Exact sequence for relative cohomology + normal crossing divisors

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.

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

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.
• @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 $0 \to j_! j^* \mathbb Q \to \mathbb Q \to i_* i^* \mathbb Q \to 0$. Oct 29, 2023 at 1:03