Sheaves and localization
The purpose of the paper is to calculate projective limits of localization functors. Let \(\sigma\) and \(\tau\) be two idempotent kernel functors in \(R\)-mod. In general \(\sigma\) and \(\tau\) are not necessarily compatible. The author points out that \(\sigma\) and \(\tau\) are compatible iff the canonical sequence of localization functors \(0\to Q_{\sigma\wedge\tau}\to Q_\sigma\oplus Q_\tau\to Q_{\sigma\vee\tau}\) is exact. He also proves that for any finite family \(\{\sigma_\alpha:\alpha\in A\}\) of idempotent kernel functors over \(R\), \(Q_{\wedge\sigma_\alpha}\) is the projective limit of the family \(\{Q_{\sigma_\alpha}\to Q_{\sigma_\alpha} Q_{\sigma_\beta},\;Q_{\sigma_\beta}\to Q_{\sigma_\alpha}Q_{\sigma_\beta};\;\alpha,\beta \in A\}\). This result is then strengthened to encompass projective limits of ``localization functors'' with respect to words in the free monoid over the set of all Gabriel filters for \(R\).
projective limits of localization functors
scientific article; zbMATH DE number 922308