nLab
Pierce spectrum

Contents

Idea

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.

References

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.

Revised on November 21, 2013 11:42:07 by Urs Schreiber (188.200.54.65)