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formal spectrum

Contents

Idea

For usual (commutative) rings, Grothendieck introduced the notion of a prime spectrum. In order to accomodate not only polynomials but also formal power series, it is convenient to consider completions and topological rings. Order nn nilpotent elements in an ordinary ring compare to completions as truncations of general power series and geometrically represent certain nn-th infinitesimal neighborhood. Completions represent certain pro-objects in the category of rings. Adic completion corresponds to have all infinitesimal neighborhoods at once.

A formal spectrum is a generalization of prime spectrum to adic noetherian rings, therefore containing information on all infinitesimal neighborhoods, corresponding to the ideal of completion.

Definition

Assume RR is a commutative ring and IRI \subset R is an ideal, such that its powers make a fundamental system of neighborhoods of zero of a complete Hausdorff topology (we say that RR is an separated complete ring in the II-adic topology).

The formal spectrum SpfRSpf R of (R,I)(R,I) is the inductive limit of the prime spectra

Spf(R):=colim nSpec(R/I n). Spf(R) :=colim_n Spec (R/I^n) \,.

where the connecting morphisms are the closed nilpotent immersions Spec(R/I n)Spec(R/I n+1)Spec(R/I^n)\hookrightarrow Spec(R/I^{n+1}) of affine schemes and the colimit is taken in the category of topologically ringed spaces.

Regarding that all affine schemes X n:=Spec(R/I n)X_n := Spec(R/I^n) for n1n\geq 1 have the same underlying topological space χ\chi because nilpotents in I nI^n do not affect the underlying reduced scheme, so does SpfR=(χ,𝒪 χ)Spf R = (\chi,\mathcal{O}_\chi). With our assumptions on II-adic topology, in fact χ\chi contains all closed points of SpecRSpec R and any open subset of SpecRSpec R containing χ\chi is the whole of SpecRSpec R. The structure sheaf 𝒪 χ\mathcal{O}_\chi has the ring of sections 𝒪 χ(U)=lim n𝒪 X n\mathcal{O}_\chi(U) = lim_n \mathcal{O}_{X_n} where the limit is taken in the category of topological rings, and 𝒪 X n\mathcal{O}_{X_n} have a discrete topology. For example, the ring of global sections is 𝒪 χ(χ)=R\mathcal{O}_\chi(\chi) = R.

A formal spectrum is an example of a formal scheme. Formal schemes in general form certain subcategory of the category of ind-schemes.

The formal spectrum of a separated complete topological II-adic ring RR depends just on the underlying topology on RR and not on a choice of the ideal II generating this topology.

References

Revised on July 21, 2014 06:03:39 by Urs Schreiber (89.15.239.100)