I am trying to understand the definition of an integral Hodge structure. Apparently, for $X$ a compact Kahler manifold, $H^n(X,\mathbb R)$, the lattice $H^n(X,\mathbb Z)$ and the Hodge filtration give a Hodge structure.
In fact, the definition of a Hodge structure consists of a lattice $\Lambda$ in a real vector space together with a "Hodge filtration".
But aren't lattices free modules? So what if $H^n(X,\mathbb Z)$ has torsion? Is the above really an example of a Hodge structure? Or should we consider $H^n(X,\mathbb Z)$ modulo torsion?
Sometimes an integral Hodge structure begins with a f.g. abelian group, and other times with a lattice. If you want to include $H^n(X,\mathbb Z)$ as an example, you should use the first definition. But since all the relevant extra structure involves passing to the tensor product with $\mathbb R$, which kills torsion, the torsion plays no role in Hodge theory, and so people often forget about it (either by defining it way, perhaps, or sometimes just by tending to ignore it when discussing this stuff).