# Homological properties of the perfect and absolute integral closures of Noetherian domains - Mathematics > Commutative Algebra

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Abstract: For a Noetherian local domain $R$ let $R^+$ be the absolute integral closureof $R$ and let $R {\infty}$ be the perfect closure of $R$, when $R$ has primecharacteristic. In this paper we investigate the projective dimension ofresidue rings of certain ideals of $R^+$ and $R {\infty}$. In particular, weshow that any prime ideal of $R {\infty}$ has a bounded free resolution ofcountably generated free $R {\infty}$-modules. Also, we show that the analogueof this result is true for the maximal ideals of $R^+$, when $R$ has residueprime characteristic. We compute global dimensions of $R^+$ and $R {\infty}$ insome cases. Some applications of these results are given.