cohomology of $X$ relative to a closed set $Z$ is the cohomology of $j_! \mathbb Q$ for $\mathbb Q$ the constant sheaf and $j$ the open immersion of $X\setminus Z$ into $X$. So if $Z_1$ and $Z_2$ are closed sets in $X$, $j_{12} \colon X \setminus(Z_1 \cup Z_2) \to X$ is the open immersion, $j_1 \colon X \setminus Z_1\to X$ is the open immersion, $j_1^2 \colon Z_2 \setminus (Z_1 \cap Z_2 ) \to Z_2$ is the open immersion, and $i_2 \colon Z_2 \to X$ is the closed immesion, then your claim follows from the short exact sequence of sheaves:

$$ 0 \to j_{12!} \mathbb Q \to j_{1!} \mathbb Q \to i_{2*} j_{1!}^2 \mathbb Q \to 0$$

which is possible to check: The first map is the adjunction map $j_{12!} j_{12}^* j_{1!} \mathbb Q\to j_{1!} \mathbb Q$, the second is the adjunction map $j_{1!} \mathbb Q\to i_{2*} i_2^* j_{1!} \mathbb Q$, and the exactness may be checked on stalks.

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  • $\begingroup$ Thank you for the answer! Is it possible to prove in this way Gyzin localisation sequence? $\endgroup$ Oct 26, 2023 at 17:47
  • $\begingroup$ @Galoisgroup I think the Gysin sequence is obtained from the derived dual $0 \to R j_* j^* \mathbb Q \to \mathbb Q \to i_* i^! \mathbb Q $ to the elementary short exact sequence $0 \to j_! j^* \mathbb Q \to \mathbb Q \to i_* i^* \mathbb Q \to 0$. $\endgroup$
    – Will Sawin
    Oct 29, 2023 at 1:03

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