In “Compact ringed spaces” [J. Algebra 52, 411-436 (1978; Zbl 0418.18009)], C. J. Mulvey initiated the study of compact sheaf representations of a unital ring R. These are ringed spaces where the stalks are quotients of R and the base space is compact, Hausdorff (together with some further requirements). He showed that R has a universal compact representation, which is determined by a certain universal quotient of the maximal ideal space, max R, of R.
The aim of the paper under review is to describe this construction lattice theoretically using the lattice \(\Lambda\) (R) of 2-sided ideals of R. The general lattice theoretic notion, which forms the central object of study, is that of a “2-sided carrier”. This can be most efficiently described as a 2-sided quantale (for a discussion of quantales, see the reviewer’s book: Quantales and their applications (1990; Zbl 0703.06007)) together with the condition that meets distribute over arbitrary directed joins. A notion of regularity is developed for carriers. Regular carriers are in fact frames and it is shown that each 2-sided carrier has a largest regular subframe, called its regular core.
The theory of representations of a carrier \(\Lambda\) over a base space S is developed in terms of Galois connections between \(\Lambda\) and \({\mathcal O}(S)\), the frame of open sets of S. This leads to the main theorem showing that compact representations of \(\Lambda\) are in bijective correspondence in the regular subframes of \(\Lambda\) and the universal compact representation corresponds to the regular core of \(\Lambda\). This result is then applied to the case of \(\Lambda\) (R), where R is a unital ring, and thus the ring theoretic case is recovered as an example of the general theorem. As with the author’s other work, the paper is clearly written and the requisite lattice theoretic and representation theoretic background is provided, making the paper accessible to a wide audience.