Take $X$ a hypersurface in $\mathbb{C}P^{n+1}$ of degree $d$, denote $\Lambda\_X=H^n(X,\mathbb{Z})$. We know that $\Lambda\_X$ is a finitely generated Abelian group. I was wondering whether $\Lambda\_X$ is always free (for any $n$, $d$). Anyone know an answer or a refrence of this? Thanks a lot!



Freeness of the integral middle cohomology of a smooth hypersurface in $\mathbb{C}P^{n+1}$
If $X$ is smooth, then by the Lefschetz hyperplane theorem, $H\_k(X; \mathbb{Z}) \cong H\_k(\mathbb{CP}^{n+1}; \mathbb{Z})$ for $k < n$. By the Universal Coefficient Theorem, 


$$H^n(X; \mathbb{Z}) \cong \operatorname{Hom}(H\_n(X;\mathbb{Z}), \mathbb{Z})\oplus\operatorname{Ext}^1(H\_{n-1}(X; \mathbb{Z}), \mathbb{Z}).$$


As $H\_{n-1}(X; \mathbb{Z}) \cong H\_{n-1}(\mathbb{CP}^{n+1}; \mathbb{Z})$ is either $0$ or $\mathbb{Z}$ depending on the parity of $n$, we see that $\operatorname{Ext}^1(H\_{n-1}(X; \mathbb{Z}), \mathbb{Z}) = 0$. Therefore $H^n(X; \mathbb{Z}) \cong \operatorname{Hom}(H\_n(X; \mathbb{Z}), \mathbb{Z})$ which is free.


