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Diers spectrum

Idea

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Definition

Suppose U:ABU\colon A\to B is a functor that has a left multiadjoint.

The Diers spectrum of UU is the functor B opSetB^{op}\to Set that sends an object XBX\in B to the set II that indexes the value (g i:BU(A i)) iI(g_i\colon B\to U(A_i))_{i\in I} of the left multiadjoint of UU at XX. A morphism f:XXf\colon X\to X' is sent to the induced map III'\to I that assigns to iIi'\in I' the unique iIi\in I such that g if:XU(A i)g_{i'}\circ f\colon X\to U(A'_i) factors through g ig_i.

Example

The category of integral domains and injective homomorphisms is a multireflexive subcategory of the category of commutative rings and ring homomorphisms.

The associated Diers spectrum functor is a functor

CRing opSetCRing^{op} \to Set

that sends a commutative ring to its prime spectrum.

Example

The category of reduced rings and injective homomorphisms is a multireflexive subcategory of the category of commutative rings and ring homomorphisms.

The associated Diers spectrum functor is a functor

CRing opSetCRing^{op} \to Set

that sends a commutative ring to its poset of radical ideals. This is precisely the underlying poset of the localic Zariski spectrum.

Reference

  • Yves Diers, Some spectra relative to functors. Journal of Pure and Applied Algebra 22:1 (1981), 57–74. doi

  • Axel Osmond: On Diers theory of Spectrum I: Stable functors and right multi-adjoints, arXiv.

  • Axel Osmond: On Diers theory of Spectrum II Geometries and dualities, arXiv.

Last revised on April 8, 2021 at 14:36:44. See the history of this page for a list of all contributions to it.