= Nonzero Ext of the rationals with integer coefficients
{title2=$\operatorname{Ext}_{\mathbb Z}^1(\mathbb Q,\mathbb Z)\ne0$}
A <free resolution> of $\mathbb Q$ has generators $a_n=1/n!$ and independent relations $a_n-(n+1)a_{n+1}$. Thus the <Ext functor> gives the cokernel of the map on integer sequence products
$$
(u_n)\longmapsto(u_n-(n+1)u_{n+1}).
$$
The constant sequence one is not in its image. A preimage would require $u_1\equiv\sum_{j=1}^N j!\pmod{(N+1)!}$ for every $N$. For $N\geq3$, the sum is below $2N!$, tends to infinity, and its distance from $(N+1)!$ also tends to infinity. No fixed integer satisfies all these congruences. Hence $\operatorname{Ext}_{\mathbb Z}^1(\mathbb Q,\mathbb Z)\ne0$.
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