# A book on locally ringed spaces?

Are there enough interesting results that hold for general locally ringed spaces for a book to have been written? If there are, do you know of a book? If you do, pelase post it, one per answer and a short description.

I think that the tags are relevant, but feel free to change them.

Also, have there been any attempts to classify locally ringed spaces? Certainly, two large classes of locally ringed spaces are schemes and manifolds, but this still doesn't cover all locally ringed spaces.

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I'm also interested in this question. Many Things, which have been defined/constructed for schemes, work also for locally ringed spaces, for example fibred products (wolkenkratzerseite.de/pdf/faserprodukte.pdf), fibres, closed immersions, reduced spaces, tangent space, sheaves of differentials, ... – Martin Brandenburg Feb 1 '10 at 8:06

## 3 Answers

In addition to the examples mentioned in the question, of manifolds and schemes, other commonly occuring types of locally ringed spaces are formal schemes and complex analytic spaces.

I don't know how extensive the taxonomy of locally ringed spaces is. For example, if $A$ is a local ring, we can form the locally ringed space consisting of a single point, with $A$ sitting on top of it. These are the topologically simplest locally ringed spaces (after the empty space). If $A$ is a field, one obtains a scheme. If $A$ is a complete local ring, one obtains a formal scheme. In general, this doesn't fit into any particular taxonomic grouping that I know of.

Incidentally, it might be worth mentioning that the various taxonomic classes can interact: for example, analytification of schemes over ${\mathbb C}$ is conveniently described in terms of maps (in the category of locally ringed spaces) to complex analytic spaces.

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Is the use of "formal" in "formal scheme" related to "formally smooth" or "formally etale"? – Harry Gindi Feb 1 '10 at 3:41
In some sense. The latter have to do with deformation theoretic questions (formal smoothness is related to the unobstructedness of deformations of maps into the scheme), and formal schemes also arise naturally in the context of deformation theory. But the adjective formal'' for formal schemes also goes back to Zariski's theory of formal holomorphic functions. This is the same use of formal as when one speaks of formal powers series; i.e. the consideration of power series without regard to convergence issues. Thus this usage significantly predates Grothendieck's definitions. – Emerton Feb 1 '10 at 3:59
-1: you don't address the question (=book on LRS). – Martin Brandenburg Feb 1 '10 at 8:07
+1. @MB: this answer is quite relevant to the question asked in the third paragraph. – Pete L. Clark Feb 1 '10 at 8:45
Dear Martin and Pete, It was indeed the third para. of the question that I was trying to address. Regarding the existence of a book, I don't know of any. – Emerton Feb 1 '10 at 14:33

A locally ringed space is nothing but a local ring object (in the internal sense) in a category of sheaves over a topological space, which happens to be an example of a topos.

So in order to study general properties of locally ringed spaces, one could proceed by studying properties of local ring objects in arbitrary topoi. As any proposition (in the internal language) on a local ring (a commutative ring with $0 \neq 1$ and $s + t = 1 \implies s \in R^\times \lor t \in R^\times$) is true for any topos if and only if it can be derived intuitionistically (which is more or less the same as constructively), the original questions seems to boil down to:

"What are the constructively valid properties and constructions for a local ring?"

For example, the construction of Kähler differentials makes also sense constructively, which immediately implies (by the above reasoning) that every morphism $X \to Y$ of locally ringed spaces has an associated module $\Omega_{X/Y}$ of Kähler differentials.

And there is a lot of literature on constructive algebra. The book of Mines, Richman and Ruitenburg as well as many of the preprints on Fred Richman's homepage are a start. Some material can also be found in "Sheaves in Geometry and Logic" by Mac Lane and Moerdijk.

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Could you please provide some links to such literature? – Peter Arndt Dec 16 '10 at 12:12
@Peter. Did so. – Marc Nieper-Wißkirchen Dec 16 '10 at 12:38
Very nice answer, 1+. Anyone who is interested in this sort of reasoning may consult the introductory paper "Intuitionistic algebra and representations of rings" by Christopher Mulvey. – Martin Brandenburg Dec 16 '10 at 13:50
Mines, Richman, Ruitenburg looks really nice - thank you! – Peter Arndt Dec 16 '10 at 22:07
One can also mention the paper "Generic Galois theory of local rings" of G.Wraith which is, from this perspective, essentially about constructing the etale topos of a general locally ringed space. – Simon Henry Oct 16 '14 at 7:46

An Introduction to Families, Deformations and Moduli, by T.E. Venkata Balaji, has a beautiful little appendix that introduces smooth manifolds, complex manifolds, schemes, and complex analytic spaces in a unified way as locally ringed spaces. Although it doesn't say much in general about arbitrary locally ringed spaces, I enjoyed reading it and seeing how the stuff I knew about particular classes of locally ringed spaces fit into the general framework. One thing that particularly struck me, although it's obvious in hindsight, was the remark (A.5.5) that for all the categories of spaces mentioned above, a morphism as defined classically is the same thing as a morphism of locally ringed spaces.

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