What does it mean to tensor with $\mathbb{Q}$? At our algebraic geometry seminar I often hear that something is 'tensored with $\mathbb{Q}$', e.g. a ring of endomorphisms. This phrase seems to have some intuitive meaning that I don't know. What does 'tensoring with $\mathbb{Q}$' mean? 

Let $M$ be a $\mathbb{Z}$-module. As $\mathbb{Q}$ is a $\mathbb{Z}$-module (i.e. it is an abelian group), we can form the tensor product $M\otimes_{\mathbb{Z}}\mathbb{Q}$ which is a $\mathbb{Z}$-module (and also a $\mathbb{Q}$-module). It consists of finite sums of expressions of the form $m\otimes q$ where $m \in M$ and $q \in \mathbb{Q}$ and we impose the rules


*

*$(m_1 + m_2)\otimes q = m_1\otimes q + m_2\otimes q$

*$m\otimes(q_1 + q_2) = m\otimes q_1 + m\otimes q_2$

*$(km)\otimes q = m\otimes (kq) = k(m\otimes q)$ where $k \in \mathbb{Z}$.


If $t \in M$ is a torsion element, that is $nt = 0$ for some $n \in \mathbb{N}\setminus\{0\}$, consider $t\otimes q$. Note that 
$$t\otimes q = t\otimes\frac{nq}{n} = t\otimes \left(n\frac{q}{n}\right) = (nt)\otimes \frac{q}{n} = 0\otimes\frac{q}{n} = 0.$$
So any torsion elements in $M$ vanish when $M$ is tensored by $\mathbb{Q}$. That is, they contribute nothing to the module $M\otimes_{\mathbb{Z}}\mathbb{Q}$.
