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In Borceaux and Janelidze's Galois Theories, a construction of the Pierce spectrum is given. It is the poset of ideals in a Boolean ring. It's construction is reminiscent of the Zariski spectrum in commutative ring theory, however I've not seen it used elsewhere.

Q. Is the Pierce spectrum important elsewhere in mathematics?

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    $\begingroup$ It's used by some people in ring theory. For a boolean ring it is the Zariski spectrum. For a began ring it plays the role of the central primitive idempotents in an Artinian ring in that it encodes direct product decompositions by representing the end in a sheaf but not a sheaf of local rings. If your ring is von Neumann regular it can be used to represent your ring as a sheaf of directly indecomposable rings. You should see Pierce's memoir ams.org/books/memo/0070. $\endgroup$
    – Benjamin Steinberg
    Commented Dec 1, 2020 at 2:53
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    $\begingroup$ I believe Magid uses it in his galois theory for commutative rings. We use it in arxiv.org/abs/1906.06952 to represent skew inverse semigroup rings as convolution algebras on groupoids $\endgroup$
    – Benjamin Steinberg
    Commented Dec 1, 2020 at 2:56
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    $\begingroup$ It is the Zariski spectrum (of a Boolean ring)! $\endgroup$
    – Qiaochu Yuan
    Commented Dec 1, 2020 at 3:19
  • $\begingroup$ Actually if the ring is von Neumann regular you get a sheaf of fields. For a general ring you get it represented as the global sections of a sheaf of directly indecomposable rings over a compact totally disconnected space $\endgroup$
    – Benjamin Steinberg
    Commented Dec 1, 2020 at 3:48
  • $\begingroup$ @BenjaminSteinberg "began ring" ? $\endgroup$
    – David Roberts ♦
    Commented Dec 1, 2020 at 5:13

1 Answer 1

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For Boolean rings, the Pierce spectrum coincides with the Zariski spectrum and is one of the functors implementing the Stone duality between Boolean algebras and compact totally disconnected Hausdorff spaces and Stonean duality between complete Boolean algebras and compact extremally disconnected Hausdorff spaces.

Restricting further to complete Boolean algebras of projections of commutative von Neumann algebras produces an equivalence of categories of commutative von Neumann algebras and compact strictly localizable enhanced measurable spaces, i.e., a version of the Gelfand duality in the setting of measure theory.

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