Cosupports and minimal pure-injective resolutions of affine rings

Abstract

We prove that any affine ring $R$ over a field $k$ has full cosupport, i.e., the cosupport of $R$ is equal to $\mathrm{Spec}R$. Using this fact, we give a complete description of all terms in a minimal pure-injective resolution of $R$, provided that $|k|={\aleph }_1$ and $\dim R\ge 2$, or $|k|\ge {\aleph }_1$ and $\dim R=2$. As a corollary, we obtain a partial answer to a conjecture by Gruson.

9 pages, Section 3 has been revised, to appear in J. Algebra

References in corpus (3)

• Minimal complexes of cotorsion flat modules
• Cosupport computations for finitely generated modules over commutative noetherian rings
• Minimal semi-flat-cotorsion replacements and cosupport