Cosupports and minimal pure-injective resolutions of affine rings (Q2330440)

Throughout this paper, \(R\) is a commutative noetherian ring. The cosupport of an \(R\)-complex \(X\), introduced by Benson, Iyengar and Krause [\textit{D. J. Benson} et al., J. Reine Angew. Math. 673, 161--207 (2012; Zbl 1271.18012)], is defined as \(\mathrm{cosupp}_RX=\{p\in \mathrm{Spec}R\ |\ \mathrm{RHom}_R(k(p),X)\neq 0\}\). It is known that \(\mathrm{cosupp}_RR\subseteq V(c_R)\), where \(c_R\) denotes the sum of all ideals \(I\) such that \(R\) is \(I\)-adically complete. \textit{P. Thompson} proved in [J. Algebra 511, 249--269 (2018; Zbl 1439.13035)] that the cosupport of \(R=k[[X]][Y]\) is strictly contained in \(V(c_R)\), where \(k\) is a field. Let \(R=k[X_1,\ldots,X_n]\). It is clear that \(c_R=(0)\). In the paper under review the author proves that the cosupport of \(R=k[X_1,\ldots,X_n]\) is equal to \(\mathrm{Spec}R\). It follows from this result that the cosupport of any affine ring \(R\) over a field \(k\) is equal to \(\mathrm{Spec}R\). Using this fact, the author gives a complete description of all terms in a minimal pure-injective resolution of \(R\), provided that \(|k|=\aleph_1\) and \(\mathrm{dim}R\geq2\), or \(|k|\geq\aleph_1\) and \(\mathrm{dim}R=2\). As a corollary, it is shown that a conjecture by Gruson (see [\textit{A. Thorup}, J. Pure Appl. Algebra 219, No. 4, 1278--1283 (2015; Zbl 1320.13032)]) is partially true.