# Structure theorem of symplectic modules

My question comes from the content presented on slide 31 of the following presentation given by Jean-Pierre Tignol (unfortunately, I do not have access to the main reference on the subject, that is Tignol and Amitsur's paper on symplectic modules).

A symplectic module is a finite abelian group $$M$$ with a nondegenerate alternating bilinear pairing $$b:M\times M \rightarrow \mathbb Q/\mathbb Z$$.

On the slide mentioned above, the following structure theorem is refered to as "De Rham theorem, 1931".

Every symplectic module $$M$$ has a symplectic basis $$M\cong (\mathbb Z/n_1\mathbb Z)^2\times\ldots\times (\mathbb Z/n_s\mathbb Z)^2$$ with generators $$e_1,f_1,\ldots ,e_s,f_s$$ such that $$b(e_i,f_i)=\frac{1}{n_i}+\mathbb Z$$ and $$b(e_i,f_j)=0$$ for $$i\not = j$$ and $$b(e_i,e_j)=b(f_i,f_j)=0$$ for any $$i,j$$.

I have two questions in regards to this.
First, can we assume furthermore that $$n_1|n_2|\ldots |n_s$$ ?
Second, what does this statement have to do with the renowned De Rham theorem of 1931, expressing that for a smooth manifold $$M$$, we have an isomorphism between De Rham cohomology groups $$H_{dR}^p(M)$$ and singular cohomology groups $$H^p(M;\mathbb R)$$ ?

For your first question, the answer is yes, and follows directly from the following lemma: if $$M\simeq (\mathbb{Z}/n\mathbb{Z})^2\times (\mathbb{Z}/m\mathbb{Z})^2$$ with $$n$$ and $$m$$ coprime, and $$(e_1,f_1,e_2,f_2)$$ is a symplectic basis, then there is a symplectic basis $$(e_3,f_3)$$ for the natural decomposition $$M\simeq (\mathbb{Z}/nm\mathbb{Z})^2$$.
It is not hard to find, I'll let you fill in the details: just take $$u\in \mathbb{Z}$$ such that $$u(a+b)\equiv 1$$ mod $$ab$$, and define $$e_3=u(e_1+e_2)$$ and $$f_3=f_1+f_2$$.