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The theory of group schemes seems to be well developed: there are many applications and examples, and the literature is vast.

On the other hand, a quick google search with some obvious keywords (ring, schemes, algebraic) does not yield any good reference on Ring Schemes, nor any interesting examples (other than Witt schemes) or applications. My question, out of pure curiosity, is:

Is there any good introductory reference(s) to the subject?

I'd be most interested in knowing what the important questions and applications of Ring schemes are.

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    $\begingroup$ I don't know what algebraic geometers have done, but there's a nontrivial amount of applicative algebraic topology literature about affine formal ring schemes / coalgebraic rings. Such objects appear when building the co/homology of the destabilization of another co/homology theory. Searching on any of 'Hopf ring', 'coalgebraic ring', 'biring', 'ring-ring', and potentially 'Tall-Wraith monoid' will probably turn topology-application type stuff up. $\endgroup$
    – Eric Peterson
    Commented Feb 4, 2012 at 8:50
  • $\begingroup$ $\mathbb{A}^n_S$ is a ring scheme over $S$, or does this count as a non interesting example? $\endgroup$
    – Martin Brandenburg
    Commented Feb 4, 2012 at 8:59
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    $\begingroup$ Martin's example is called $\mathbf{O}^n$ in SGA3. If we forget the multiplicative structure, the additive group scheme is called $\mathbf{G}_a^n$. In general, should avoid using the notation describing plain affine space when there is extra structure in play. $\endgroup$
    – S. Carnahan ♦
    Commented Feb 5, 2012 at 7:20
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    $\begingroup$ You might to read Hans Schoutens notes from his class this semester: websupport1.citytech.cuny.edu/faculty/hschoutens/… He draws a connection between equivalent understandings of a direct limit of rings through ultra products and an algebraic definition involving stalks. $\endgroup$
    – user10290
    Commented Feb 5, 2012 at 12:59
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    $\begingroup$ @Scott: Thanks for this clarification. Where are ring schemes such as $\mathbf{O}^n$ treated in SGA3 (I cannot read ~ 1700 pages)? @Erin: I think this is offtopic. $\endgroup$
    – Martin Brandenburg
    Commented Feb 17, 2012 at 9:10

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Some general results about ring schemes and algebraic rings can be found in the following papers by M. J. Greenberg: "Schemata over local rings" and "Algebraic rings", respectively. The concept of algebraic ring also appeared earlier in some work of Serre and Weil (see the references in the second paper above). There was also some early input from the Japanese school of ring theory.

Let's consider algebraic rings over an algebraically closed field $k$. It has been shown by Greenberg that an irreducible algebraic ring over $k$ is Artinian. Conversely, every commutative Artinian local ring carries the structure of (irreducible) algebraic ring. Moreover, as a variety a ring variety is isomorphic to an affine space, so there isn't as much variation as for algebraic groups. However, if $G$ is a group scheme (let's say affine and smooth) over an Artinian local ring $A$ with residue field $k$, then because $A$ is an algebraic ring, the so called Greenberg functor lets us view $G(A)$ as an algebraic group over $k$, and the structure of these groups can be quite complicated. This in turn has applications for instance in the representation theory of reductive groups over finite rings. Algebraic rings have also found some applications in group theory and appear in a paper by M. Kassabov and M. Sapir called "Nonlinearity of matrix groups".

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This is not a complete reference of course, but there is Lecture 26 (Ring Schemes; The Witt Scheme) in Mumford's book Lectures on curves on an algebraic surface

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    $\begingroup$ If I remember, this lecture was written by George Bergman. $\endgroup$
    – JBorger
    Commented Feb 6, 2015 at 1:30

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