nLab Pierce spectrum




Let RR be a commutative ring. By E(R)E(R) denote the Boolean algebra of idempotents of RR whose meet operation is given by the multiplication of RR. the Pierce spectrum Idl(E(R))\mathrm{Idl}(E(R)) of RR is the poset (in fact locale) of ideals of E(R)E(R). There is a sheaf R¯\bar{R} of indecomposable rings (rings whose only idempotents are 00 and 11) on Idl(E(R))\mathrm{Idl}(E(R)), called the Pierce sheaf, whose ring of sections over the principal ideal (e)(e) (where ee is a prime filter in E(R)E(R)) is R eR_e.


F. Borceux, G. Janelidze, Galois Theories, Cambridge studies in advanced mathematics, 72, Cambridge University Press 2001.

P. T. Johnstone, Stone Spaces, Cambridge studies in advanced mathematics 3, Cambrdge Univ. Press 1982.

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