  Consider the abelian category $\mathcal{C}_k$ of commutative group schemes of
finite type over a field $k$. By results of Serre and Oort, $\mathcal{C}_k$ has
homological dimension $1$ (resp. $2$) if $k$ is algebraically closed of
characteristic $0$ (resp. positive). In this article, we explore the abelian
category of commutative algebraic groups up to isogeny, defined as the quotient
of $\mathcal{C}_k$ by the full subcategory $\mathcal{F}_k$ of finite $k$-group
schemes. We show that $\mathcal{C}_k/\mathcal{F}_k$ has homological dimension
$1$, and we determine its projective or injective objects. We also obtain
structure results for $\mathcal{C}_k/\mathcal{F}_k$, which take a simpler form
in positive characteristics.
