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We prove that any affine ring $R$ over a field $k$ has full cosupport, i.e.,the cosupport of $R$ is equal to ${\rm Spec}\, R$. Using this fact, we give acomplete description of all terms in a minimal pure-injective resolution of$R$, provided that $|k|=\aleph_1$ and ${\rm dim}\, R\geq 2$, or $|k|\geq\aleph_1$ and ${\rm dim}\, R=2$. As a corollary, we obtain a partial answer toa conjecture by Gruson.

Cosupports and minimal pure-injective resolutions of affine rings