# Confusion regarding the vanishing of certain relative cohomology groups

Let $$X$$ be a projective variety of dimension $$n$$ and $$D \subset X$$ is a proper subvariety. Embed $$X$$ into a projective space $$\mathbb{P}^{3n}$$. The following argument implies that $$H^i(X,X\backslash D)=0$$ for all $$i>1$$, which is not always true. Unfortunately, I am not able to find the mistake. I am afraid that there is something basic that I am misunderstanding. It will be very helpful if somebody could point out the gap in my understanding. I now give my argument:

We know that for every $$i$$ there are exact sequences of the form:

$$H^i(\mathbb{P}^{3n}, X) \to H^i(\mathbb{P}^{3n}) \to H^i(X) \to H^{i+1}(\mathbb{P}^{3n}, X ) \to H^{i+1}(\mathbb{P}^{3n})$$ and

$$H^i(\mathbb{P}^{3n} \backslash D, X \backslash D) \to H^i(\mathbb{P}^{3n} \backslash D) \to H^i(X \backslash D) \to H^{i+1}(\mathbb{P}^{3n}, X \backslash D ) \to H^{i+1}(\mathbb{P}^{3n} \backslash D)$$

There is a natural morphism of exact sequences from the first exact sequence to the second one. Note that, by Excision theorem $$H^i(\mathbb{P}^{3n}, X) \cong H^i(\mathbb{P}^{3n}\backslash D, X \backslash D)\, \mbox{ and } H^{i+1}(\mathbb{P}^{3n}, X) \cong H^{i+1}(\mathbb{P}^{3n} \backslash D, X\backslash D).$$ By Thom isomorphism as given in Fulton's page 371, we have (note that $$\dim(D)) $$H^i(\mathbb{P}^{3n},\mathbb{P}^{3n}\backslash D) \cong H_{6n-i}^{\mathrm{BM}}(D)=0\, \mbox{ for all } 0 \le i \le 2n$$ This implies that the morphism from $$H^i(\mathbb{P}^{3n})$$ to $$H^i(\mathbb{P}^{3n}\backslash D)$$ are isomorphisms for all $$0 \le i \le 2n$$. Finally, using five lemma, we conclude that the morphism from $$H^i(X)$$ to $$H^i(X\backslash D)$$ is an isomorphism from all $$i>0$$. This implies that $$H^i(X,X\backslash D)=0$$ for all $$i>0$$.

Where am I making a mistake?

The hypothesis of the excision theorem that the closure of $$D$$ is contained in the interior of $$X$$ is not satisfied, since the interior of $$X$$ is empty.
• I see. Is this because X is not an open subscheme in $\mathbb{P}^{3n}$? I thought D is in the interior of X because D is a closed subvariety in X. Commented May 2 at 13:57
• @user45397 Exactly. The interior of a subset is the largest open subset but $X$ itself is not open and in fact does not contain any nonempty open subset. Commented May 2 at 14:12