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formal spectrum

Idea

For usual 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 $n$ nilpotent elements in an ordinary ring compare to completions as truncations of general power series and geometrically represent certain $n$-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 $R$ is a commutative ring and $I\subset R$ is an ideal, such that its powers make a fundamental system of neighborhoods of zero of a complete Hausdorff topology (we say that $R$ is an separated complete ring in $I$-adic topology).

The formal spectrum $\mathrm{Spf}R$ of $(R,I)$ is the inductive limit of the prime spectra

$$\mathrm{Spf}R:={\mathrm{colim}}_n\mathrm{Spec}(R/I^n).$$Spf R :=colim_n Spec (R/I^n) \,.

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

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

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 $I$-adic ring $R$ depends just on the underlying topology on $R$ and not on a choice of the ideal $I$ generating this topology.

References

• A Survey of Elliptic Cohomology - formal groups and cohomology)

• standard references are EGA, Hartshorne

• Luc Illusie, Grothendieck existence theorem in formal geometry, in FGA explained (draft: pdf)